% Copyright (c) 2026 Scott Armstrong. % Released under Apache 2.0 license. \documentclass[11pt,letterpaper]{article} \usepackage[T1]{fontenc} \usepackage{amsmath,amsthm,amssymb,mathtools} \usepackage[margin=1.05in]{geometry} \usepackage{microtype} \setlength{\emergencystretch}{1em} \usepackage{xcolor} \definecolor{paperlink}{RGB}{0,0,114} \usepackage{hyperref} \hypersetup{colorlinks=true,linktoc=all, linkcolor=paperlink,citecolor=paperlink,urlcolor=paperlink,pdfstartview=Fit, pdftitle={The Escauriaza--Seregin--Sverak theorem: version for the Lean formalization}, pdfauthor={}} \usepackage{enumitem} \setlist[itemize]{leftmargin=2em,itemsep=0.25em,topsep=0.35em} \setlist[enumerate]{leftmargin=2.25em,itemsep=0.25em,topsep=0.35em} \setlength{\parindent}{1.25em} \setlength{\parskip}{0pt} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \fancyhead[L]{\small The Escauriaza--Seregin--\v{S}ver\'ak theorem} \fancyhead[R]{\small\thepage} \setlength{\headheight}{14pt} \numberwithin{equation}{section} \setcounter{tocdepth}{1} \date{} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{deviation}[theorem]{Deviation} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \input{macros} \title{The Escauriaza--Seregin--\v{S}ver\'ak theorem:\\ version for the Lean formalization} \author{} \begin{document} \maketitle \begin{abstract} We state a suitable form of Leray's global existence theorem for the three-dimensional Navier--Stokes equations on $\R^3$, which is proved in the companion Caffarelli--Kohn--Nirenberg paper, and we prove the endpoint regularity theorem of Escauriaza, Seregin, and \v{S}ver\'ak. Every divergence-free initial datum in $L^2(\R^3)$ has a global Leray--Hopf solution that is suitable and has a canonical pressure in $L^{5/3}(\R^3\times(0,T))$ on each finite time interval. The formalized Caffarelli--Kohn--Nirenberg theorem then gives zero one-dimensional parabolic Hausdorff measure for its singular set. Every Leray--Hopf solution with finite $L^\infty(0,T;L^3(\R^3))$ norm is regular at every point of $\R^3\times(0,T)$, belongs to $L^5(\R^3\times(0,T))$, and is unique among Leray--Hopf solutions with the same datum. The endpoint argument combines an open-top smallness criterion, vorticity regularity, and explicit Carleman estimates for unique continuation and backward uniqueness. We also prove the Ladyzhenskaya--Prodi--Serrin theorem for the full range $3\nf52$ (Theorem~\ref{thm:leray}). \item \textbf{The CKN headline.} For every $a\in\J$ there is a global Leray--Hopf solution $(u,Du)$ with datum $a$ whose singular set in $\R^3\times(0,\infty)$ has zero one-dimensional parabolic Hausdorff measure (Corollary~\ref{cor:ckn-headline}). \item \textbf{Associated pressure.} Let $(u,Du)$ be a Leray--Hopf solution on $\R^3\times[0,T]$ with datum $a\in\J$. There is $p\in L^{5/3}(\R^3\times(0,T))$ such that (S3) of Definition~\ref{def:sws} holds on $\R^3\times(0,T)$ with $f=0$. If in addition $\esssup_{t\in(0,T)}\norm{u(\cdot,t)}_{L^3(\R^3)}<\infty$, then this $p$ also belongs to $L^\infty((0,T);L^{3/2}(\R^3))$ (Theorem~\ref{thm:assoc-pressure}). \item \textbf{Local endpoint regularity.} Let $u$, $Du$, $p$ be measurable on $Q=B_1\times(-1,0)$ with \begin{gather*} \esssup_{t\in(-1,0)}\int_{B_1}\abs{u(x,t)}^2\,dx<\infty,\qquad \iint_Q\abs{u}^2+\abs{Du}^2<\infty,\qquad p\in L^{3/2}(Q),\\ \esssup_{t\in(-1,0)}\int_{B_1}\abs{u(x,t)}^3\,dx<\infty, \end{gather*} such that $Du(\cdot,t)$ is the weak gradient of $u(\cdot,t)$ on $B_1$ for a.e. $t\in(-1,0)$, and (S2) and (S3) of Definition~\ref{def:sws} hold on $Q$ with $f=0$. Then there is $\gamma\in(0,1]$ and a function $\tilde u$ equal to $u$ almost everywhere on $B_{1/2}\times(-\nf14,0)$ that is parabolically H\"older continuous with exponent $\gamma$ on the closure of $B_{1/2}\times(-\nf14,0)$ (Theorem~\ref{thm:ess-local}). \item \textbf{Global endpoint regularity.} Let $(u,Du)$ be a Leray--Hopf solution on $\R^3\times[0,T]$ with datum $a\in\J$, and suppose $\esssup_{t\in(0,T)}\int_{\R^3}\abs{u(x,t)}^3\,dx<\infty$. Then every point of $\R^3\times(0,T)$ is a regular point of $u$ (Theorem~\ref{thm:ess-global}). \item \textbf{$L^5$ and uniqueness.} Under the same hypotheses, $u\in L^5(\R^3\times(0,T))$, and every other Leray--Hopf solution $(v,Dv)$ on $[0,T]$ with datum $a$ satisfies $u=v$ almost everywhere on $\R^3\times(0,T)$ (Theorem~\ref{thm:ess-l5-unique}). \item \textbf{Forced Leray existence.} For data and forces in the class of Theorem~\ref{thm:leray-forced}, there is one global forced Leray--Hopf solution that is suitable for every $q>5/2$ for which $f$ is locally in $L^q$; its singular set has zero one-dimensional parabolic Hausdorff measure, and its associated pressure has the local split stated in Theorems~\ref{thm:leray-forced-singularSet} and \ref{thm:assoc-pressure-forced}. \item \textbf{Ladyzhenskaya--Prodi--Serrin.} Every Leray--Hopf solution in $L^\ell(0,T;L^s(\R^3))$ with $3/s+2/\ell=1$ and $30$, write \[ \begin{aligned} Q_r^-(z_0) &= B_r(x_0)\times(t_0-r^2,t_0),\\ \overline{Q_r^-(z_0)} &= \overline{B_r(x_0)}\times[t_0-r^2,t_0]. \end{aligned} \] \paragraph{Space-time weak derivatives.} Let $U\subset\R^3\times\R$ be open and $w\in\Lloc{1}(U;\R^3)$. Fields $Dw=(Dw_{ij})$, $D^2w=(D^2w_{ijk})$ and $D_tw=(D_tw_i)$ in $\Lloc{1}(U)$ are the \emph{space-time weak derivatives} $\partial_j w_i$, $\partial_j\partial_k w_i$ and $\dt w_i$ of $w$ on $U$ if, for every $\varphi\in\Cc(U)$ and all indices, \[ \int_U w_i\,\partial_j\varphi=-\int_U Dw_{ij}\,\varphi,\qquad \int_U Dw_{ij}\,\partial_k\varphi=-\int_U D^2w_{ijk}\,\varphi,\qquad \int_U w_i\,\dt\varphi=-\int_U D_tw_i\,\varphi . \] Such data are unique almost everywhere. We then write $\nabla w$, $\nabla^2 w$, $\dt w$ for them and $\Lop w=\dt w+\Delta w$ for the field $D_tw_i+\sum_j D^2w_{ijj}$; inequalities involving these are understood almost everywhere. The class $W^{2,1}_2(U)$ consists of those $w$ with $\abs{w}+\abs{\nabla w}+\abs{\nabla^2w}+\abs{\dt w}\in L^2(U)$. \section{Solution classes and main results}\label{sec:leray-hopf} This section fixes the solution notions and states the main results. The suitable weak solutions are exactly those of the formalized Caffarelli--Kohn--Nirenberg theorem~\cite{CKNLean}; the Leray--Hopf solutions transcribe (1.3)--(1.7) of~\cite{ESS2003}. \paragraph{Test functions and suitable weak solutions.} For open $\Omega\subset\R^3$ and an open interval $I\subset\R$ let $\Cc(\Omega\times I)$ denote the smooth functions with compact support in $\Omega\times I$. A velocity is a map $u\colon\R^3\times\R\to\R^3$, and its weak spatial gradient is carried as separate data $Du\colon\R^3\times\R\to\R^{3\times3}$ with $Du_{ij}=\partial_j u_i$. \begin{definition}[Suitable weak solution; {\cite[Definition~1.5]{CKNLean}}]\label{def:sws} Let $q>\nf52$, $\Omega\subset\R^3$ open, $I\subset\R$ an open interval, and $f\in\Lloc{q}(\Omega\times I;\R^3)$. The quadruple $(u,Du,p,f)$ is a \emph{suitable weak solution on $\Omega\times I$} if the following hold. \begin{enumerate}[label=(S\arabic*)] \item For every open $\Omega'$ with compact closure $\overline{\Omega'}\subset\Omega$ and every interval $J$ with compact closure $\overline J\subset I$: $u$, $Du$, $p$, $f$ are measurable on $\Omega'\times J$; $\esssup_{t\in J}\int_{\Omega'}\abs{u(x,t)}^2\,dx<\infty$; $\iint_{\Omega'\times J}\abs{u}^2+\abs{Du}^2<\infty$; $p\in L^{3/2}(\Omega'\times J)$; $f\in L^q(\Omega'\times J)$; and for a.e.\ $t\in J$, $Du(\cdot,t)$ is the weak gradient of $u(\cdot,t)$ on $\Omega'$. \item $\iint u\cdot\nabla\psi=0$ for every $\psi\in\Cc(\Omega\times I)$. \item For every $\varphi\in\Cc(\Omega\times I;\R^3)$, \[ \iint\Bigl[-u\cdot\dt\varphi-u_iu_j\,\partial_j\varphi_i+Du_{ij}\,\partial_j\varphi_i -p\,\dv\varphi-f\cdot\varphi\Bigr]=0 . \] \item For every $\psi\in\Cc(\Omega\times I)$ with $\psi\ge0$, \[ 2\iint\abs{Du}^2\psi\le\iint\Bigl[\abs{u}^2(\dt\psi+\Delta\psi) +(\abs{u}^2+2p)\,u\cdot\nabla\psi+2(f\cdot u)\,\psi\Bigr]. \] \end{enumerate} \end{definition} A point $z_0\in\Omega\times I$ is \emph{regular} if $u$ agrees almost everywhere on some open neighborhood $N\subset\Omega\times I$ of $z_0$ with a parabolically H\"older continuous function~\cite[Definition~2.2]{CKNLean}; the \emph{singular set} is the set of points of $\Omega\times I$ that are not regular. \paragraph{Divergence-free data.} Following~\cite{ESS2003}, $\J$ is the closure in $L^2(\R^3;\R^3)$ of the smooth compactly supported divergence-free fields. We use the equivalent description by the weak divergence. \begin{lemma}\label{lem:J-weak-div} A field $a\in L^2(\R^3;\R^3)$ belongs to $\J$ if and only if $\int_{\R^3}a\cdot\nabla\psi=0$ for every $\psi\in\Cc(\R^3)$. \end{lemma} \begin{proof}[Proof of Lemma~\ref{lem:J-weak-div}] If $a\in\J$, choose $a_k\in C_c^\infty(\R^3;\R^3)$ with $\dv a_k=0$ and $a_k\to a$ in $L^2$. For $\psi\in C_c^\infty(\R^3)$, integration by parts gives $\int a_k\cdot\nabla\psi=0$, and $\abs{\int(a_k-a)\cdot\nabla\psi}\le \norm{a_k-a}_2 \norm{\nabla\psi}_2\to0$. This proves the weak-divergence identity. Conversely, suppose that $a\in L^2$ satisfies that identity. With the Fourier convention $\widehat f(\xi)=\int e^{-2\pi i x\cdot\xi}f(x)\,dx$, for $\delta>0$ define a vector potential $A_\delta$ and its curl $b_\delta$ by \begin{equation}\label{eq:J-vector-potential} \widehat{A_\delta}(\xi)= \frac{i\,\xi\times\widehat a(\xi)}{2\pi(|\xi|^2+\delta^2)}, \qquad \widehat{b_\delta}(\xi)= \frac{|\xi|^2}{|\xi|^2+\delta^2}\widehat a(\xi). \end{equation} The weak divergence identity gives $\xi\cdot\widehat a(\xi)=0$ almost everywhere, so $b_\delta=\curl A_\delta$. The multipliers show that $A_\delta$ and its first weak derivatives belong to $L^2$. Spatially mollifying this $H^1$ potential gives smooth $L^2$ potentials whose curls converge to $b_\delta$ in $L^2$; cutting off each smooth potential on expanding balls shows that its curl belongs to $\J$. Thus $b_\delta\in\J$. For $\delta_n=1/(n+1)$ the curl multiplier is bounded by one and converges to one almost everywhere. Plancherel and dominated convergence therefore give \begin{equation}\label{eq:J-cutoff-convergence} \|b_{\delta_n}-a\|_2\longrightarrow0. \end{equation} The closedness of $\J$ in $L^2$ now gives $a\in\J$. This route regularizes the Fourier potential and passes to its curl in $L^2$; it does not require an $L^6$ potential estimate or a prescribed cutoff-curl sequence for $a$. \end{proof} \paragraph{Leray--Hopf solutions.} The velocity of a Leray--Hopf solution is a function defined at every time, and it is this function, not an equivalence class, that the definition constrains at every time. The same function is used as the velocity of a suitable weak solution. \begin{definition}[Leray--Hopf solution]\label{def:leray-hopf} Let $00$. \end{definition} \begin{remark}[The global notion]\label{rem:global-LH} \cite{ESS2003} define the global notion by replacing $[0,T]$ with $[0,\infty)$ in (1.3)--(1.7). Read literally, (1.3) would then require $u\in L^2(0,\infty;W^{1,2})$, that is, $\int_0^\infty\!\int\abs{u}^2<\infty$, which Leray's solutions need not satisfy; the energy inequality controls only $\sup_t\int\abs{u}^2$ and $\int_0^\infty\!\int\abs{\nabla u}^2$. We therefore define global solutions through every finite interval, which is the reading under which~\cite[Theorem~1.1]{ESS2003} holds. \end{remark} \paragraph{Main results.} The first theorem is Leray's existence theorem, with the constructed solution suitable in the sense of Definition~\ref{def:sws}. \begin{theorem}[Existence]\label{thm:leray} For every $a\in\J$ there are $u$, $Du$ and $p$ such that $(u,Du)$ is a global Leray--Hopf solution with datum $a$, and $(u,Du,p,0)$ is a suitable weak solution on $\R^3\times(0,\infty)$ for every $q>\nf52$. \end{theorem} Combined with the Caffarelli--Kohn--Nirenberg theorem~\cite[Theorem~2.10]{CKNLean}, this gives the headline of~\cite{CKN1982}. \begin{corollary}\label{cor:ckn-headline} For every $a\in\J$ there is a global Leray--Hopf solution $(u,Du)$ with datum $a$ whose singular set in $\R^3\times(0,\infty)$ has zero one-dimensional parabolic Hausdorff measure. \end{corollary} Every Leray--Hopf solution has a pressure. \begin{theorem}[Associated pressure]\label{thm:assoc-pressure} Let $(u,Du)$ be a Leray--Hopf solution on $\R^3\times[0,T]$ with datum $a\in\J$. There is $p\in L^{5/3}(\R^3\times(0,T))$ such that (S3) of Definition~\ref{def:sws} holds on $\R^3\times(0,T)$ with $f=0$. If in addition $\esssup_{t\in(0,T)}\norm{u(\cdot,t)}_{L^3(\R^3)}<\infty$, then this $p$ also belongs to $L^\infty((0,T);L^{3/2}(\R^3))$. \end{theorem} \paragraph{Forced Leray--Hopf solutions.} The same all-time velocity representative is used in the forced equation, the work term, and the initial trace. \begin{definition}[Forced Leray--Hopf solution]\label{def:forced-leray-hopf} Let $05/2$. There are one velocity $u$, weak gradient $Du$, and pressure $p$ such that $(u,Du)$ is a global forced Leray--Hopf solution with datum $a$. For every $q>5/2$ for which $f\in L^q_{\mathrm{loc}}(\R^3\times(0,\infty))$, the same quadruple $(u,Du,p,f)$ is a suitable weak solution on $\R^3\times(0,\infty)$ in the sense of Definition~\ref{def:sws}. In particular, its local energy inequality contains the term $2(f\cdot u)\psi$. \end{theorem} \begin{theorem}[Forced singular-set conclusion] \label{thm:leray-forced-singularSet} Under the hypotheses of Theorem~\ref{thm:leray-forced}, for every $q>5/2$ for which $f\in L^q_{\mathrm{loc}}$, the witness there satisfies \[ \mathcal H^1_{\mathrm{par}}\bigl(\operatorname{Sing}(u)\bigr)=0\,. \] \end{theorem} \begin{theorem}[Forced associated pressure]\label{thm:assoc-pressure-forced} Let $(u,Du)$ be a forced Leray--Hopf solution on $\R^3\times[0,T]$ with datum $a\in\J$ and $f\in L^2(\R^3\times(0,T);\R^3)$. Let $p_N=P[u\otimes u]$ and let $p_f$ be the force pressure of \cite[Lemma~13.1]{CKNLean}. Then \[ p_N\in L^{5/3}(\R^3\times(0,T)),\qquad p_f\in L^2((0,T);L^6(\R^3)),\qquad p=p_N+p_f\in L^{3/2}_{\mathrm{loc}}(\R^3\times(0,T)). \] For every $\varphi\in C_c^\infty(\R^3\times(0,T);\R^3)$, \[ \iint\Bigl[-u\cdot\partial_t\varphi-u_i u_j\,\partial_j\varphi_i +Du_{ij}\,\partial_j\varphi_i-p\,\dv\varphi -f\cdot\varphi\Bigr]\,dx\,dt=0\,. \] When $f=0$, this pressure is the pressure of Theorem~\ref{thm:assoc-pressure} and has all of its stated global and conditional integrability properties. \end{theorem} The next theorem is the local result~\cite[Theorem~1.4]{ESS2003}. Its hypotheses are those of a suitable weak solution without the local energy inequality, imposed on the whole cylinder $B_1\times(-1,0)$ rather than on compactly contained subcylinders, together with the $L^\infty L^3$ bound. \begin{theorem}[Local regularity]\label{thm:ess-local} Let $u$, $Du$, $p$ be measurable on $Q=B_1\times(-1,0)$ with \begin{gather*} \esssup_{t\in(-1,0)}\int_{B_1}\abs{u(x,t)}^2dx<\infty,\qquad \iint_Q\abs{u}^2+\abs{Du}^2<\infty,\qquad p\in L^{3/2}(Q),\\ \esssup_{t\in(-1,0)}\int_{B_1}\abs{u(x,t)}^3dx<\infty , \end{gather*} such that $Du(\cdot,t)$ is the weak gradient of $u(\cdot,t)$ on $B_1$ for a.e.\ $t\in(-1,0)$, and (S2) and (S3) of Definition~\ref{def:sws} hold on $Q$ with $f=0$. Then there is $\gamma\in(0,1]$ and a function $\tilde u$ equal to $u$ almost everywhere on $B_{1/2}\times(-\nf14,0)$ that is parabolically H\"older continuous with exponent $\gamma$ on the closure of $B_{1/2}\times(-\nf14,0)$. \end{theorem} The global result is~\cite[Theorem~1.3]{ESS2003} in the form of its step (3.5): the solution has no singular points. \begin{theorem}[Regularity of $L^\infty L^3$ Leray--Hopf solutions]\label{thm:ess-global} Let $(u,Du)$ be a Leray--Hopf solution on $\R^3\times[0,T]$ with datum $a\in\J$, and suppose $\esssup_{t\in(0,T)}\int_{\R^3}\abs{u(x,t)}^3dx<\infty$. Then every point of $\R^3\times(0,T)$ is a regular point of $u$. \end{theorem} \begin{remark}[Smoothness on positive-time intervals]\label{rem:not-claimed} Theorem~\ref{thm:ess-l5-unique} records the $L^5$ and uniqueness conclusions of~\cite[Theorem~1.3]{ESS2003}. The Prodi--Serrin theorem~\cite[Theorem~8.17, printed p.~173]{RobinsonRodrigoSadowski2016} further implies that its $L^5$ solution is smooth on every closed interval $[\delta,T]$ with $0<\delta0$ and every $w\in \Cc(\R^3\times(0,2);\R^3)$, with $h(t)=t e^{(1-t)/3}$, one has \begin{equation}\label{eq:carleman-gauss} \begin{aligned} \int_{\R^3\times(0,2)} h^{-2a}(t)e^{-|x|^2/(4t)} \left(\frac{a}{t}|w|^2+|\nabla w|^2\right)\,dx\,dt\\ \le c_0\int_{\R^3\times(0,2)}h^{-2a}(t)e^{-|x|^2/(4t)}|Lw|^2\,dx\,dt\,. \end{aligned} \end{equation} \end{proposition} \begin{proof} Set $q=a+1$, $\phi(x,t)=-|x|^2/(8t)-q\log h(t)$, and $v=e^\phi w$. The field $v$ is smooth and compactly supported in $\R^3\times(0,2)$. Define the conjugated operator \[ \mathcal L_\phi v:=e^\phi Lw =\dt v-\dv(v\otimes\nabla\phi)-\nabla v\nabla\phi+\Delta v +( |\nabla\phi|^2-\dt\phi)v\,. \] Here $\dv(v\otimes\nabla\phi)_i=\sum_j\partial_j(v_i\partial_j\phi)$ and $(\nabla v\nabla\phi)_i=\sum_j(\partial_jv_i)(\partial_j\phi)$. In particular, \[ \mathcal L_\phi v=(\dt+\Delta)v-2\nabla\phi\cdot\nabla v +( |\nabla\phi|^2-\dt\phi-\Delta\phi)v\,. \] For this compactly supported field, write $t\mathcal L_\phi=S+A$, where \begin{align} Sv&=t\Delta v+t( |\nabla\phi|^2-\dt\phi)v-\tfrac12v,\label{eq:carleman-S}\\ Av&=\tfrac12\big(\dt(tv)+t\dt v\big) -t\big(\dv(v\otimes\nabla\phi)+\nabla v\nabla\phi\big)\,.\label{eq:carleman-A} \end{align} For compactly supported $v$, the product rule and the flux identity below compute the integrated cross term directly. The time integrations have no endpoint terms because $v$ is supported in a compact subset of $(0,2)$, and the spatial integrations have no boundary terms because $v$ has compact spatial support. All integrals in this proof are over $\R^3\times(0,2)$ unless specified otherwise. Write $\mathcal D_\phi(v)$ for the density in the following direct product-rule computation, with $i,j\in\{1,2,3\}$ summed: \begin{align}\label{eq:carleman-commutator} I:=\int\mathcal D_\phi(v)\,dx\,dt &=4\int t^2\left(\phi_{,ij}\,\partial_i v\cdot\partial_jv +\phi_{,ij}\phi_{,i}\phi_{,j}|v|^2\right)\\ &\quad+\int t^2|v|^2 \left(\dt^2\phi-2\dt|\nabla\phi|^2-\Delta^2\phi\right) +\int t|\nabla v|^2 -\int t|v|^2\left(|\nabla\phi|^2-\dt\phi\right)\,.\nonumber \end{align} Consequently, expanding the square and using the integrated flux identity below gives \begin{equation}\label{eq:carleman-commutator-energy} P:=\int t^2|\mathcal L_\phi v|^2\,dx\,dt =\int|Sv|^2\,dx\,dt+\int|Av|^2\,dx\,dt+I. \end{equation} For vector fields, sum the scalar fluxes over components: \begin{align*} F_0={}&t^2\bigl(|\nabla\phi|^2-\dt\phi\bigr)|v|^2 -t^2|\nabla v|^2-\tfrac12t|v|^2,\\ F_i={}&2t^2\partial_i v\cdot\dt v+t\,v\cdot\partial_i v -4t^2\partial_i v\cdot\left(\sum_{j=1}^3\phi_{,j}\partial_jv\right) +2t^2\phi_{,i}|\nabla v|^2\\ &\quad-2t^2(\Delta\phi)\,v\cdot\partial_i v +t^2\partial_i(\Delta\phi)|v|^2 -2t^2\bigl(|\nabla\phi|^2-\dt\phi\bigr)\phi_{,i}|v|^2 +t\phi_{,i}|v|^2,\qquad i=1,2,3. \end{align*} Expanding $2(Sv)\cdot(Av)$ and differentiating the displayed fluxes by the product rule, with equality of mixed space-time derivatives, gives the pointwise identity \begin{equation*} 2(Sv)\cdot(Av)=\mathcal D_\phi(v)+\dt F_0+\sum_{i=1}^3\partial_iF_i\,. \end{equation*} The fluxes and their derivatives are supported in $\supp v$, a compact subset of the open cylinder, so their time and spatial integrals vanish. Thus $I=2\int Sv\cdot Av$. Expanding $|t\mathcal L_\phi v|^2=|Sv+Av|^2$ pointwise gives the displayed identity for $P$; nonnegativity of the two square integrals yields $I\leq P$. The explicit value below gives $I\geq0$. This integrated estimate requires no separate space-time pairing identities for $S$ and $A$. For the present $\phi$, let $r=h'/h=1/t-1/3$. Then \[ \nabla\phi=-\frac{x}{4t},\qquad \phi_{,ij}=-\frac{\delta_{ij}}{4t}, \qquad |\nabla\phi|^2=\frac{|x|^2}{16t^2}, \qquad |\nabla\phi|^2-\dt\phi=-|\nabla\phi|^2+q r, \] while $\Delta^2\phi=0$ and \[ \dt^2\phi-2\dt|\nabla\phi|^2=\frac{q}{t^2}. \] Substituting these identities into~\eqref{eq:carleman-commutator}, the $\nabla v$ terms cancel, as do the two terms containing $t|\nabla\phi|^2|v|^2$. The remainder is \begin{equation}\label{eq:carleman-I} I=q\int(1-tr)|v|^2\,dx\,dt =\frac{q}{3}\int t|v|^2\,dx\,dt\,. \end{equation} In particular,~\eqref{eq:carleman-commutator-energy} gives $0\le I\le P$. The pointwise identity \begin{equation}\label{eq:carleman-gradient-pointwise} |\nabla v|^2=\tfrac12(\dt+\Delta)|v|^2-v\cdot(\dt v+\Delta v) \end{equation} holds componentwise. To integrate it, use the expression for $\mathcal L_\phi v$ above. Spatial integration by parts yields $\int v\cdot(2\nabla\phi\cdot\nabla v)=-\int(\Delta\phi)|v|^2$; this has no boundary term because $v$ is compactly supported. The $\Delta\phi$ terms then cancel, giving \begin{equation}\label{eq:carleman-gradient-integrated} \int t^2|\nabla v|^2 =-\int t|v|^2-\int t^2v\cdot\mathcal L_\phi v +\int t^2|v|^2( |\nabla\phi|^2-\dt\phi)\,. \end{equation} Here the time integration by parts has no endpoint terms because the support is compact in $(0,2)$, and the spatial Laplacian integration by parts has no boundary term because the support is compact in $\R^3$. Let \[ E=\int t^2\left(|\nabla v|^2+|v|^2|\nabla\phi|^2\right)\,. \] Combining~\eqref{eq:carleman-gradient-integrated} with $|\nabla\phi|^2-\dt\phi=-|\nabla\phi|^2+q h'/h$ gives \[ E=-\int t|v|^2-\int t^2v\cdot\mathcal L_\phi v +q\int t^2\frac{h'}h|v|^2 \le 3I+\left|\int t^2v\cdot\mathcal L_\phi v\right|\,. \] Indeed, the coefficient of $|v|^2$ before the last cross term is $-t+qt^2h'/h=at-qt^2/3\le qt$, and $q\int t|v|^2=3I$ by~\eqref{eq:carleman-I}. Since $00$ and $c_*(\alpha)>0$ such that, for every $a>a_0(\alpha)$ and every $w\in \Cc(\{x_3>1\}\times(0,1);\R^3)$, the following estimate holds with $x'=(x_1,x_2)$ and \[ \phi(x,t)=-\frac{|x'|^2}{8t}+a(1-t)x_3^{2\alpha}t^{-\alpha}\,: \] \begin{equation}\label{eq:carleman-halfspace} \begin{aligned} \int_{\{x_3>1\}\times(0,1)}t^2e^{2\phi} \left(a\frac{|w|^2}{t^2}+\frac{|\nabla w|^2}{t}\right)\,dx\,dt\\ \le c_*(\alpha)\int_{\{x_3>1\}\times(0,1)}t^2e^{2\phi}|Lw|^2\,dx\,dt\,. \end{aligned} \end{equation} One may take $a_0(\alpha)=2$ and \[ c_*(\alpha)=5+\frac{7}{2(2\alpha-1)}\,. \] \end{proposition} \begin{proof} Fix $\alpha\in(1/2,1)$, $a\ge2$, and a test field as in the statement. Set $Q=\{x_3>1\}\times(0,1)$, $v=e^\phi w$, and $\mathcal L_\phi v=e^\phi Lw$. The function $x_3^{2\alpha}$ is smooth on $x_3>1$, and $v$ has compact support in $Q$. Apply the conjugation and commutator identities above on $Q$, with all their integrations by parts justified by this compact support: there are no contributions from spatial infinity, the plane $x_3=1$, or the time endpoints $t=0,1$. The gradients and Hessians of the two parts of $\phi$ are \begin{align} \nabla\phi^{(1)}&=-\frac{x'}{4t},& \nabla\phi^{(2)}&=2\alpha a(1-t)x_3^{2\alpha-1}t^{-\alpha}e_3, \qquad \phi^{(1)}=-\frac{|x'|^2}{8t},\quad \phi^{(2)}=a(1-t)x_3^{2\alpha}t^{-\alpha}. \label{eq:carleman-half-gradients} \end{align} Their gradients are orthogonal, so \begin{equation}\label{eq:carleman-half-orthogonal} \nabla\phi^{(1)}\cdot\nabla\phi^{(2)}=0, \qquad |\nabla\phi|^2=|\nabla\phi^{(1)}|^2+|\nabla\phi^{(2)}|^2\,. \end{equation} The only nonzero entries of the Hessians are \begin{equation}\label{eq:carleman-half-hessians} \phi^{(1)}_{,ij}=-\frac{\delta_{ij}}{4t}\quad(1\le i,j\le2), \qquad \phi^{(2)}_{,33}=2\alpha(2\alpha-1)a(1-t)x_3^{2\alpha-2}t^{-\alpha}. \end{equation} It follows that \begin{equation}\label{eq:carleman-half-hess-grad} \phi_{,ij}\phi_{,i}\phi_{,j} =-\frac1{4t}|\nabla\phi^{(1)}|^2 +\frac{2\alpha(2\alpha-1)a(1-t)}{t^\alpha} x_3^{2\alpha-2}|\nabla\phi^{(2)}|^2 \ge-\frac{|x'|^2}{64t^3}\,. \end{equation} To split the commutator~\eqref{eq:carleman-commutator}, define, for $s=1,2$, \begin{align*} I_s={}&4\int_Q t^2\left(\phi^{(s)}_{,ij}\partial_i v\cdot\partial_jv +\phi^{(s)}_{,ij}\phi^{(s)}_{,i}\phi^{(s)}_{,j}|v|^2\right)\,dx\,dt\\ &+\int_Q t^2|v|^2\left(\dt^2\phi^{(s)}-2\dt|\nabla\phi^{(s)}|^2-\Delta^2\phi^{(s)} -\frac1t|\nabla\phi^{(s)}|^2+\frac1t\dt\phi^{(s)}\right)\,dx\,dt\,. \end{align*} The Hessians and gradients in~\eqref{eq:carleman-half-gradients}--\eqref{eq:carleman-half-hessians} have disjoint tangential and normal coordinates. Thus the mixed terms vanish, and~\eqref{eq:carleman-commutator} becomes \begin{equation}\label{eq:carleman-half-I-split} I=I_1+I_2+\int_Q t|\nabla v|^2\,dx\,dt\,. \end{equation} For $\phi^{(1)}$, direct differentiation gives \begin{align*} 4t^2\phi^{(1)}_{,ij}\partial_i v\cdot\partial_jv &=-t|\nabla'v|^2,\\ 4t^2\phi^{(1)}_{,ij}\phi^{(1)}_{,i}\phi^{(1)}_{,j} &=-t|\nabla\phi^{(1)}|^2,\\ t^2\left(\dt^2\phi^{(1)}-2\dt|\nabla\phi^{(1)}|^2-\Delta^2\phi^{(1)} -t^{-1}|\nabla\phi^{(1)}|^2+t^{-1}\dt\phi^{(1)}\right) &=t|\nabla\phi^{(1)}|^2\,. \end{align*} Therefore $I_1=-\int_Qt|\nabla'v|^2$ and \begin{equation}\label{eq:carleman-half-I1} I=\int_Q t|\partial_3v|^2\,dx\,dt+I_2\,. \end{equation} For the second part of the weight, $\phi^{(2)}_{,33}>0$ on $Q$, and $\phi^{(2)}_{,33}(\partial_3\phi^{(2)})^2\ge0$. Dropping these nonnegative terms from $I_2$ gives \begin{equation}\label{eq:carleman-half-A} I_2\ge\int_Q t^2|v|^2(A_1+A_2+A_3)\,dx\,dt, \end{equation} where \[ A_1=-\dt|\nabla\phi^{(2)}|^2,\qquad A_2=A_1-\Delta^2\phi^{(2)}-\frac1t|\nabla\phi^{(2)}|^2,\qquad A_3=\dt^2\phi^{(2)}+\frac1t\dt\phi^{(2)}\,. \] Differentiating $x_3^{2\alpha}$ four times yields \[ \Delta^2\phi^{(2)}=2\alpha(2\alpha-1)(2\alpha-2)(2\alpha-3) a(1-t)x_3^{2\alpha-4}t^{-\alpha}. \] Using the exact derivative of $|\nabla\phi^{(2)}|^2$, and $2t+(2\alpha-1)(1-t)\ge2\alpha-1$, we obtain \begin{equation}\label{eq:carleman-half-A2} A_2\ge\frac{1-t}{t^\alpha}x_3^{2\alpha-4}a(2\alpha-1) \left[\frac{4\alpha^2a x_3^{2\alpha+2}}{t^{\alpha+1}} -2\alpha(2\alpha-2)(2\alpha-3)\right]\,. \end{equation} On $Q$, $x_3^{2\alpha+2}t^{-(\alpha+1)}\ge1$. The bracket in~\eqref{eq:carleman-half-A2} is therefore at least \[ 8\alpha^2-2\alpha(2\alpha-2)(2\alpha-3) =-4\alpha(2\alpha-1)(\alpha-3)>0, \] where the last inequality uses $1/2<\alpha<1$. Hence $A_2>0$ on $Q$ for every $a\ge2$. A direct differentiation also gives \begin{equation}\label{eq:carleman-half-A3} A_3=a x_3^{2\alpha}t^{-\alpha-2} \left(\alpha^2-(1-\alpha)^2t\right) \ge a(2\alpha-1)\frac{x_3^{2\alpha}}{t^{\alpha+2}}\,. \end{equation} For $A_1$, direct calculation gives \begin{align*} A_1-\frac1t|\nabla\phi^{(2)}|^2 &=4\alpha^2a^2x_3^{4\alpha-2}(1-t)t^{-2\alpha-1} \left(2t+(2\alpha-1)(1-t)\right)\\ &\ge(2\alpha-1)4\alpha^2a^2(1-t)x_3^{4\alpha-2}t^{-2\alpha-1}\ge0. \end{align*} Thus $A_1\ge t^{-1}|\nabla\phi^{(2)}|^2$. From~\eqref{eq:carleman-half-I1}--\eqref{eq:carleman-half-A3}, the commutator energy identity, and $x_3^{2\alpha}t^{-\alpha}\ge1$, we obtain \begin{equation}\label{eq:carleman-half-mass} \begin{aligned} P&:=\int_Q t^2|\mathcal L_\phi v|^2\,dx\,dt \ge I\ge a(2\alpha-1)U+G,\\ U&:=\int_Q\frac{x_3^{2\alpha}}{t^\alpha}|v|^2\,dx\,dt, \qquad G:=\int_Qt|v|^2|\nabla\phi^{(2)}|^2\,dx\,dt\,. \end{aligned} \end{equation} In particular, for $\theta=2\alpha-1>0$ and $V=\int_Q|v|^2\,dx\,dt$, we have $V\le U$ and $P\ge a\theta U+G$. Integrating~\eqref{eq:carleman-gradient-pointwise} with the factor $t$ gives \begin{equation}\label{eq:carleman-half-gradient-id} \int_Qt|\nabla v|^2\,dx\,dt =-\frac12\int_Q|v|^2\,dx\,dt-\int_Qtv\cdot\mathcal L_\phi v\,dx\,dt +\int_Qt|v|^2(|\nabla\phi|^2-\dt\phi)\,dx\,dt\,. \end{equation} The time integration by parts has no endpoint terms because $v$ is compactly supported in $(0,1)$; the spatial integrations by parts have no boundary terms because $v$ is compactly supported in $\{x_3>1\}$. Since $|\nabla\phi^{(1)}|^2-\dt\phi^{(1)}=-|\nabla\phi^{(1)}|^2$, rearranging~\eqref{eq:carleman-half-gradient-id} yields \begin{align}\label{eq:carleman-half-gradient-rewrite} \int_Q t\left(|\nabla v|^2+|v|^2|\nabla\phi|^2\right)\,dx\,dt ={}&-\frac12\int_Q|v|^2-\int_Qtv\cdot\mathcal L_\phi v\\ &+2\int_Qt|v|^2|\nabla\phi^{(2)}|^2 -\int_Qt|v|^2\dt\phi^{(2)}\,dx\,dt\,. \end{align} Moreover, \[ -t\dt\phi^{(2)} =a\frac{x_3^{2\alpha}}{t^\alpha}\big(\alpha+(1-\alpha)t\big) \le a\frac{x_3^{2\alpha}}{t^\alpha}, \] since $01\}\times(0,1)$. The following extensions retain the constants of the stated propositions: \begin{enumerate} \item For every $a>0$, the inequality in Proposition~\ref{prop:carleman-gauss} holds for all $w\in W^{2,1}_2(U_G;\R^3)$ compactly supported in $U_G$, with the same absolute constant $c_0$. \item For every $\alpha\in(1/2,1)$ and every $a>a_0(\alpha)$, the inequality in Proposition~\ref{prop:carleman-halfspace} holds for all $w\in W^{2,1}_2(U_H;\R^3)$ compactly supported in $U_H$, with the same constant $c_*(\alpha)$. \end{enumerate} \end{lemma} \begin{proof} Fix one of the two open sets $U$ in the statement and let $K=\supp w\Subset U$. Extend $w$ by zero to $\R^3\times\R$. Because $w$ vanishes in a neighborhood of $\partial U$, this extension has as its weak derivatives the zero extensions of $Dw$, $D^2w$, and $D_tw$: testing against a smooth function on $\R^3\times\R$ reduces, after inserting a cutoff equal to one near $K$, to the weak-derivative identities on $U$. Thus, for a standard space-time mollifier $\rho_\delta$, \[ \partial_j(w*\rho_\delta)=(Dw)_{\cdot j}*\rho_\delta,\qquad \partial_k\partial_j(w*\rho_\delta)=(D^2w)_{\cdot jk}*\rho_\delta,\qquad \dt(w*\rho_\delta)=(D_tw)*\rho_\delta\,. \] These identities use only the stated spatial weak derivatives through order two and the time weak derivative; no mixed derivative is needed. If $\delta$ is smaller than the distance from $K$ to $\partial U$, then $w*\rho_\delta\in \Cc(U;\R^3)$. All these mollifications are supported in a fixed compact set $K'\Subset U$, on which every scalar weight and coefficient in the two inequalities is bounded. The approximate-identity theorem gives convergence of $w*\rho_\delta$, its first and second spatial derivatives, and its time derivative to the corresponding fields in $L^2(K')$. Hence the weighted $L^2$ norms of the fields and of $\Lop(w*\rho_\delta)$ converge to those of $w$ and $\Lop w$. Apply the appropriate proposition to $w*\rho_\delta$ and pass to the limit. The constants do not depend on $\delta$. \end{proof} \begin{remark}[Departure from \cite{ESS2003}] Propositions 6.1 and 6.2 of~\cite{ESS2003}, printed pp.~238--243, are stated for smooth compactly supported fields. The argument above supplies the density passage needed to use those estimates for compactly supported $W^{2,1}_2$ fields. \end{remark} \begin{lemma}\label{lem:caccioppoli} Let $00$. Suppose that $w:B\times(0,\tau)\to\R^3$ has space-time weak time derivative $D_tw\in L^2(B\times(0,\tau))$, extends continuously to $B\times[0,\tau)$, and satisfies $w(x,0)=0$ for every $x\in B$. Then, for every $00$ with $T\leq1$, let $x_0\in\R^3$, and let $w\colon B_R(x_0)\times[0,T)\to\R^3$ be continuous. Suppose that $w$ has space-time weak derivatives $\nabla w$, $\nabla^2w$, and $\dt w$ on $B_R(x_0)\times(0,T)$ and \[ \abs{w}+\abs{\nabla w}+\abs{\nabla^2w}+\abs{\dt w} \in L^2(B_R(x_0)\times(0,T)),\qquad \abs{\Lop w}\leq c_1(\abs{w}+\abs{\nabla w}) \quad\text{a.e.} \] Assume also that for every $m\in\N$ there are $C_m,r_m>0$ such that \begin{equation}\label{eq:uc-integral-vanishing} \int_0^{r^2}\int_{B_r(x_0)}\abs{w(y,s)}^2\,dy\,ds \leq C_m r^m\qquad(01$, $a>0$. Choose smooth cutoffs $\theta_\rho(y)$ and $\eta(s)$ with $\theta_\rho=1$ on $B_{\rho-1}$ and supported in $B_\rho$, and $\eta=1$ on $(0,3/2)$ and $\eta=0$ for $s\geq7/4$. Their derivatives through order two are bounded independently of $\rho\geq4$. Choose $\chi_\varepsilon\in C^\infty(\R)$ equal to zero for $s\leq\varepsilon$, equal to one for $s\geq2\varepsilon$, and satisfying $\abs{\chi_\varepsilon'}\leq C\varepsilon^{-1}$. For sufficiently small $\varepsilon>0$, set $z_\varepsilon=\chi_\varepsilon\theta_\rho\eta v$, extending it by zero outside $Q(\rho,2)$. The weak product rule, obtained by applying the weak derivative identities to the product of a test function and a smooth cutoff, shows that $z_\varepsilon$ and its first spatial, second spatial, and time weak derivatives are in $L^2$ and that it has compact support in $\R^3\times(0,2)$. The density extension Lemma~\ref{lem:carleman-sobolev} therefore permits applying Proposition~\ref{prop:carleman-gauss} to $z_\varepsilon$. Let \[ K=\bigl(B_\rho\times(0,2)\bigr) \setminus\bigl(B_{\rho-1}\times(0,3/2)\bigr). \] Using the weak product rule twice and the inequality for $Lv$, one obtains \[ \abs{\Lop z_\varepsilon}^2\leq Cc_1^2\lambda^2(\abs{z_\varepsilon}^2+ \abs{\nabla z_\varepsilon}^2) +C(c_1)\mathbf 1_K(\abs v^2+\abs{\nabla v}^2) +C\varepsilon^{-2}\mathbf 1_{(0,2\varepsilon)}\abs v^2. \] Indeed, the derivatives of $\theta_\rho\eta$ are supported in $K$; in the leading term, replace $\chi_\varepsilon\theta_\rho\eta\nabla v$ by $\nabla z_\varepsilon$ minus the cutoff-gradient term, which is again supported in $K$. The last term contains the derivative of $\chi_\varepsilon$. For $\rho\geq4$, $a\geq a_*:=8\beta/\log h(3/2)>0$. The left side of the Carleman inequality controls \[ \kappa\int\omega_a(\abs{z_\varepsilon}^2+ \abs{\nabla z_\varepsilon}^2), \qquad \kappa=\min\{1,a_*/2\}, \] because $s<2$ on the support. Let $C$ be the absolute coefficient of $c_1^2\lambda^2$ in the preceding bound and let $c_0$ be the constant in Proposition~\ref{prop:carleman-gauss}. Since $\lambda^2=2t\leq2\gamma$, choose \[ 0<\gamma(c_1)<\min\left\{\frac{3}{16}, \frac{\kappa}{4Cc_0c_1^2}\right\}. \] Then the first term on the right is absorbed by the Carleman left side. The result is \begin{align} \int\omega_a\left(\frac{a}{s}\abs{z_\varepsilon}^2+ \abs{\nabla z_\varepsilon}^2\right) &\leq C(c_1)\int_K\omega_a(\abs v^2+\abs{\nabla v}^2)\notag\\ &\quad+C\varepsilon^{-2} \int_0^{2\varepsilon}\int_{B_\rho}\omega_a\abs v^2. \label{eq:uc-cutoff-carleman} \end{align} We verify that the last term tends to zero. Fix an integer $m>4a+4$ and a radius $r_*>0$ small enough that the scaled form of \eqref{eq:uc-integral-vanishing} with this exponent $m$ holds whenever $00$. Put $\tau_j=2^{1-j}\varepsilon$ and $I_j=(\tau_j/2,\tau_j)$, for $j=0,1,\ldots$; these intervals cover $(0,2\varepsilon)$ up to endpoints. Partition space into $S_{j,0}=B_{\sqrt{\tau_j}}$ and, for $k\geq1$, \[ S_{j,k}=\{y:2^{k-1}\sqrt{\tau_j}\leq\abs y< 2^k\sqrt{\tau_j}\}. \] If $2^k\sqrt{\tau_j}< r_*$, the set $S_{j,k}\times I_j$ is contained in $B_{2^k\sqrt{\tau_j}}\times]0,(2^k\sqrt{\tau_j})^2[$. On this shell the Gaussian is at most $e^{-4^k/16}$ for $k\geq1$ (and at most one for $k=0$), so its weighted integral is bounded by \[ C_m\tau_j^{m/2-2a}2^{km}e^{-4^k/16}, \] with the exponential factor omitted at $k=0$. Summing these shells gives at most $C_{m,a}\tau_j^{m/2-2a}$. On the remaining shells, $\abs y\geq r_*/2$; their total contribution is at most \[ C\norm{v}_{L^2(Q(\rho,2))}^2\tau_j^{-2a} e^{-r_*^2/(16\tau_j)}. \] Consequently the final term of~\eqref{eq:uc-cutoff-carleman} is bounded by \[ C_{m,a}\varepsilon^{-2}\sum_{j\geq0}\tau_j^{m/2-2a} +C\varepsilon^{-2}\norm{v}_{L^2(Q(\rho,2))}^2 \sum_{j\geq0}\tau_j^{-2a}e^{-r_*^2/(16\tau_j)}. \] The first sum is $O(\varepsilon^{m/2-2a-2})=o(1)$. For every $q>0$ the exponential dominates $\tau^{-q}$ as $\tau\downarrow0$; choose $q>2a+2$. The second sum after multiplication by $\varepsilon^{-2}$ is then $O(\varepsilon^{q-2a-2})=o(1)$. This proves the required limit of the initial-time error. For the remaining cutoff term, we claim \begin{equation}\label{eq:uc-collar-weight} \sup_K\omega_a\leq e^{-\beta\rho^2}. \end{equation} On the part where $s\geq3/2$, monotonicity of $h$ on $[3/2,2]$ gives $h(s)^{-2a}\leq h(3/2)^{-2a}=e^{-\beta\rho^2}$. On the spatial collar $\abs y\geq\rho-1$, $s<3/2$, and $(\rho-1)^2\geq9\rho^2/16$. The function \[ g(s)=h(s)^{-2a}e^{-\rho^2/(16s)} \] is increasing on $(0,2)$: differentiating its logarithm and substituting $a=\beta\rho^2/(2\log h(3/2))$ reduces the sign to $1/16+(\beta/\log h(3/2))(-s+s^2/3)\geq0$. Since $s-s^2/3\leq3/4$ for all $s$, with equality at $s=3/2$, this follows from $\beta\leq\log h(3/2)/12$; here $\log h(3/2)=\log(3/2)-1/6>0.23$. Moreover, $g(2)\leq h(2)^{-2a}\leq h(3/2)^{-2a}=e^{-\beta\rho^2}$, since $h(2)>h(3/2)$. On the spatial collar, $\abs y^2\geq9\rho^2/16$ gives $e^{-\abs y^2/(4s)}\leq e^{-9\rho^2/(64s)}\leq e^{-\rho^2/(16s)}$, so $\omega_a(y,s)\leq g(s)\leq g(2)\leq e^{-\beta\rho^2}$. This proves~\eqref{eq:uc-collar-weight}. Thus the weighted collar integral is at most \[ e^{-\beta\rho^2}\int_{Q(\rho,2)}(\abs v^2+\abs{\nabla v}^2). \] Fatou's lemma applied to the nonnegative Carleman integrand in \eqref{eq:uc-cutoff-carleman}, together with the vanishing initial-time term, now yields \begin{equation}\label{eq:uc-interior-weight} D:=\int_0^{3/2}\int_{B_{\rho-1}}\omega_a (\abs v^2+\abs{\nabla v}^2) \leq C(c_1)e^{-\beta\rho^2} \int_{Q(\rho,2)}(\abs v^2+\abs{\nabla v}^2). \end{equation} The change of variables gives \[ \int_{Q(\rho,2)}(\abs v^2+\abs{\nabla v}^2) =\lambda^{-5}\int_{\lambda Q(\rho,2)}\abs w^2 +\lambda^{-3}\int_{\lambda Q(\rho,2)}\abs{\nabla w}^2 \leq 2\lambda^{-5}\mathcal N_0(R,T;w), \] where $\lambda Q(\rho,2)$ denotes $B_{\lambda\rho}\times(0,2\lambda^2)$ and we used $\lambda^2\leq2T$. Put $\mu=\sqrt{2\beta}$ and $\Omega=B_{\mu X/\lambda}\times(1/2,1)$. The condition $\rho\geq4$ implies $B_{\mu X/\lambda}(1)\subset B_{\rho-1}$, since $\mu\leq1$ and $\abs{X}/\lambda=\rho/2$. On $\Omega$, $h(s)\leq1$ and $\abs y^2\leq2\mu^2\abs X^2/\lambda^2+2$, so \[ \int_\Omega\abs v^2\leq e^{\mu^2\abs X^2/\lambda^2+1}D. \] Rescaling this box back to the original variables, using $\rho=2\abs X/\lambda$ and $\lambda^2=2t$, gives \[ \int_t^{2t}\int_{B_\lambda(\mu X)}\abs{w(y,s)}^2\,dy\,ds \leq C(c_1)\mathcal N_0(R,T;w) e^{-\beta\abs X^2/t}. \] By construction $x=\mu X$, so the target ball is $B_\lambda(x)$ and $\beta\abs X^2/t=\abs x^2/(2t)$. This is \eqref{eq:uc-gaussian-box}. The average statement follows because the box has measure $2^{3/2}|B_1|t^{5/2}$. \end{proof} \begin{theorem}[Unique continuation across a spatial boundary]\label{thm:uc} Let $R,T,c_1>0$ and let $w\colon B_R\times[0,T)\to\R^3$ be continuous. Suppose that $w$ has space-time weak derivatives $\nabla w$, $\nabla^2w$, and $\dt w$ on $Q(R,T)=B_R\times(0,T)$ and \begin{equation}\label{eq:uc-w212} \abs{w}+\abs{\nabla w}+\abs{\nabla^2w}+\abs{\dt w}\in L^2(Q(R,T)), \qquad \abs{\Lop w}\leq c_1(\abs w+\abs{\nabla w})\quad\text{a.e.} \end{equation} Assume that for every $k\in\N$ there is $C_k>0$ such that \begin{equation}\label{eq:uc-pointwise-vanishing} \abs{w(x,t)}\leq C_k(\abs x+\sqrt t)^k \qquad ((x,t)\in Q(R,T)). \end{equation} Then $w(x,0)=0$ for every $x\in B_R$. \end{theorem} \begin{proof} Set $T_0=\min\{T,1\}$ and restrict $w$ to $B_R\times[0,T_0)$. For $00$ its conditions hold. The average of $|w|^2$ over $B_{\sqrt{2t}}(x)\times(t,2t)$ is bounded by \[ C(c_1)\mathcal N_0(R,T_0;w)t^{-5/2} e^{-|x|^2/(2t)}, \] which tends to zero. By continuity of $w$ at $(x,0)$, these averages tend to $|w(x,0)|^2$; hence $w(x,0)=0$. At $x=0$, (\ref{eq:uc-pointwise-vanishing}) and continuity give $w(0,0)=0$. We record why the same Gaussian estimate supplies the integral vanishing needed to move the center. Consider any application of Lemma~\ref{lem:uc-gaussian} centered at $x_0$ on a ball of radius $R_0$, and let $y_0$ satisfy $00$ small enough that $B(y_0,3r)$ lies in $B(x_0,\beta_1R_0)$, $|y-x_0|\geq d/2$ for $y\in B(y_0,3r)$, $\beta_2r^20$. Therefore \[ \int_0^{r^2}\int_{B_r(y_0)}|w|^2 \leq C\mathcal N_{x_0}(R_0,T_0;w)r^3 \sum_{j\geq0}\tau_j^{-3/2}e^{-c d^2/\tau_j}. \] For every $p>0$, $e^{-cd^2/\tau}\leq C(p,d)\tau^p$ for sufficiently small $\tau$. Taking $p$ arbitrarily large shows that the last expression is $O(r^m)$ for every $m$. Thus $w$ has integral vanishing of infinite order at $(y_0,0)$. If $y_0=x_0$, that property is the hypothesis of the application itself. This argument uses the $L^2$ box estimate only and does not assume pointwise spatial regularity at positive times. It remains to reach every point of $B_R$. Define \[ r_0=\beta_1R,\qquad r_{j+1}=r_j+\beta_1(R-r_j). \] We have proved both $w(y,0)=0$ and integral vanishing at every $|y|0$ there is $\gamma=\gamma(c_1)\in(0,1/12)$ with the following property. Let $A\in[0,A_0]$, and suppose that $w$ is continuous on $\Hplus\times[0,1)$, has space-time weak derivatives $Dw,D^2w,D_tw\in L^2$ on every bounded subset of $\Qp$, and satisfies \begin{align*} |D_tw+\Delta w|&\le c_1(|Dw|+|w|) &&\text{a.e. in }\Qp,\\ w(x,0)&=0 &&\text{for }x\in\Hplus,\\ |w(x,t)|&\le e^{A|x|^2} &&\text{for }(x,t)\in\Qp. \end{align*} Writing $\bar A=\max\{A,A_0/2\}$, there is $C=C(c_1,A)>0$ such that, for every $x_3>2$ and $02$ and $0\frac16. \] For $01/(3t)>H/\beta$, so $a>1$ and $\rho>4$; also $\lambda^2=3t\le3\gamma$ absorbs the first term in the estimate for $I_\varepsilon$. For the chosen $a=\beta\rho^2/H$, put $T_\rho=e^{8\bar A|x|^2}e^{-2\beta\rho^2}$. The spatial-shell errors lie on $S_\rho=\{13\rho/20\leq|y|\leq3\rho/4\}$; the late-time mass error lies in $B_{3\rho/4}$ over Carleman times $[\sigma+3/2,\sigma+7/4]$, where the unshifted rescaled time $s-\sigma$ lies in $[3/2,7/4]$. Take $r=(16\sqrt a)^{-1}$. A finite Besicovitch cover of $S_\rho$ by balls of radius $r$ has spatial overlap at most $N$, and its doubled balls lie in $B_\rho$. For Caccioppoli in the rescaled source time, set $a_j=\sigma+jr^2/2$, use inner intervals $(a_j,a_j+r^2)$, and enlarge them to $J_j=(a_j,a_j+4r^2)$. These cover $(\sigma,\sigma+7/4)$ up to a null set, remain below $2$, and have overlap at most $8$; the resulting outer cylinders therefore have overlap at most $8N$. The Gaussian weight varies by a bounded factor on each outer cylinder because $\rho r=\sqrt{H/\beta}/16$ is fixed. Caccioppoli on each inner cylinder, including its initial-time version at $s=\sigma$, controls the shell gradient by outer-cylinder radial mass. The radial mass tail bound, the late-time bound, and the cutoff derivative bounds give shell and late errors bounded by $C(c_1,A)(1+a)(1+\rho)^3T_\rho$. Choose $0<\varepsilon<1/12$ small enough that the weighted initial error in the cutoff estimate above is at most $T_\rho$; the initial-strip limit above permits this choice. Absorbing the term proportional to $c_1^2\lambda^2$ then yields the direct one-cutoff estimate \begin{equation}\label{eq:bu-gaussian-collar} I_\varepsilon\leq C(c_1,A)(1+a)(1+\rho)^3 e^{8\bar A|x|^2}e^{-2\beta\rho^2}. \end{equation} The spatial cutoff equals one on $B_1$, and for this choice of $\varepsilon$ the time cutoff equals one on the averaging core $B_1\times(1/2,1)$. Since $\omega_a\geq e^{-3/2}$ there, the core mass is directly controlled: \[ D:=\int_{1/2}^1\int_{B_1}|v(y,s)|^2\,dy\,ds\leq e^{3/2}I_\varepsilon. \] Finally, $a=\beta\rho^2/H$ implies that the polynomial factor is absorbed by the Gaussian tail: $ (1+a)(1+\rho)^3e^{-2\beta\rho^2}\leq C e^{-\beta\rho^2}$. Thus $D\leq C(c_1,A)e^{8\bar A|x|^2}e^{-\beta\rho^2}.$ Changing variables back on $B_1\times(1/2,1)$ gives the left side of \eqref{eq:bu-gaussian-average} up to the absolute factor $3^{5/2}$. Finally, $\rho^2=(x_3-1)^2/(3t)\ge x_3^2/(12t)$ for $x_3>2$. \end{proof} \begin{remark}[Departure from \cite{ESS2003}] The initial time is not extended by zero. A cutoff above $s=1/6+\varepsilon$ and Lemma~\ref{lem:ftc-small-time} replace that extension; its differentiated error tends to zero by the displayed $L^2$ estimate. The proof uses Lemma~\ref{lem:caccioppoli} for the collar gradients and proves an average estimate, so it does not use the pointwise parabolic regularity bounds or estimate (5.17) of ESS~\cite[§5, pp.~234--236]{ESS2003}. The explicit choice $A_0=10^{-12}$, $\beta=10^{-6}$ also supplies the uniform positive Gaussian exponent needed when the input value $A$ is zero; it satisfies $A_0<1/32$, $8A_0<1/256$, $16A_0/24<1/48$, $16A_0/24<\beta/48$, $\beta<1/256$, and $\beta1/6$. \end{remark} \begin{lemma}[Vanishing on a short time interval]\label{lem:bu-small-time} Fix $c_1>0$ and let $A\in[0,A_0]$. Suppose $w$ satisfies the continuity, weak-derivative, differential-inequality and zero-trace hypotheses of Lemma~\ref{lem:bu-gaussian}, as well as $|w(x,t)|\le e^{A|x|^2}$ on $\Qp$. There is $\gamma_1=\gamma_1(c_1)\in(0,\gamma(c_1)/2]$ such that \[ w(x,t)=0\qquad\text{for every }x\in\Hplus\text{ and }00$ and $1/20$. Indeed, $F$ increases in $y_3$ and decreases in $s$, so the support of $\psi_1'$ has $\Phi\le\Phi(y_+,1/2)=-D/2$. On the support of $\psi_2'$, $\Phi\le-2B/3$. Since $\lambda<1/2$, \[ \frac{D}{2B}=1-\frac12\left(\frac{y_+}{y_0}\right)^{3/2} <1-\frac{45}{128}<\frac23, \] which gives the same bound there. Also $\Phi>0$ implies $y_3>y_+$, so $\eta=1$ on that region. For $R>1$, choose a smooth spatial cutoff $\chi_R$ equal to one on $B_R$, zero outside $B_{2R}$, and satisfying $|D\chi_R|\le C/R$, $|D^2\chi_R|\le C/R^2$. For $\varepsilon>0$, choose $\vartheta_\varepsilon$ equal to zero for $s\le1/2+\varepsilon$, equal to one for $s\ge1/2+2\varepsilon$, and with $|\vartheta_\varepsilon'|\le C/\varepsilon$. The field $V_{R,\varepsilon}=\chi_R\eta\vartheta_\varepsilon v$ has compact support in $(\R^3_++e_3)\times(0,1)$: its spatial support stays above $y_3>1$, and on the bounded set $B_{2R}$ the condition $\psi_2(\Phi/B)\ne0$ keeps $s$ a positive distance below $1$. It belongs to the Sobolev class of Lemma~\ref{lem:carleman-sobolev} by the weak product rule. Apply Proposition~\ref{prop:carleman-halfspace} to $V_{R,\varepsilon}$ with parameter $a>\max\{a_0(3/4),1\}$. The terms containing $D_sv+\Delta_yv$ are bounded by $C c_1^2\lambda^2$ times the Carleman left side, plus cutoff errors. Choose $\gamma_1$ so that \[ 2C c_*(3/4)c_1^2\gamma_1\le\frac14, \qquad \gamma_1\le\frac{\gamma(c_1)}2, \] where $C$ is the absolute product-rule constant. These terms are then absorbed. The errors from $D\eta$ are supported in \[ \omega=\{y_3>y_-,\ 1/21$. The lower-time error vanishes as $\varepsilon\downarrow0$. On the bounded spatial support $B_{2R}$ the Carleman weight is bounded for fixed $a,R$, and Lemma~\ref{lem:ftc-small-time} gives \[ E_{\varepsilon,a}\le C(a,R,\lambda) \varepsilon^{-2}\int_{1/2+\varepsilon}^{1/2+2\varepsilon} \int_{B_{2R}}|v|^2\,dy\,ds \le C(a,R,\lambda)\int_0^{2\lambda^2\varepsilon} \int_{B_{2\lambda R}}|D_tw|^2\,dx\,dt\longrightarrow0. \] We next control the noncompact collar by shifted dyadic cells. For $d=2^{-k}$, each layer $d/2<\delta=s-1/2R\}$, and the weighted integral is finite because the negative quadratic Gaussian dominates the positive normal weight $e^{2a y_3^{2\alpha}}$, with $2\alpha<2$. For every fixed $R$, the lower-time error tends to zero as $\varepsilon\downarrow0$ by Lemma~\ref{lem:ftc-small-time}. Thus, for fixed $a$, the cutoff estimate has the form $I_a\leq C(c_1)J e^{-Da}+S_R+E_R(\varepsilon)$, where $S_R\to0$ as $R\to\infty$ and $E_R(\varepsilon)\to0$ for each fixed $R$. Applying the two-parameter limit in that order, first $\varepsilon\downarrow0$ and then $R\to\infty$, gives \[ I_a\leq C(c_1,A,\lambda)e^{-Da}, \] with a constant independent of $a$. For any compact $K\subset\{y_3>0,\ 1/20\}$, $\eta=1$ and $\Phi$ has a positive minimum on $K$. Hence $a e^{2a\min_K\Phi}\int_K|v|^2\le I_a\to0$ as $a\to\infty$. Continuity gives $v=0$ on this open curved region. It remains to fill the half-space at each fixed time $s_0\in(1/2,1)$. Write \[ g(s)=\left(\frac{B s^\alpha}{1-s}\right)^{1/(2\alpha)}, \] so that $\Phi(y_3,s)>0$ exactly when $y_3>g(s)$. Given any target $y=(y',y_3)\in\Hplus$, choose $H>\max\{2g(s_0),y_3\}$ and let $c=(y',H)$, $R=H-y_3/2$. Then $|y-c|=H-y_30$ such that $s_0+T<1$ and $g(s)0$. On the rest of the compact cylinder $u$ is bounded, so the condition holds with $C_k=r_0^{-k}\sup|u|$. Theorem \ref{thm:uc} therefore gives $v(\cdot,s_0)=0$ on $B_R(c)$, in particular $v(y,s_0)=0$. As $y$ and $s_0$ were arbitrary, these balls cover the connected half-space at every $s_0\in(1/2,1)$. No time reversal is used. Returning to $(x,t)$ proves the lemma. \end{proof} \begin{remark}[Departure from \cite{ESS2003}] The application of Proposition~\ref{prop:carleman-halfspace} to the noncompact cutoff is justified by first inserting $\chi_R$; its shell terms vanish by the Gaussian-average estimate and Caccioppoli. A cutoff above $s=1/2+\varepsilon$, controlled by Lemma~\ref{lem:ftc-small-time}, avoids extending the field by zero at its initial trace. The source invokes Theorem~\ref{thm:uc} to fill the curved zero region in one sentence~\cite[§5, pp.~236--238]{ESS2003}; here the center $c=(y',H)$ and radius $R=H-y_3/2$ are chosen for each target point, and Theorem~\ref{thm:uc} is applied forward from the time $s_0$. The modified transition interval for $\psi_2$ ensures every cutoff derivative lies in $\Phi\le-D/2$. \end{remark} \begin{lemma}[Iteration in time]\label{lem:bu-iterate} Suppose $w$ satisfies the hypotheses of Lemma~\ref{lem:bu-small-time} with some $A\in[0,A_0]$. Then $w\equiv0$ on $\Qp$. \end{lemma} \begin{proof} Set $\gamma_0=0$ and $\gamma_{k+1}=\gamma_k+(1-\gamma_k)\gamma_1$. If $w=0$ for $00$, $M\in\R$, and $\Qp=\Hplus\times(0,1)$. Suppose that $w$ is continuous on $\Hplus\times[0,1)$, has space-time weak derivatives $Dw,D^2w,D_tw$ on $\Qp$, and satisfies \begin{align*} w(x,0)&=0 &&(x\in\Hplus),\\ |D_tw+\Delta w|&\le c_1(|Dw|+|w|) &&\text{a.e. on }\Qp,\\ |w(x,t)|&\le e^{M|x|^2} &&((x,t)\in\Qp), \end{align*} with $w,Dw,D^2w,D_tw\in L^2$ on every bounded subset of $\Qp$. Then $w\equiv0$ on $\Qp$. \end{theorem} \begin{proof} If $M\le A_0$, replace $M$ by $A=\max\{M,0\}$; the growth bound then implies $|w(x,t)|\le e^{A|x|^2}$ and $A\in[0,A_0]$. Lemma~\ref{lem:bu-iterate} proves the conclusion. Suppose next that $M>A_0$, and set $\lambda^2=A_0/(2M)<1/2$. Define $v_0(y,s)=w(\lambda y,\lambda^2s)$. The weak derivatives transform by the affine chain rule, giving \[ |D_sv_0+\Delta_yv_0| \le c_1\big(\lambda|D_yv_0|+\lambda^2|v_0|\big) \le c_1(|D_yv_0|+|v_0|),\qquad |v_0(y,s)|\le e^{(A_0/2)|y|^2}. \] The zero trace, continuity, and local $L^2$ conditions are preserved, so Lemma~\ref{lem:bu-iterate} gives $v_0\equiv0$. Hence $w(x,t)=0$ for $00$, $\overline{\mathcal Q_r(z_0)}\subset D$, and \begin{equation}\label{eq:thmA-rescaled-smallness} r^{-2}\iint_{\mathcal Q_r(z_0)} \bigl(|u|^3+|p|^{3/2}\bigr)\,dx\,dt<\varepsilon_0. \end{equation} Then there is a representative $u_{z_0,r}$ of $u$ on $\mathcal Q_{r/2}(z_0)$ which extends to its closure and satisfies \begin{align*} \sup_{\overline{\mathcal Q_{r/2}(z_0)}}|u_{z_0,r}|&\le C_4r^{-1},\\ |u_{z_0,r}(z)-u_{z_0,r}(z')|&\le C_4r^{-1-\gamma_0}d_{\rm par}(z,z')^{\gamma_0} \quad(z,z'\in\overline{\mathcal Q_{r/2}(z_0)}). \end{align*} \end{lemma} \begin{proof} Set $\Phi(y,s)=(x_0+ry,t_0+r^2s)$ and define $u_r(y,s)=r u(\Phi(y,s))$, $Du_r(y,s)=r^2Du(\Phi(y,s))$, and $p_r(y,s)=r^2p(\Phi(y,s))$. The rescaled domain is $D_r=\Phi^{-1}(D)$. The CKN rescaling theorem, together with the equivalence of its suitable and integrable-suitable predicates, makes $(u_r,Du_r,p_r,0)$ suitable on $D_r$. The affine map $\Phi$ sends the closed unit past cylinder onto $\overline{\mathcal Q_r(z_0)}$, so the assumed domain inclusion gives the closed-cylinder hypothesis of Theorem A. The Jacobian of $\Phi$ is $r^5$, and hence \[ \iint_{\mathcal Q_1(0)}|u_r|^3 =r^{-2}\iint_{\mathcal Q_r(z_0)}|u|^3,\qquad \iint_{\mathcal Q_1(0)}|p_r|^{3/2} =r^{-2}\iint_{\mathcal Q_r(z_0)}|p|^{3/2}. \] The strict real-valued bound in \eqref{eq:thmA-rescaled-smallness} implies the non-strict extended-integral bound required by Theorem A; the force term vanishes. That theorem gives a representative $w$ on the closed half unit-cylinder with supremum at most $C_4$ and parabolic Hölder seminorm at most $C_4$. Define $u_{z_0,r}(z)=r^{-1}w(\Phi^{-1}(z))$. Since $\Phi$ preserves null sets, this is a representative of $u$ on the half-cylinder. Moreover, $d_{\rm par}(\Phi^{-1}z,\Phi^{-1}z')=r^{-1}d_{\rm par}(z,z')$. The two stated bounds follow. \end{proof} \begin{lemma}[Top-boundary $\varepsilon$-regularity]\label{lem:thmA-top} Let $\rho>r>0$, $z_0=(x_0,t_0)$, and suppose $(u,Du,p,0)$ is suitable with exponent $q=3$ on $B_\rho(x_0)\times(t_0-\rho^2,t_0)$. If \begin{equation}\label{eq:thmA-top-smallness} r^{-2}\iint_{B_r(x_0)\times(t_0-r^2,t_0)} (|u|^3+|p|^{3/2})\,dx\,dt<\varepsilon_0/8, \end{equation} then, with $V=B_{r/2}(x_0)\times(t_0-r^2/4,t_0)$, $u$ has a representative on $V$ which extends to the closure and obeys \begin{align*} \sup_{\overline V}|u_{\rm top}|&\le 2C_4/r,\\ |u_{\rm top}(z)-u_{\rm top}(z')| &\le 4\cdot 8^{\gamma_0}C_4r^{-1-\gamma_0} d_{\rm par}(z,z')^{\gamma_0} \quad(z,z'\in\overline V). \end{align*} \end{lemma} \begin{proof} For $x\in B_{r/2}(x_0)$ and $00$. Then both deficits lie strictly between $c$ and $c+r^2/16$, because $a_{\rm early}-a_{\rm late}0$ such that \[ B_r(x_0)\times(t_0-r^2,t_0)\subset D,\qquad \overline B_r(x_0)\times[t_0-r^2,t_0)\subset\widehat D, \] and \[ r^{-2}\iint_{B_r(x_0)\times(t_0-r^2,t_0)} (|u|^3+|p|^{3/2})\,dx\,dt<\varepsilon_0/8. \] \end{definition} \begin{lemma}[Good points and Hölder gluing]\label{lem:good-open-glue} The good-point set is relatively open in $\widehat D$. Let $K\subset\overline{B_{1/2}(0)\times(-1/4,0)}$ be compact and suppose every point of $K$ is good. Then there are a relatively open neighborhood $N\subset\widehat D$ of $K$ and a function $\widetilde u:N\to\mathbb R^3$ such that $\widetilde u=u$ almost everywhere on $N\cap D$ and $\widetilde u|_K$ is parabolically Hölder with exponent $\gamma_0$. In particular, this representative is Hölder on $K$ including its top face $t=0$, and extends continuously there. Moreover, if $z_0=(x_0,t_0)\in K$ is not good, then for every $R>0$ such that $B_R(x_0)\times(t_0-R^2,t_0)\subset D$, \begin{equation}\label{eq:bad-point-lower-bound} R^{-2}\iint_{B_R(x_0)\times(t_0-R^2,t_0)} (|u|^3+|p|^{3/2})\,dx\,dt\ge\varepsilon_0/8. \end{equation} \end{lemma} \begin{proof} Write $G=|u|^3+|p|^{3/2}$. Fix a good point with witness radius $r$. As $r'\uparrow r$, the cylinders $B_{r'}(x_0)\times(t_0-r'^2,t_0)$ increase to the witness cylinder. Monotone convergence and the strict smallness allow us to replace $r$ by some $r'0$ small enough that \[ K_0=\overline B_{r'+\eta}(x_0)\times [t_0-r'^2-\eta,t_0+\eta]\Subset D. \] Every translated cylinder with center sufficiently close to $z_0$ is contained in $K_0$. In both cases the cylinder indicators converge almost everywhere as the center varies, since their boundaries have measure zero. Dominated convergence, using $G$ on the witness cylinder at the top and $\mathbf 1_{K_0}G$ at interior points, gives continuity of the translated integral. The strict inequality persists, proving relative openness. We next construct local representatives around points of $K$. Fix a good point $z_0=(x_0,t_0)$ and a witness radius. Shrink it as in the preceding paragraph to a radius $r'0$, the CKN cylinder of radius $r'/2$ centered at $(x_0,t_0+\sigma)$ has normalized integral less than $\varepsilon_0$ and closed full cylinder in $D$. Choose also $\sigma0$ for the parabolic metric. Let $H$ be the maximum of the local Hölder constants and let $M$ bound the glued function on $K$; both are finite. Pairs in $K$ at distance less than $\delta$ lie in one patch and satisfy its Hölder estimate. For pairs at distance at least $\delta$, the difference is at most $2M$, which is at most $2M\delta^{-\gamma_0}d_{\rm par}(z,z')^{\gamma_0}$. This proves the claimed Hölder bound on all of $K$. Finally, suppose the normalized integral in \eqref{eq:bad-point-lower-bound} were strictly less than $\varepsilon_0/8$ for some such $R$. As $r'\uparrow R$ through smaller radii, the open-top cylinders increase to the cylinder of radius $R$. Monotone convergence and the strict margin give an $r'0$ there is a $\delta>0$ such that \[ \overline B_\rho(0)\times(t-\delta,t+\delta) \quad\text{consists entirely of regular points.} \] \end{lemma} \begin{proof} The CKN parabolic Hausdorff measure uses the diameter gauge for $d_{\rm par}$. If a set in a parabolic cover has diameter $d$, its time projection has Euclidean diameter at most $d^2$, because $|t-s|\le d_{\rm par}((x,t),(y,s))^2$. Thus a parabolic cover of $S$ with diameters $d_i$ induces a time cover whose one-half-dimensional diameter cost is at most $\sum_i d_i$. Taking covers with arbitrarily small total cost shows that the Euclidean Hausdorff measure of dimension $1/2$ of $\pi_tS$ is zero. To see this implies Lebesgue nullity, enclose each time cover set in an interval of length $\ell_i$; for covers with $\ell_i\le\eta$, the total interval length is bounded by $\sqrt{\eta}\sum_i\sqrt{\ell_i}$. The latter sum can be made arbitrarily small, proving $\mathcal L^1(\pi_tS)=0$. For a CKN-suitable solution in the stated class, the CKN version of Theorem C gives zero one-dimensional parabolic Hausdorff measure of its singular set. Also, its regular-point locus is open: if a point has an open neighborhood with a Hölder representative, that same neighborhood and representative witness regularity for each of its points. Hence $S$ is relatively closed in $\mathbb R^3\times I$. Fix $t\in I\setminus\pi_tS$ and $\rho>0$. The compact slice $\overline B_\rho(0)\times\{t\}$ is disjoint from $S$, and so is covered by finitely many product neighborhoods contained in the regular-point locus. The minimum of their time radii is positive; decrease it if necessary so that $(t-\delta,t+\delta)\subset I$. These neighborhoods cover $\overline B_\rho(0)\times(t-\delta,t+\delta)$, as required. \end{proof} \begin{remark} Lemma~\ref{lem:time-projection} supplies neighborhoods consisting of regular points. By itself it does not supply the higher Sobolev bounds, vorticity differential inequality, or infinite-order vanishing hypotheses of ESS Theorem~4.1; those inputs must be established before that theorem is applied (ESS 2003, printed p.~230, (4.1)--(4.3)). \end{remark} \begin{lemma}[Shifted-cylinder regularity]\label{lem:regular-point-shift} Let $\Omega\subset\mathbb R^3$ be open and let $I\subset\mathbb R$ be an open interval. Set $D=\Omega\times I$, and let $z_0=(x_0,t_0)\in D$ and $R>0$. Set $z_1=(x_0,t_0+R^2/8)$ and suppose $\overline{\mathcal Q_{R/2}(z_1)}\subset D$. If $u$ has a representative Hölder on $\overline{\mathcal Q_{R/2}(z_1)}$, equal to $u$ almost everywhere on its interior, then $z_0$ is a regular point in the CKN sense. \end{lemma} \begin{proof} The closed half-cylinder is \[ \overline B_{R/2}(x_0)\times [t_0-R^2/8,t_0+R^2/8]. \] Thus $z_0$ is strictly inside it in both space and time. Choose $0<\delta0$ and $t_0+R^2/8\nf52$, $M\ge0$, $(u,Du,p,0)$ is suitable on $U$, and $|u|\le M$ almost everywhere on $Q_1^-(z_0)$. Set \[ E=\left(\iint_{Q_1^-(z_0)}|Du|^2\,dx\,dt\right)^{1/2}\,. \] Then there is a constant $C=C(M,E)$ such that, on $Q_{1/2}^-(z_0)$, \begin{align} \operatorname*{ess\,sup}_{Q_{1/2}^-(z_0)}\big(|u|+|Du|\big)&\le C,\label{eq:vorticity-regularity-gradient}\\ \|\omega\|_{W^{2,1}_2(Q_{1/2}^-(z_0))} +\sup_{t\in[t_0-1/4,t_0]}\|\widetilde\omega(\cdot,t)\|_{H^2(B_{1/2}(x_0))} +\|\widetilde\omega\|_{C(\overline{Q_{1/2}^-(z_0)})}&\le C, \label{eq:vorticity-regularity-omega} \end{align} where $\omega=\curl u$ is the weak vorticity and $\widetilde\omega$ is a representative continuous on the closed past half-cylinder. In addition, \begin{equation}\label{eq:vorticity-regularity-inequality} |\dt\omega-\Delta\omega| \le C\big(|\omega|+|\nabla\omega|\big) \quad\text{almost everywhere on }Q_{1/2}^-(z_0)\,. \end{equation} The bounds for $u$ and $Du$ are essential bounds on the open past half-cylinder, uniform as $t\uparrow t_0$; no time-continuous representative of $u$ or $Du$ is asserted. \end{theorem} \begin{proof} The suitable-solution energy clause gives $E<\infty$ and $u\in L^2_tH^1_x$ on compact subcylinders. The weak vorticity equation is~\eqref{eq:vorticity-weak-eq}. We use nested radii \[ 1>\frac78>\frac34>\frac58>\frac9{16}>\frac{17}{32}>\frac{33}{64}>\frac{65}{128}>\frac12\,. \] All cutoff constants below depend only on these fixed gaps. We denote by $C(M,E)$ a finite bound depending only on the displayed parameters and the fixed radii; its value may increase from line to line. \emph{Initial energy level.} On $Q_1^-$, $\|\omega\|_{L^2}\le C E$. Let $J=(t_0-1,t_0)$, and choose a spatial cutoff $\varphi\in C_c^\infty(B_1(x_0))$ equal to one on $\overline{B_{7/8}(x_0)}$ and a time cutoff $\theta$ that vanishes near $t_0-1$ and equals one on $[t_0-(7/8)^2,t_0]$. Set $\chi(x,t)=\varphi(x)\theta(t)$. Define $v=u$ on $B_1(x_0)\times J$ and extend it by zero to $J\times\R^3$; extend $z=\chi\omega$ by zero outside $B_1(x_0)$. Then $\|v\|_\infty\le M$ and $v_jz_i-z_jv_i=\chi C_{ji}$. Multiplying the weak equation by $\chi$ gives \[ \dt(\chi\omega_i)-\Delta(\chi\omega_i) =-\partial_j\big(\chi C_{ji}+2\omega_i\partial_j\chi\big) +\partial_j\chi\,C_{ji}+(\dt\chi+\Delta\chi)\omega_i\,. \] Every term on the right is in $L^2_tH^{-1}_x$; the two non-divergence terms are in $L^2_{t,x}$, and the divergence flux is in $L^2_{t,x}$ because $|C|\le2M|\omega|$. The cutoff field has zero initial trace. Lemma~\ref{lem:localized-vorticity-energy}, with the zero-extended multiplier $v$, therefore gives \begin{equation}\label{eq:vorticity-base-energy} \omega\in C([t_0-(7/8)^2,t_0];L^2(B_{7/8})) \cap L^2((t_0-(7/8)^2,t_0);H^1(B_{7/8})), \qquad \|\omega\|_{L^\infty L^2\cap L^2H^1}\le C(M,E)\,. \end{equation} No spatial boundary term occurs because the cutoff is compactly supported in $B_1$; the energy estimate includes the terminal time $t_0$. \emph{Velocity through order two.} For almost every time, $\operatorname{div}u=0$ by (S2) and $\curl u=\omega$ by definition. Lemma~\ref{lem:local-div-curl} with $m=0$, together with $\|u(t)\|_{L^2(B_{7/8})}\le C M$, gives \[ \begin{aligned} u&\in L^\infty((t_0-(3/4)^2,t_0);H^1(B_{3/4}))\\ &\cap L^2((t_0-(3/4)^2,t_0);H^2(B_{3/4})),\\ \|u\|_{L^\infty H^1\cap L^2H^2}&\le C(M,E)\,. \end{aligned} \] Choose a spatial cutoff supported in $B_{3/4}$ and equal to one on $B_{5/8}$. Applying~\eqref{eq:vorticity-product-h2} to the cutoff velocity gives $u\otimes u\in L^2_tH^2_x(B_{5/8})$, with norm at most $C(M,E)$. Since $\operatorname{div}u=0$, the weak vorticity equation is also \[ \dt\omega_i-\Delta\omega_i =-\epsilon_{ik\ell}\partial_k\partial_j(u_\ell u_j)\,. \] This identity follows by taking the curl of the weak momentum equation; it is valid distributionally because $u\otimes u\in L^1_{\rm loc}$ and its localized derivatives just obtained are in $L^2_tH^2_x(B_{5/8})$. Its right side belongs to $L^2_tL^2_x(B_{5/8})$. Apply Lemma~\ref{lem:local-heat-gain} with $m=1$, outer ball $B_{5/8}$, inner ball $B_{9/16}$, initial time $t_0-(3/4)^2$, and terminal time $t_0$. The prior $L^2_tH^1_x$ bound follows from~\eqref{eq:vorticity-base-energy}. We obtain \[ \begin{aligned} \omega&\in C([t_0-(9/16)^2,t_0];H^1(B_{9/16}))\\ &\cap L^2((t_0-(9/16)^2,t_0);H^2(B_{9/16})),\\ \|\omega\|_{C H^1\cap L^2H^2}&\le C(M,E)\,. \end{aligned} \] A first application of Lemma~\ref{lem:local-div-curl} with $m=1$ from $B_{9/16}$ to $B_{17/32}$ uses the $L^\infty_tH^1_x$ bound on $\omega$ and gives $u\in L^\infty_tH^2_x(B_{17/32})$. A separate application with $m=2$ on the same pair of balls uses $\omega\in L^2_tH^2_x(B_{9/16})$ from the heat gain and $u\in L^2_tH^2_x(B_{9/16})$ from the earlier $m=0$ step; it gives $u\in L^2_tH^3_x(B_{17/32})$. Combining these outputs yields \[ \begin{aligned} u&\in L^\infty((t_0-(9/16)^2,t_0);H^2(B_{17/32}))\\ &\cap L^2((t_0-(9/16)^2,t_0);H^3(B_{17/32})),\\ \|u\|_{L^\infty H^2\cap L^2H^3}&\le C(M,E)\,. \end{aligned} \] \emph{One more vorticity level.} A cutoff of $u$ supported in $B_{17/32}$ and equal to one on $B_{33/64}$ is bounded in $L^\infty_tH^2_x$ and $L^2_tH^3_x$. Estimate~\eqref{eq:vorticity-product-h3} therefore gives $u\otimes u\in L^2_tH^3_x(B_{33/64})$, with norm bounded by $C(M,E)$. The right side of the last vorticity equation is now in $L^2_tH^1_x(B_{33/64})$. The prior estimate gives $\omega\in L^2_tH^2_x$ there, so Lemma~\ref{lem:local-heat-gain} with $m=2$, outer ball $B_{33/64}$, inner ball $B_{65/128}$, initial time $t_0-(17/32)^2$ and terminal time $t_0$ yields \[ \begin{aligned} \omega&\in C([t_0-(65/128)^2,t_0];H^2(B_{65/128}))\\ &\cap L^2((t_0-(65/128)^2,t_0);H^3(B_{65/128})),\\ \|\omega\|_{C H^2\cap L^2H^3}&\le C(M,E)\,. \end{aligned} \] Applying Lemma~\ref{lem:local-div-curl} with $m=2$ from $B_{65/128}$ to $B_{1/2}$ gives \[ u\in L^\infty((t_0-1/4,t_0);H^3(B_{1/2})), \qquad \|u\|_{L^\infty H^3}\le C(M,E)\,. \] The Sobolev embedding $H^3(B_{1/2})\hookrightarrow W^{1,\infty}(B_{1/2})$ proves the gradient bound in~\eqref{eq:vorticity-regularity-gradient}. The same level gives $\|\widetilde\omega(t)\|_{H^2(B_{1/2})}\le C(M,E)$ for every $t$ in the closed time interval. The continuous embedding $H^2(B_{65/128})\hookrightarrow C(\overline{B_{1/2}})$ and continuity in time with values in $H^2$ supply the representative $\widetilde\omega\in C(\overline{Q_{1/2}^-})$. Finally, on $Q_{1/2}^-$ the vorticity equation gives \[ \dt\omega=\Delta\omega-\partial_j(u_j\omega_i-\omega_j u_i)\,. \] The level-two bounds put $\omega,\nabla\omega,\nabla^2\omega$ in $L^2$ and the right side in $L^2$. Indeed, $\Delta\omega\in L^2$; expanding the flux derivative is legitimate since $u$ and $Du$ are essentially bounded, $\operatorname{div}u=\operatorname{div}\omega=0$, and it is bounded pointwise by $|u||\nabla\omega|+|Du||\omega|$. Thus $\omega\in W^{2,1}_2(Q_{1/2}^-)$ and \[ |\dt\omega-\Delta\omega| \le |u|\,|\nabla\omega|+|Du|\,|\omega| \le C(M,E)(|\omega|+|\nabla\omega|) \] a.e. This proves~\eqref{eq:vorticity-regularity-omega}--\eqref{eq:vorticity-regularity-inequality}. \emph{A variant of the bootstrap.} The same conclusions also follow from $L^2$ space-time bounds alone, on nested boxes $B_\rho(x_0)\times(t_0-\sigma,t_0)$ with the same top time, without Lemma~\ref{lem:localized-vorticity-energy}. All smoothing uses the backward mollifications from the proof of Lemma~\ref{lem:local-heat-gain}, which are smooth up to $t_0$. Each box below is slightly smaller than the previous one, and every bound depends only on $M$, $E$ and the fixed gaps. Write $C_{ji}=u_j\omega_i-\omega_ju_i$ as in Lemma~\ref{lem:vorticity-weak-eq}. Then $|C|\le2M|\omega|$ and $|\omega|\le\sqrt2|Du|$. The vorticity solves $\dt\omega_i-\Delta\omega_i=-\partial_jC_{ji}$ with $\omega,C\in L^2$, so the first-order heat gain gives $\nabla\omega\in L^2$. For each $j$, the field $\partial_ju$ is divergence free, and its antisymmetric derivatives are given by $\partial_j\omega$. The integrated div--curl bound therefore gives $D^2u\in L^2$, and the $L^4$ bound gives $Du\in L^4$. Differentiating the weak equation shows that $\partial_m\omega$ solves the heat equation with flux $\partial_mC$. Since $|\partial_mC|\le2|Du||\omega|+2M|\nabla\omega|$, this flux is in $L^2$, and the first-order heat gain gives $\nabla^2\omega\in L^2$. By the integrability of $|Du|^2(|D^2u|^2+|\nabla\omega|^2)$, the flux $\partial_k\partial_mC$ in the equation for $\partial_k\partial_m\omega$ is also in $L^2$. Each of $\omega$, $\nabla\omega$ and $\nabla^2\omega$ therefore solves a heat equation with $L^2$ flux. Their backward mollifications are Cauchy in $L^2$ of a ball, uniformly in time. By $H^2\hookrightarrow L^\infty$ on a slightly larger ball, the mollifications of $\omega$ are uniformly Cauchy on $\overline{Q_{1/2}^-(z_0)}$. The limit is the continuous representative $\widetilde\omega$, and the same uniform bounds give $\sup_t\|\widetilde\omega(t)\|_{H^2(B_{1/2})}+\|\widetilde\omega\|_{C(\overline{Q_{1/2}^-})}\le C(M,E)$. Next, fix a time. The embedding $H^2\hookrightarrow L^\infty$ for $\nabla u_\varepsilon(t)$ and the smooth div--curl estimate at that time bound $|\nabla u_\varepsilon|$ on $Q_{1/2}^-$ by $C(M,E)$. These use $|u_\varepsilon|\le M$ and the uniform-in-time bounds of the mollified $\omega$, $\nabla\omega$ and $\nabla^2\omega$. Since $\nabla u_\varepsilon\to Du$ in $L^2$, a subsequence converges almost everywhere, which gives~\eqref{eq:vorticity-regularity-gradient}. Finally, $\dt\omega_i=\Delta\omega_i-\partial_jC_{ji}$ has right side in $L^2$. Hence $\omega\in W^{2,1}_2(Q_{1/2}^-)$ with the stated bound, and the pointwise flux estimate above gives~\eqref{eq:vorticity-regularity-inequality}. This variant does not use time continuity of $u$ or $Du$. \end{proof} \begin{remark}[Departure from \cite{ESS2003}] ESS Lemma~2.2 and Remark~2.3~\cite[Lemma~2.2, Remark~2.3, pp.~216, 220--221]{ESS2003} refer the higher spatial orders to a smooth-solution estimate and linear Stokes regularity; that citation does not by itself justify the passage from a suitable weak solution to the regularity used here. We give the finite weak-to-regular bootstrap needed for this theorem, following the vorticity and div--curl route of Lemari\'e-Rieusset~\cite[Theorems~13.1--13.2, pp.~397--403]{LemarieRieusset2016}. The proof retains the dependence on the local gradient energy $E$ needed to start the vorticity energy estimate, and establishes joint continuity only for $\omega$, which is the trace used below. \end{remark} \begin{remark}[Dependence of the bounds] Every constant $C$ in the theorem depends only on $(M,E)$ and the fixed radii in the proof. To make the dependence on $E=\|Du\|_{L^2(Q_1^-)}$ uniform in a family, take a larger cylinder $\mathcal Q=B_R(x_0)\times(t_0-T_-,t_0+T_+)\Subset U$ containing $\overline{Q_1^-(z_0)}$ in its interior, and choose $\psi\in C_c^\infty(\mathcal Q)$ with $\psi\ge0$ and $\psi=1$ on $\overline{Q_1^-(z_0)}$. Set $M_{\mathcal Q}=\operatorname*{ess\,sup}_{\mathcal Q}|u|$. With $f=0$, the local energy inequality (S4) and H\"older's inequality give \[ E^2\le\frac12\Bigl[M_{\mathcal Q}^2\bigl(\|\dt\psi\|_\infty+\|\Delta\psi\|_\infty\bigr)|\mathcal Q| +\|\nabla\psi\|_\infty\bigl(M_{\mathcal Q}^3|\mathcal Q|+2M_{\mathcal Q}|\mathcal Q|^{1/3}\|p\|_{L^{3/2}(\mathcal Q)}\bigr)\Bigr]. \] Thus fixed larger-cylinder geometry and uniform bounds for $\sup_{\mathcal Q}|u|$ and $\|p\|_{L^{3/2}(\mathcal Q)}$ give a uniform $E$, and hence a uniform theorem constant when $M$ is also uniform. Finiteness of $E$ for each suitable solution follows from (S1) in Definition~\ref{def:sws}. The argument proves continuity up to the top face for $\omega$; it does not assert joint continuity of $u$ or $Du$ in time. \end{remark} \section{Stability of suitable weak solutions and the blow-up limit}\label{sec:blowup} The hypotheses of Theorem~\ref{thm:ess-local} give energy bounds and an $L^\infty_tL^3_x$ bound, but they do not include the local energy inequality required by Definition~\ref{def:sws}. We first recover an energy identity and a representative continuous under spatial $L^{3/2}$ tests. We then fix a pressure decomposition on the original cylinder; its harmonic part has enough interior control to vanish under blow-up scaling. \subsection{Energy equality and weak continuity} \begin{lemma}[Local energy equality]\label{lem:lei-L4} Suppose that $u$, $Du$, and $p$ satisfy the hypotheses of Theorem~\ref{thm:ess-local}, and set \begin{equation}\label{eq:lei-L4-M} M:=\esssup_{t\in(-1,0)}\norm{u(\cdot,t)}_{L^3(B_1)}<\infty\,. \end{equation} For every $r\in(0,1)$, $u\in L^4(B_r\times(-1,0))$ and, for every $\psi\in C_c^\infty(B_r\times(-1,0))$, \begin{equation}\label{eq:local-energy-equality} 2\iint_{B_r\times(-1,0)}\abs{Du}^2\psi\,dx\,dt =\iint_{B_r\times(-1,0)}\left[\abs{u}^2(\dt\psi+\Delta\psi) +(\abs{u}^2+2p)u\cdot\nabla\psi\right]dx\,dt\,. \end{equation} Consequently $(u,Du,p,0)$ is a suitable weak solution on $B_r\times(-1,0)$, in the sense of Definition~\ref{def:sws} with $q=3$. \end{lemma} \begin{proof} For almost every $t\in(-1,0)$, each component of $u(\cdot,t)$ belongs to $H^1(B_r)$ with weak gradient given by the corresponding row of $Du(\cdot,t)$. The ball Sobolev--Poincar\'e inequality applied after subtracting the component means, followed by H\"older's inequality for those means, gives the scale-explicit estimate \begin{equation}\label{eq:local-sobolev-L6} \norm{u(\cdot,t)}_{L^6(B_r)}\le C\left(\norm{Du(\cdot,t)}_{L^2(B_r)} +r^{-1}\norm{u(\cdot,t)}_{L^2(B_r)}\right)\,, \end{equation} where $C$ is an absolute constant in dimension three for three components (CKN library). Interpolation between $L^3$ and $L^6$ gives \begin{equation}\label{eq:local-interpolate-L4} \int_{B_r}\abs{u(x,t)}^4\,dx\le\norm{u(\cdot,t)}_{L^3(B_r)}^2 \norm{u(\cdot,t)}_{L^6(B_r)}^2\le CM^2\left(\norm{Du(\cdot,t)}_{L^2(B_r)}^2 +r^{-2}\norm{u(\cdot,t)}_{L^2(B_r)}^2\right)\,. \end{equation} Integrating in time and using the energy bound in Theorem~\ref{thm:ess-local} proves $u\in L^4(B_r\times(-1,0))$. In particular $F:=u\otimes u$ belongs to $L^2$ on this cylinder. Fix a compact subset of $B_r\times(-1,0)$ containing the support of $\psi$ in its interior, and choose a nonnegative smooth space-time mollifier $\rho$ of integral one, with $\rho_\epsilon(z)=\epsilon^{-4}\rho(z/\epsilon)$. For sufficiently small $\epsilon$, convolution is taken only where its translated support remains in $B_1\times(-1,0)$. Write $u_\epsilon=u*\rho_\epsilon$, $F_\epsilon=F*\rho_\epsilon$, and $p_\epsilon=p*\rho_\epsilon$ there. Spatial divergence commutes with this convolution, so $\nabla\cdot u_\epsilon=0$. The local $L^s$ convergence of mollification for finite $s$ follows from translation continuity in $L^s$: $\norm{g*\rho_\epsilon-g}_{L^s}\le\int\rho_\epsilon(h)\norm{g(\cdot-h)-g}_{L^s}\,dh\to0$ after restricting to a slightly larger compact subset. Convolving the weak momentum identity (S3) gives \begin{equation}\label{eq:mollified-momentum} \dt u_\epsilon+\nabla\cdot F_\epsilon-\Delta u_\epsilon+\nabla p_\epsilon=0\,. \end{equation} Set $R_\epsilon=F_\epsilon-u_\epsilon\otimes u_\epsilon$. Testing \eqref{eq:mollified-momentum}, rewritten with $u_\epsilon\otimes u_\epsilon$ on the left and $-\nabla\cdot R_\epsilon$ on the right, against $2\psi u_\epsilon$ and integrating by parts yields \begin{align}\label{eq:mollified-energy} 2\iint\abs{Du_\epsilon}^2\psi ={}&\iint\left[\abs{u_\epsilon}^2(\dt\psi+\Delta\psi) +(\abs{u_\epsilon}^2+2p_\epsilon)u_\epsilon\cdot\nabla\psi\right]\notag\\ &+2\iint R_\epsilon:Du_\epsilon\,\psi +2\iint R_{\epsilon,ij}(u_\epsilon)_i\partial_j\psi\,. \end{align} All these integrations by parts have no boundary terms because $\psi$ is compactly supported in the region where the convolutions are defined. The pressure term reduces to $p_\epsilon u_\epsilon\cdot\nabla\psi$ because $\nabla\cdot u_\epsilon=0$. The approximate-identity convergence gives $u_\epsilon\to u$ in $L^4$, $Du_\epsilon\to Du$ in $L^2$, and $F_\epsilon\to F$ in $L^2$ on the support of $\psi$. Also $u_\epsilon\otimes u_\epsilon\to u\otimes u$ in $L^2$, so $R_\epsilon\to0$ in $L^2$. Thus the two error integrals in \eqref{eq:mollified-energy} tend to zero by Cauchy--Schwarz. Since $p_\epsilon\to p$ in $L^{3/2}$ and $u_\epsilon\to u$ in $L^3$ on this finite-measure set, $p_\epsilon u_\epsilon\to pu$ in $L^1$. The remaining terms converge in $L^1$ by the stated $L^4$ and $L^2$ convergences. Passing to the limit in \eqref{eq:mollified-energy} proves \eqref{eq:local-energy-equality}. The energy, pressure, measurability, and slice weak-gradient clauses of Definition~\ref{def:sws} on $B_r\times(-1,0)$ follow by restricting the hypotheses of Theorem~\ref{thm:ess-local} to compactly contained boxes. Conditions (S2) and (S3) restrict to tests supported there, and (S4) follows from \eqref{eq:local-energy-equality} for nonnegative tests. This proves suitability with $q=3$ and $f=0$. \end{proof} \begin{lemma}[Weakly continuous $L^3$ representative]\label{lem:weak-cont-L3} Under the hypotheses of Theorem~\ref{thm:ess-local}, there is a map \begin{equation}\label{eq:weak-cont-domain} \bar u\colon[-(3/4)^2,0]\longrightarrow L^3(B_{3/4};\R^3) \end{equation} that agrees with $u(\cdot,t)$ for almost every $t$ and satisfies, for every $w\in L^{3/2}(B_{3/4};\R^3)$, continuity of \begin{equation}\label{eq:weak-cont-pairing} t\longmapsto\int_{B_{3/4}}\bar u(x,t)\cdot w(x)\,dx\,, \end{equation} together with \begin{equation}\label{eq:weak-cont-bound} \sup_{-(3/4)^2\le t\le0}\norm{\bar u(\cdot,t)}_{L^3(B_{3/4})}\le M\,, \end{equation} where $M$ is defined in \eqref{eq:lei-L4-M}. \end{lemma} \begin{proof} Fix $\varphi\in C_c^\infty(B_{3/4};\R^3)$. Testing (S3) with $\eta(t)\varphi(x)$ shows that the distributional derivative of $t\mapsto\int u(x,t)\cdot\varphi(x)\,dx$ is the locally integrable function \begin{equation}\label{eq:weak-cont-derivative} \int_{B_{3/4}}\left[u_i u_j\partial_j\varphi_i -Du_{ij}\partial_j\varphi_i+p\,\nabla\cdot\varphi\right](x,t)\,dx\,. \end{equation} The first term has absolute value at most $M^2\norm{\nabla\varphi}_{L^3(B_{3/4})}$ for almost every $t$. For $s0$, set \begin{equation}\label{eq:pressure-split-rescalings} \begin{aligned} v^R(x,t)&:=R\widetilde u(x_0+Rx,t_0+R^2t),\\ p_1^R(x,t)&:=R^2\widetilde p_1(x_0+Rx,t_0+R^2t),\\ p_2^R(x,t)&:=R^2\widetilde p_2(x_0+Rx,t_0+R^2t)\,. \end{aligned} \end{equation} For almost every $t\in\R$ these satisfy \begin{align}\label{eq:pressure-split-scaling} \int_{\R^3}\abs{v^R(x,t)}^3\,dx &=\int_{\R^3}\abs{\widetilde u(x,t_0+R^2t)}^3\,dx\,,\notag\\ \int_{\R^3}\abs{p_1^R(x,t)}^{3/2}\,dx &=\int_{\R^3}\abs{\widetilde p_1(x,t_0+R^2t)}^{3/2}\,dx\,. \end{align} For every bounded measurable $\Omega\subset\R^3$, all sufficiently small $R$ also satisfy $x_0+R\Omega\subset B_{3/4}$ and \begin{equation}\label{eq:pressure-split-p2-scaling} \int_{\R}\norm{p_2^R(\cdot,t)}_{L^\infty(\Omega)}^{3/2}\,dt =R\int_{\R}\norm{\widetilde p_2(\cdot,s)}_{L^\infty(x_0+R\Omega)}^{3/2}\,ds \le CR\left(\norm{p}_{L^{3/2}(Q)}^{3/2}+M^3\right)\,. \end{equation} In particular, $p_2^R\to0$ in $L^{3/2}(\R;L^\infty(\Omega))$ as $R\downarrow0$. \end{lemma} \begin{proof} For almost every $t$, the zero-extended tensor $F(\cdot,t)$ belongs to $L^{3/2}(\R^3)$ and \begin{equation}\label{eq:pressure-split-cz} \sum_{i,j=1}^3\norm{F_{ij}(\cdot,t)}_{L^{3/2}(\R^3)} \le 9\norm{u(\cdot,t)}_{L^3(B_1)}^2\,. \end{equation} The slice estimate and joint measurability in Definition~\ref{def:riesz-pressure} give the first bound in \eqref{eq:pressure-split-bounds}. Since $p\in L^{3/2}(Q)$ and $p_1$ is in that space on $B_1\times(-1,0)$, their difference $p_2$ also belongs to $L^{3/2}$ there. For $\zeta\in C_c^\infty(Q)$, use $\nabla\zeta$ as the vector test in (S3). The time term vanishes by (S2) applied to $\partial_t\zeta$. In the viscous term, integration by parts in $x_j$ gives $-\iint u_i\partial_i\Delta\zeta$, which vanishes by (S2) applied to $\Delta\zeta$. These integrations have no boundary terms because $\zeta$ is compactly supported in $Q$. The remaining terms yield \begin{equation}\label{eq:pressure-poisson} \iint_Q p\,\Delta\zeta\,dx\,dt =-\sum_{i,j=1}^3\iint_Q u_i u_j\partial_i\partial_j\zeta\,dx\,dt\,. \end{equation} By Definition~\ref{def:riesz-pressure}, $-\Delta p_1=\sum_{i,j}\partial_i\partial_jF_{ij}$ in spatial distributions. Since $F=u\otimes u$ on $B_1$, subtracting this identity from \eqref{eq:pressure-poisson} gives $\Delta_xp_2=0$ in distributions on $B_1\times(-1,0)$. To obtain the slice assertion, test with products $\zeta(x)\eta(t)$, first for a countable dense family of $\zeta\in C_c^\infty(B_1)$; Fubini's theorem gives a common full-measure set of times on which the harmonic identity holds for that family. The $L^{3/2}$ slice bound for $p_2$ extends the identity to every spatial test function, so $p_2(\cdot,t)$ is weakly harmonic for almost every $t$. The CKN weakly harmonic interior estimate gives, for almost every $t$, \begin{equation}\label{eq:pressure-harmonic-sup} \norm{p_2(\cdot,t)}_{L^\infty(B_{3/4})} \le C_{17}\norm{p_2(\cdot,t)}_{L^{3/2}(B_1)}\,, \end{equation} where $C_{17}$ is absolute in dimension three (CKN library). The triangle inequality and the first estimate in~\eqref{eq:pressure-split-bounds} give \begin{align*} \int_{-1}^0\norm{p_2(\cdot,t)}_{L^\infty(B_{3/4})}^{3/2}\,dt &\le C\int_{-1}^0\int_{B_1}\left(\abs{p(x,t)}^{3/2}+\abs{p_1(x,t)}^{3/2}\right)dx\,dt\\ &\le C\left(\norm{p}_{L^{3/2}(Q)}^{3/2}+M^3\right)\,, \end{align*} which proves the second estimate in \eqref{eq:pressure-split-bounds}. The first scaling identity in \eqref{eq:pressure-split-scaling} follows by changing variables $y=x_0+Rx$ and using $|R\widetilde u|^3dx=|\widetilde u|^3dy$. The second follows from the same change of variables and $|R^2\widetilde p_1|^{3/2}dx=|\widetilde p_1|^{3/2}dy$. For the last identity, the same spatial change and then $s=t_0+R^2t$ give the factor $R^3R^{-2}=R$ in \eqref{eq:pressure-split-p2-scaling}. Because $x_0\in\overline{B_{1/2}}$ and $\Omega$ is bounded, $x_0+R\Omega\subset B_{3/4}$ for all sufficiently small $R$, with the threshold depending only on $\sup_{x\in\Omega}|x|$. The second estimate of \eqref{eq:pressure-split-bounds} then proves \eqref{eq:pressure-split-p2-scaling}; its right side tends to zero with $R$. \end{proof} \begin{remark}[Departure from \cite{ESS2003}]\label{rem:pb1-departure} ESS define their local $p_1$ by the weak Dirichlet problem in (3.11)--(3.12) on printed pp.~222--223 of~\cite{ESS2003}, and take $p_2$ to be the resulting harmonic remainder; here $p_1$ is the whole-space Riesz pressure of the zero extension of $u\otimes u$. The new $p_2$ is harmonic on $B_1$, and the needed interior bound and rescaled vanishing are proved directly from this fixed decomposition and the assumed $L^{3/2}(Q)$ pressure norm. Thus the proof does not transfer (3.19) on printed p.~223 of~\cite{ESS2003} to a pressure remainder redefined after rescaling. ESS derive the time continuity in (3.8) through the stronger estimate (3.10) on printed p.~222; here the pairing modulus follows directly from (S3), the energy bound, and the $L^{3/2}$ pressure bound. In the local energy equality, the full pressure remains in $L^{3/2}$: pairing it with the mollified velocity converges in $L^1$ by the $L^{3/2}$--$L^3$ H\"older inequality, so no $L^2$ hypothesis on $p$ is imposed. \end{remark} \subsection{The blow-up limit}\label{subsec:blowup-limit} The compactness argument needs uniform control on each fixed region of the rescaled past. The local energy inequality supplies the gradient bound; the pressure split gives the pressure bound and, after blow-up, identifies its limit with the whole-space pressure of the limiting velocity. We use the open past cylinder $Q_r^-(z_0)$ fixed in Section~\ref{sec:notation}. The open top is useful when \(t_0=0\), since all rescaled times then remain in the original time interval. \begin{proposition}[Blow-up limit]\label{prop:blowup-limit} Suppose \(u_{\mathrm{orig}},Du_{\mathrm{orig}},p_{\mathrm{orig}}\) satisfy the hypotheses of Theorem~\ref{thm:ess-local} on \(D=B_1\times(-1,0)\). Let \begin{equation*} K=\overline{B_{1/2}}\times[-1/4,0], \end{equation*} and let \(z_0=(x_0,t_0)\in K\) be a bad point in the sense of Definition~\ref{def:good-point}, with its top-face interpretation. Let \(R_k\to0\), and discard finitely many terms so that \(Q_{R_k}^-(z_0)\subset D\) for every \(k\). For almost every \(t\in(-1,0)\), let \begin{equation*} F(x,t)=\mathbf 1_{B_1}(x)\,u_{\mathrm{orig}}(x,t)\otimes u_{\mathrm{orig}}(x,t),\qquad p_1(\cdot,t)=P[F(\cdot,t)],\qquad p_2=p_{\mathrm{orig}}-p_1\quad\hbox{on }D, \end{equation*} as in Lemma~\ref{lem:pressure-split}. Extend \(\tilde u_{\mathrm{orig}}=\mathbf1_Du_{\mathrm{orig}}\), \(\widetilde{Du}_{\mathrm{orig}}=\mathbf1_DDu_{\mathrm{orig}}\), and \(\tilde p_{\mathrm{orig}}=\mathbf1_Dp_{\mathrm{orig}}\) by zero to \(\mathbb R^3\times\mathbb R\). Extend \(p_1\) by zero in time only, retaining its whole-space spatial values, and extend \(p_2\) by zero outside \(D\); call these extensions \(\tilde p_1,\tilde p_2\). Define \begin{align*} v^k(x,t)&=R_k\tilde u_{\mathrm{orig}}(x_0+R_kx,t_0+R_k^2t),& Dv^k(x,t)&=R_k^2\widetilde{Du}_{\mathrm{orig}}(x_0+R_kx,t_0+R_k^2t),\\ p^k(x,t)&=R_k^2\tilde p_{\mathrm{orig}}(x_0+R_kx,t_0+R_k^2t),& p_1^k(x,t)&=R_k^2\tilde p_1(x_0+R_kx,t_0+R_k^2t),\\ &&p_2^k(x,t)&=R_k^2\tilde p_2(x_0+R_kx,t_0+R_k^2t). \end{align*} Let \(D_k=\{(x,t):(x_0+R_kx,t_0+R_k^2t)\in D\}\). Then \(p_1^k=P[v^k\otimes v^k]\), and \(p^k=p_1^k+p_2^k\) on \(D_k\). For time-slice assertions, use the \(L^3(B_{3/4})\) equivalence class \(\bar u(\cdot,\tau)\) from Lemma~\ref{lem:weak-cont-L3} whenever \(\tau=t_0+R_k^2t\in[-(3/4)^2,0]\) and the spatial region under consideration rescales into \(B_{3/4}\). The weak-continuity lemma identifies its smooth-test pairings with the absolutely continuous representatives from the momentum equation. This choice agrees with the zero-extended spacetime field almost everywhere and applies throughout each fixed bounded cylinder \(B_R\times[a,0]\) for all sufficiently large \(k\); in particular it specifies the terminal slice at \(t=0\). There is a subsequence, still denoted \(k\), and fields \(u,Du,q\) on \(\mathbb R^3\times(-\infty,0)\) with the following properties. \begin{enumerate}[label=(\alph*)] \item For every bounded cylinder \(C=B_R\times(a,0)\), with \(R<\infty\) and \(a<0\), a constant depending only on \(C\) and the norms in Theorem~\ref{thm:ess-local} satisfies \begin{equation*} \iint_C\bigl(|p^k|^{3/2}+|Dv^k|^2\bigr)\,dx\,dt\le C_C \end{equation*} for all sufficiently large \(k\). \item On each such cylinder, \(v^k\) has the pairing modulus required in \cite[Lemma~11.1]{CKNLean}: for every compact \(K_0\subset\mathbb R^3\), every finite interval \(J=[a,0]\) with \(a<0\), every \(w\in C_c^\infty(\mathbb R^3;\mathbb R^3)\) supported in \(K_0\), and all \(s,t\in J\), \begin{align*} \left|\int_{\mathbb R^3}(v^k(x,t)-v^k(x,s))\cdot w(x)\,dx\right| \le C_{K_0,J}\Bigl(&|t-s|\bigl(\|\Delta w\|_{L^2(\mathbb R^3)} +\|\nabla w\|_{L^\infty(\mathbb R^3)}\bigr)\\ &+|t-s|^{1/3}\|\operatorname{div}w\|_{L^\infty(\mathbb R^3)}\Bigr), \end{align*} where the constants are independent of \(k,s,t,w\) for all sufficiently large \(k\). For \(J\subset(-\infty,0)\) this is the hypothesis of \cite[Lemma~11.1]{CKNLean}; the same estimate extends to \(t=0\) using the trace just defined. Also the \(L^{10/3}(C)\) norms are uniformly bounded for all sufficiently large \(k\) on every bounded cylinder \(C=B_R\times(a,0)\). \item The subsequence satisfies \begin{equation*} v^k\longrightarrow u\quad\hbox{strongly in }L^3(C) \end{equation*} for every bounded cylinder \(C=B_R\times(a,0)\), and \(v^k(\cdot,t)\rightharpoonup u(\cdot,t)\) weakly in \(L^2_{\mathrm{loc}}(\mathbb R^3)\) for every \(t<0\). Also \(Dv^k\rightharpoonup Du\) weakly in \(L^2(C)\) for every such \(C\), where \(Du\) is the weak spatial gradient of \(u\). Moreover, \begin{align*} u&\in L^\infty((-\infty,0);L^3(\mathbb R^3)),\\ p_1^k&\longrightarrow q\quad\hbox{strongly in }L^{3/2}(C),\\ p_2^k&\longrightarrow0\quad\hbox{strongly in }L^{3/2}(C),\\ p^k&\longrightarrow q\quad\hbox{strongly in }L^{3/2}(C). \end{align*} \item The limit \((u,Du,q,0)\) is a suitable weak solution with exponent \(3\) on every \(B_\rho\times(a,0)\), where \(\rho<\infty\) and \(a<0\). The limiting pressure is the whole-space associated pressure \begin{equation*} q=P[u\otimes u]\quad\hbox{on }\mathbb R^3\times(-\infty,0). \end{equation*} \item There is a terminal slice \(u(\cdot,0)\in L^2_{\mathrm{loc}}(\mathbb R^3)\) such that \(v^k(\cdot,0)\rightharpoonup u(\cdot,0)\) weakly in \(L^2_{\mathrm{loc}}(\mathbb R^3)\), and \begin{equation*} \int_{B_1(x)}|u(y,0)|^2\,dy=0\qquad\hbox{for every }x\in\mathbb R^3. \end{equation*} \item For every \(k\), \begin{equation*} \iint_{Q_1^-(0)}\bigl(|v^k|^3+|p^k|^{3/2}\bigr)\,dx\,dt\ge\varepsilon_0/8. \end{equation*} If \(u=0\) almost everywhere on \(\mathbb R^3\times(-T,0)\) for some \(T>1\), then \begin{equation*} \lim_{k\to\infty}\iint_{Q_1^-(0)} \bigl(|v^k|^3+|p^k|^{3/2}\bigr)\,dx\,dt=0. \end{equation*} \end{enumerate} \end{proposition} \begin{proof} Put \(M=\esssup_{t\in(-1,0)}\|u_{\mathrm{orig}}(\cdot,t)\|_{L^3(B_1)}\). Lemma~\ref{lem:lei-L4} makes \((u_{\mathrm{orig}},Du_{\mathrm{orig}},p_{\mathrm{orig}},0)\) suitable on \(B_{3/4}\times(-1,0)\), while Lemma~\ref{lem:weak-cont-L3} supplies the all-time local \(L^3\) representative up to \(t=0\). For each fixed bounded cylinder in the rescaled variables, its inverse image lies in \(B_{3/4}\times(-1,0)\) for all sufficiently large \(k\), except for its top face, which has measure zero. Since \(t_0+R_k^2t\le t_0\le0\) whenever \(t\le0\), these rescalings use only past times. \paragraph{Local energy and pressure bounds.} The velocity scaling and Lemma~\ref{lem:pressure-split} give the bound $M$ for the global spatial $L^3$ norm of $v^k(\cdot,t)$ for almost every rescaled time. For almost every such $t$, both $p_1^k(\cdot,t)$ and $P[v^k(\cdot,t)\otimes v^k(\cdot,t)]$ belong to $L^{3/2}(\mathbb R^3)$ and solve the Poisson equation with right-hand side $\sum_{i,j=1}^3\partial_i\partial_j(v_i^kv_j^k)$. Their difference is harmonic, hence smooth, and belongs to $L^{3/2}(\mathbb R^3)$. The mean-value property gives $|h(x)|\le |B_R|^{-2/3}\|h\|_{L^{3/2}(\mathbb R^3)}$ for this difference $h$ and every $R>0$; letting $R\to\infty$ shows that $h=0$. Hence $p_1^k=P[v^k\otimes v^k]$ on almost every slice, and the $L^{3/2}$ boundedness of $P$ gives \begin{equation*} \|p_1^k(\cdot,t)\|_{L^{3/2}(\mathbb R^3)}\le C M^2. \end{equation*} For every bounded spatial set \(\Omega\), \begin{equation*} \int_{\mathbb R}\|p_2^k(\cdot,t)\|_{L^\infty(\Omega)}^{3/2}\,dt \le C R_k\bigl(\|p_{\mathrm{orig}}\|_{L^{3/2}(D)}^{3/2}+M^3\bigr). \end{equation*} Here and below \(C\) may change from line to line and depends only on the displayed bounded sets and the fixed data norms. In particular, \(p^k\) is uniformly bounded in \(L^{3/2}(C)\) on each bounded cylinder \(C\). For a fixed bounded spatial set \(\Omega\), Hölder's inequality and the velocity scaling give, uniformly for \(t\) in any fixed interval \([a,0]\), \begin{equation}\label{eq:blowup-slice-bound} \int_\Omega |v^k(x,t)|^2\,dx \le R_k^{-1}\int_{x_0+R_k\Omega}|\bar u(y,t_0+R_k^2t)|^2\,dy \le |\,\Omega\,|^{1/3}M^2. \end{equation} This holds for all sufficiently large \(k\), since \(x_0+R_k\Omega\subset B_{3/4}\) and the time arguments lie in \([-(3/4)^2,0]\). Fix \(C=B_R\times(a,0)\). Choose a nonnegative spatial cutoff \(\zeta\in C_c^\infty(B_{R+1})\) that equals one on \(B_R\). For \(b<0\), choose time cutoffs \(\eta_b\in C_c^\infty((a-1,0))\), \(0\le\eta_b\le1\), equal to one on \([a,b]\), with \(\|\eta_b'\|_{L^1}\le2\), uniformly as \(b\uparrow0\). Apply the local energy inequality to \(\psi(x,t)=\zeta(x)\eta_b(t)\) for the rescaled suitable solution. The terms with \(\nabla\zeta\) are bounded by the local \(L^\infty_tL^3_x\) bound and the local \(L^{3/2}\) pressure bound; the term with \(\eta_b'\) is bounded by \(2\sup_t\int_{\operatorname{supp}\zeta}|v^k|^2\). The term with \(\Delta\zeta\) has the same slice \(L^2\) bound. These bounds do not depend on \(b\) or \(k\). Thus \(\iint_{B_R\times(a,b)}|Dv^k|^2\le C_C\); letting \(b\uparrow0\) proves (a), including its top-face endpoint. The weak momentum identity (S3), tested with a spatial field \(w\) and a scalar time test, gives the absolutely continuous pairing identity \begin{equation*} \int_{\mathbb R^3}(v^k(x,t)-v^k(x,s))\cdot w(x)\,dx =\int_s^t\int_{\mathbb R^3} \bigl(v^k\cdot\Delta w+(v^k\otimes v^k):\nabla w +p^k\operatorname{div}w\bigr)\,dx\,d\tau. \end{equation*} The integration by parts in the diffusion term has no spatial boundary contribution because \(w\) is compactly supported. The first term is bounded by \(C|t-s|\|\Delta w\|_2\) using the local slice \(L^2\) bound. The second is bounded by \(C|t-s|\|\nabla w\|_\infty\) using the \(L^\infty_tL^3_x\) bound. For the last term, the local \(L^{3/2}\) pressure bound and Hölder's inequality give \begin{equation*} \int_s^t\int_{\operatorname{supp}w}|p^k|\,|\operatorname{div}w| \le C_{K_0,J}|t-s|^{1/3}\|\operatorname{div}w\|_\infty. \end{equation*} The identity first holds for \(s,t<0\). The trace from Lemma~\ref{lem:weak-cont-L3} and the integrability of the right side let \(t\uparrow0\), proving the same modulus on \([a,0]\). For the local \(L^{10/3}\) bound, choose a spatial cutoff \(\eta\) supported in a bounded neighborhood of \(B_R\) and equal to one there. The three-dimensional Gagliardo--Nirenberg inequality applied to \(\eta v^k(\cdot,t)\), extended by zero, gives \begin{equation*} \|\eta v^k(\cdot,t)\|_{10/3}^{10/3} \le C\|\eta v^k(\cdot,t)\|_2^{4/3} \|\nabla(\eta v^k)(\cdot,t)\|_2^2. \end{equation*} The local slice bound and (a), integrated over \((a,0)\), give the asserted uniform \(L^{10/3}\) bound. For each integer $m\ge1$, choose $N(m)$ so that the local energy, pairing-modulus, and $L^{10/3}$ bounds above hold on $\overline{B}_{m+1}\times[-(m+1),0]$ for every $k\ge N(m)$. Choose a strictly increasing sequence $\nu(j)$ with $\nu(j)\ge\max_{m\le j}N(m)$. Let $\chi_j$ be a smooth spatial cutoff equal to one on $\overline{B}_j$ and supported in $B_{j+1}$, and let $\theta_j$ be a smooth time ramp vanishing before $-(j+1)$ and equal to one on $[-j,0]$. Apply \cite[Lemma~11.1]{CKNLean} once on $\mathbb R^3\times(-\infty,0)$ to the cutoff sequence $\widehat v^j=\theta_j\chi_jv^{\nu(j)}$. The preceding bounds and the cutoff derivative estimates give the required uniform hypotheses on each compact cylinder; the finitely many early indices at each stage are controlled by their individual moduli. Since $\widehat v^j=v^{\nu(j)}$ on any fixed cylinder for all sufficiently large $j$, the resulting all-slice weak $L^2_{\rm loc}$ convergence, local strong $L^2$ convergence, and local weak-gradient convergence transfer to the original rescaled fields. \cite[Corollary~11.3]{CKNLean} then gives strong $L^3$ convergence on every compact subcylinder below $t=0$. Fix \(R<\infty\) and \(a<0\). For sufficiently small \(\delta>0\), the uniform \(L^{10/3}\) bound and H\"older's inequality give \begin{equation*} \|v^k\|_{L^3(B_R\times(a,a+\delta))}^3 \le (|B_R|\delta)^{1/10}\|v^k\|_{L^{10/3}(B_R\times(a,a+\delta))}^3, \end{equation*} and the identical estimate holds on \(B_R\times(-\delta,0)\) and for \(u\) by Fatou's lemma. Both time-boundary strips therefore have \(L^3\) norms tending to zero uniformly in \(k\). On \(B_R\times(a+\delta,-\delta)\) the strong convergence follows from \cite[Corollary~11.3]{CKNLean}, since \(\overline{B_R}\times[a+\delta,-\delta]\) is compactly contained in a slightly larger space-time cylinder. Combining the interior convergence with the two strip estimates proves strong \(L^3\) convergence on \(B_R\times(a,0)\). The local weak-gradient convergence upgrades to weak convergence in \(L^2(B_R\times(a,0))\): (a) gives a uniform \(L^2\) bound on that cylinder, and compactly supported tests in its interior are dense in \(L^2\), so the local weak limit determines the weak limit on the whole cylinder. The distributional gradient identity passes to the limit by the strong local \(L^2\) convergence and identifies \(Du\) as the weak spatial gradient of \(u\). By strong local $L^3$ convergence, take a diagonal subsequence of the zero-extended fields that converges almost everywhere on every bounded space-time cylinder. Fubini's theorem gives a full-measure set of times on which this subsequence converges almost everywhere in space; intersect it with the common full-measure set on which every slice $v^k(\cdot,t)$ has global $L^3$ norm at most $M$. For each such time, Fatou's lemma on each ball gives the same bound for $u(\cdot,t)$ there. Monotone convergence over an increasing sequence of balls yields $\|u(\cdot,t)\|_{L^3(\mathbb R^3)}\le M$, and hence $u\in L^\infty((-\infty,0);L^3)$. \paragraph{The pressure limit.} Set \(F_k=v^k\otimes v^k\) and \(F=u\otimes u\). The just-proved strong \(L^3\) convergence gives \(F_k\to F\) strongly in \(L^{3/2}(B_L\times(a,0))\) for each fixed \(L<\infty\) and \(a<0\). Fix \(RL}|x-y|^{-3}|F_k(y,t)|\,dy\\ &\le C\bigl\||x-\cdot|^{-3}\bigr\|_{L^3(\mathbb R^3\setminus B_L)} \|F_k(\cdot,t)\|_{L^{3/2}(\mathbb R^3)} \le C L^{-2}M^2. \end{align*} The same estimate holds for \(F\). Taking the \(L^{3/2}\) norm on \(B_R\times(a,0)\), the far-field contribution is at most \(C|B_R|^{2/3}|a|^{2/3}L^{-2}M^2\). First let \(k\to\infty\) at fixed \(L\), then let \(L\to\infty\). This proves \(p_1^k=P[F_k]\to P[F]\) strongly in \(L^{3/2}(B_R\times(a,0))\). Lemma~\ref{lem:pressure-split} gives \(p_2^k\to0\) in \(L^{3/2}_tL^\infty_x\) on bounded spatial sets, so \(p_2^k\to0\) in \(L^{3/2}(B_R\times(a,0))\) as well. Since \(p^k=p_1^k+p_2^k\) on every fixed cylinder for all sufficiently large \(k\), the pressure limit is \(q=P[u\otimes u]\). \paragraph{Suitability of the limit.} Fix \(\rho<\infty\) and \(a<0\). On every open box compactly contained in \(B_\rho\times(a,0)\), its inverse image lies in \(B_{3/4}\times(-1,0)\) for all sufficiently large \(k\). Restriction by \cite[Lemma~11.5]{CKNLean} and the Navier--Stokes rescaling \(v^k=R_ku_{\mathrm{orig}}\), \(Dv^k=R_k^2Du_{\mathrm{orig}}\), \(p^k=R_k^2p_{\mathrm{orig}}\) make the tail of the sequence suitable there with exponent \(3\) and zero force. On each such box, (a) and the slice bound \eqref{eq:blowup-slice-bound} give the uniform essential-supremum \(L^2\) and space-time gradient bounds; (b)--(c) give strong \(L^3\), weak \(L^2\) gradients, and strong local \(L^{3/2}\) pressure convergence. Apply \cite[Theorem~11.4]{CKNLean} on each box to the corresponding tail. These conclusions for all compactly contained boxes prove suitability of \((u,Du,q,0)\) on \(B_\rho\times(a,0)\). \paragraph{The terminal slice.} For a fixed compact spatial set and a smooth compactly supported test \(w\), the modulus in (b) is uniform up to \(t=0\). Since the pairings converge for every \(t<0\), it follows that the sequence \(\int v^k(x,0)\cdot w(x)\,dx\) is Cauchy: for \(\delta>0\), compare each endpoint pairing with its value at \(-\delta\), use (b) twice, pass \(k,\ell\to\infty\) at \(-\delta\), and then let \(\delta\downarrow0\). The local \(L^2\) bounds make the limiting functionals bounded on \(L^2\); density and the Riesz representation theorem give a compatible slice \(u(\cdot,0)\in L^2_{\mathrm{loc}}\) and weak \(L^2_{\mathrm{loc}}\) convergence of \(v^k(\cdot,0)\) to it. The same modulus shows that this is the weak trace of \(u\) from times below zero. Fix \(x\in\mathbb R^3\). Since \(t_0\in[-1/4,0]\), Lemma~\ref{lem:weak-cont-L3} gives \(\bar u(\cdot,t_0)\in L^3(B_{3/4})\). For all sufficiently large \(k\), \(B(x_0+R_kx,R_k)\subset B_{3/4}\), and change of variables followed by Hölder's inequality gives \begin{align*} \int_{B_1(x)}|v^k(y,0)|^2\,dy &=R_k^{-1}\int_{B(x_0+R_kx,R_k)}|\bar u(y,t_0)|^2\,dy\\ &\le |B_1|^{1/3} \left(\int_{B(x_0+R_kx,R_k)}|\bar u(y,t_0)|^3\,dy\right)^{2/3}\\ &\longrightarrow0. \end{align*} The final limit follows from absolute continuity of the integral of the fixed \(L^3(B_{3/4})\) function on sets whose measure tends to zero. Weak convergence and lower semicontinuity of the \(L^2(B_1(x))\) norm prove the zero terminal-slice assertion in (e), for every center \(x\). Finally, by Lemma~\ref{lem:good-open-glue} and equation~\eqref{eq:bad-point-lower-bound}, the bad-point lower bound applies at every admissible open-top radius. At radius \(R_k\) it gives, for every \(k\), \begin{equation*} R_k^{-2}\iint_{Q_{R_k}^-(z_0)} (|u_{\mathrm{orig}}|^3+|p_{\mathrm{orig}}|^{3/2})\,dx\,dt\ge\varepsilon_0/8. \end{equation*} The Navier--Stokes scaling changes this exactly into the lower bound in (f). If \(u=0\) on \(\mathbb R^3\times(-T,0)\) with \(T>1\), then \(q=P[u\otimes u]=0\) there. Strong \(L^3(Q_1^-)\) convergence gives \(\int_{Q_1^-}|v^k|^3\to0\); the pressure convergence in (c) gives \(\int_{Q_1^-}|p^k|^{3/2}\to0\). This proves the final assertion. \end{proof} \begin{remark}[Departure from \cite{ESS2003}] The rescaling and the roles of the local energy and pressure bounds are those of (3.13)--(3.20), printed pp.~223--225, of~\cite{ESS2003}. The compactness argument gives strong local space-time \(L^2\), strong \(L^3\), and weak convergence of every velocity slice; it does not use the strong \(C([a,b];L^2_{\mathrm{loc}})\) convergence asserted in (3.23), printed p.~225, of~\cite{ESS2003}. The endpoint conclusion replacing (3.34), printed p.~226, follows from the uniform pairing modulus, weak slice convergence at \(t=0\), and lower semicontinuity. The pressure split is the fixed whole-space Riesz split of Lemma~\ref{lem:pressure-split}; its harmonic remainder decays by that lemma's scaling estimate, proved for this split, rather than by transferring (3.19) to a newly defined pressure remainder. \end{remark} \section{Proof of the local theorem}\label{sec:thm-local} \begin{lemma}[Flatness from a zero neighborhood]\label{lem:flat-from-vanishing} Let $R',T',r_0,M>0$, $x_\ast\in\R^3$, and let $w\colon B_{R'}(x_\ast)\times[0,T')\to\R^3$ be continuous. Suppose $\abs{w}\le M$ on this cylinder and \[ w=0\quad\text{on }B_{r_0}(x_\ast)\times(0,r_0^2), \] where $r_0\le R'$ and $r_0^2\le T'$. Then, for every integer $k\ge1$, \begin{equation}\label{eq:flat-from-vanishing} \abs{w(x,s)}\le M r_0^{-k} \bigl(\abs{x-x_\ast}+\sqrt{s}\bigr)^k \quad\text{for }(x,s)\in B_{R'}(x_\ast)\times(0,T'). \end{equation} For $k=0$ the same conclusion holds with the factor on the right interpreted as $M$. \end{lemma} \begin{proof} Put $d=\abs{x-x_\ast}+\sqrt{s}$. If $d2$ and $R_2>0$. Suppose $(u,Du,q,0)$ is suitable with exponent $3$ on $\R^3\times(-4T_2,0)$, $u\in L^\infty((-4T_2,0);L^3(\R^3))$, and its weakly continuous local trace satisfies $u(\cdot,0)=0$. Suppose there are constants $M,E<\infty$ such that for every $z_1=(x_1,t_1)$ with $\abs{x_1}>R_2$ and $t_1\in(-2,0)$, $(u,Du,q,0)$ is suitable on an open neighborhood of $\overline{Q_1^-(z_1)}$, $\abs{u}\le M$ on $Q_1^-(z_1)$, and \[ \left(\iint_{Q_1^-(z_1)}\abs{Du}^2\,dx\,dt\right)^{1/2}\le E. \] Then $\omega=\curl u$ has a representative continuous on $\{\abs{x}>R_2\}\times(-2,0]$. On every bounded spatial set whose closure lies in $\{\abs{x}>R_2\}$, this representative belongs to $W^{2,1}_2$ up to $t=0$, satisfies \begin{equation}\label{eq:exterior-vorticity-inequality} \abs{\dt\omega-\Delta\omega} \le C(M,E)\bigl(\abs{\omega}+\abs{\nabla\omega}\bigr) \quad\text{a.e. on }\{\abs{x}>R_2\}\times(-2,0), \end{equation} and has $\omega(\cdot,0)=0$ there. In addition, its representative satisfies \[ |\omega(x,t)|\le C_\omega(M,E) \quad (|x|>R_2,\ -2R_2$ and $-2R_2\}$. Cover $\overline B$ by finitely many balls $B_{1/2}(x_j)$ with $\abs{x_j}>R_2$. For the initial time portion, the interval $(-2,-7/4)$ is contained in the half-cylinder time interval centered at $-7/4$. The compact interval $[-7/4,-1/4]$ is covered by finitely many intervals $(s_\ell-1/4,s_\ell)$ with centers $s_\ell\in(-2,0)$, since this family is an open cover of that compact interval. For the terminal portion set $\tau_n=-1/(n+4)$ for $n\ge1$ and $I_n=(\tau_n-1/4,\tau_n)$. Then $\tau_n\uparrow0$ and, for every fixed $t\in(-1/4,0)$, $\mathbf1_{I_n}(t)\to1$. For each spatial center $x_j$ and each of the finitely many initial time centers, and for each $(x_j,\tau_n)$, the half-cylinder estimates give with one constant $C=C(M,E)$ \[ \esssup_{t\in I}\norm{\omega(\cdot,t)}_{H^2(B\cap B_{1/2}(x_j))} +\norm{\dt\omega}_{L^2((B\cap B_{1/2}(x_j))\times I)} \le C, \] where $I$ denotes the corresponding time interval. They also bound each spatial derivative through order two in $L^2$ on that patch and interval. The finite covers give the asserted bounds on the initial portion. On the terminal portion, Fatou's lemma applied to the indicators $\mathbf1_{I_n}$ and to the squares of $\dt\omega$ and of each spatial derivative through order two gives finite $L^2$ norms on $B\times(-1/4,0)$. Since the $I_n$ cover this terminal portion and the $H^2$ bounds are uniform, countability also gives $\esssup_{-1/40$. The kernel formula for the double Riesz transforms defining $P$, its spherical cancellation, and $\abs{K(x-y)}\le C\abs{x-y}^{-3}$ give \[ \norm{P[F(\cdot,t)]}_{L^\infty(B_{r_0/2}(x_0))}\le C(H,r_0,\chi). \] Indeed, in the principal-value integral subtract $F(x,t)$ near $x$; the remaining near-field integrand is bounded by $C H\abs{x-y}^{\alpha-3}$, which is integrable in dimension three, and the far-field integral is bounded because $F$ has fixed compact support. Set $h=q-P[F]$ on $B_{r_0/2}(x_0)$. The pressure identity from (S3) and the definition of $P$ imply $\Delta_x h=0$ there for almost every time. Also $h\in L^{3/2}$ on a slightly larger compact cylinder. The harmonic interior estimate therefore gives a function $g\in L^1$ of time such that \[ \norm{h(\cdot,t)}_{L^\infty(B_{r_0/4}(x_0))}^{3/2}\le g(t) \quad\text{for a.e. }t. \] For $r0$, \[ \abs{v_i(x)}\le \frac1{\abs{B_R}\,}\int_{B_R(x)}\abs{v_i(y)}\,dy \le \abs{B_R}^{-1/3}\norm{v_i}_{L^3(\R^3)}. \] Since $\abs{B_R}^{-1/3}=\abs{B_1}^{-1/3}R^{-1}$, letting $R\to\infty$ shows $v_i(x)=0$. This holds for each component. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:ess-local}] We argue by contradiction. By Lemma~\ref{lem:lei-L4}, the hypotheses of Theorem~\ref{thm:ess-local} make $(u,Du,p,0)$ suitable with exponent $3$ on $B_1\times(-1,0)$: every compactly supported test in this open cylinder is supported in some $B_r\times(-1,0)$, $r<1$, where that lemma gives the local energy inequality. Let \[ K=\overline{B_{1/2}}\times[-1/4,0]. \] If $u$ has no H\"older representative on $K$, the contrapositive of Lemma~\ref{lem:good-open-glue} (with its amended top-face definition of good points) gives $z_0=(x_0,t_0)\in K$ that is not good. Choose any sequence $R_k\downarrow0$ small enough that the open-top cylinders $Q^-_{R_k}(z_0)$ lie in $B_1\times(-1,0)$. Proposition~\ref{prop:blowup-limit}, using \eqref{eq:bad-point-lower-bound}, supplies a subsequential blow-up limit $(U,DU,q,0)$ on $\R^3\times(-\infty,0)$. Clause (c) of that proposition gives $U\in L^\infty((-\infty,0);L^3(\R^3))$; clause (d) gives suitability with exponent $3$ on every bounded cylinder in the open past and $q=P[U\otimes U]$; clause (e) gives a terminal slice $U(\cdot,0)=0$ in $L^2_{\mathrm{loc}}$; and clause (f) gives the lower bound \[ \iint_{Q_1^-(0)}\bigl(\abs{v^k}^3+\abs{p^k}^{3/2}\bigr)\,dx\,dt \ge\varepsilon_0/8\quad\text{for every }k, \] with convergence of these integrals to zero if $U=0$ on $\R^3\times(-T,0)$ for any $T>1$. Here $v^k,p^k$ are the rescalings in Proposition~\ref{prop:blowup-limit}; its top-face cylinders are open at $t=0$, so no future values are used. \paragraph{Exterior regularity.} Fix $T_2>2$. The bounds $U\in L^\infty L^3$ and $q=P[U\otimes U]$, together with the $L^{3/2}$ boundedness of the double Riesz transforms, imply $U\in L^3(\R^3\times(-4T_2,0))$ and $q\in L^{3/2}(\R^3\times(-4T_2,0))$. Hence \[ \iint_{(\R^3\setminus B_A)\times(-4T_2,0)} (\abs U^3+\abs q^{3/2})\,dx\,dt\longrightarrow0 \quad(A\to\infty). \] Let $\varepsilon_{\rm A}$ be the threshold in Lemma~\ref{lem:thmA-rescaled}, and set $r_{\rm e}=1/4$. Choose $A$ so large that this tail is less than $\varepsilon_{\rm A}r_{\rm e}^2$. For every center $z=(x,t)$ with $\abs x>A+1$ and $-2T_2-1t$ with $t'<0$ and $t'-t2$. Its normalized integral is less than $\varepsilon_{\rm A}$. Theorem \ref{lem:thmA-rescaled} gives a representative bounded by $C_4/r_{\rm e}$ on its half-cylinder, which contains $(x,t)$. This proves an essential bound $\abs U\le M$ on $\{\abs x>A+1\}\times(-2T_2-1,0)$, with $M=C_4/r_{\rm e}$. Set $R_2=A+3$ and $R_3=R_2+2$. For every $z_1=(x_1,t_1)$ with $\abs{x_1}>R_3$ and $t_1\in(-2T_2,0)$, the cylinder $Q_1^-(z_1)$ lies in the bounded-velocity region just obtained and in $\R^3\times(-4T_2,0)$. Its local gradient energy has a bound independent of $x_1,t_1$. To see this, apply the local energy inequality with a fixed spatial cutoff equal to one on $B_1(x_1)$ and supported in $B_2(x_1)$, and with a time cutoff that rises from zero to one on $(t_1-2,t_1-1)$, equals one on $[t_1-1,t_1]$, and falls to zero on $(t_1,t_1+\delta_1)$, where $\delta_1=\min\{-t_1/2,1/8\}$. This interval lies before $0$, and the total variation of the cutoff is $2$, uniformly in $t_1$. Its support lies in $(-4T_2,0)$ because $t_1>-2T_2$ and $T_2>2$. The terms with $U^2$ are bounded by the global $L^\infty_tL^3_x$ bound on balls of fixed volume; the terms with $U^3$ and $qU$ are bounded by the global $L^\infty_tL^3_x$ and $L^\infty_tL^{3/2}_x$ bounds and H\"older's inequality. The local energy inequality with this test bounds the gradient integral on the whole open-top cylinder by a constant uniform in $x_1,t_1$. The integrals over $B_1(x_1)\times(t_1-1,s)$ increase to this full integral as $s\uparrow t_1$, so monotone convergence includes the top endpoint. Thus \[ \left(\iint_{Q_1^-(z_1)}\abs{DU}^2\right)^{1/2}\le E \] for a constant $E$ independent of the exterior center. For each such $z_1$, the closure of $Q_1^-(z_1)$ is contained in an open cylinder compactly contained in $\R^3\times(-4T_2,0)$; suitability restricts to that neighborhood. All hypotheses of Theorem~\ref{thm:vorticity-regularity} therefore hold with the same $M,E$. We first verify the trace hypothesis of Lemma~\ref{lem:vorticity-top-extension}. For every smooth compactly supported test field $\psi$, clause (b) of Proposition~\ref{prop:blowup-limit} gives the rescaled fields a uniform pairing modulus on intervals $[a,0]$. By clause (c), their pairings converge at every negative time to those of $U$. By clause (e), their pairings at time zero converge to those of the terminal slice $U(\cdot,0)$, which is zero in $L^2_{\rm loc}$. Passing to the limit in the modulus therefore gives \[ \left|\int U(x,t)\cdot\psi(x)\,dx -\int U(x,0)\cdot\psi(x)\,dx\right| \le C_\psi\bigl(|t|+|t|^{1/3}\bigr)\longrightarrow0 \quad(t\uparrow0). \] Thus $U$ has the required weakly continuous local trace and that trace is zero. The uniform $L^\infty_tL^3_x$ bound in clause (c) and density of smooth compactly supported tests in $L^{3/2}$ extend this pairing continuity to weak continuity in local $L^3$. Lemma~\ref{lem:vorticity-top-extension} now gives a continuous representative of $\omega=\curl U$ on $\{\abs{x}>R_3\}\times(-2,0]$, locally in $W^{2,1}_2$ up to the top, with a uniform inequality \[ \abs{\dt\omega-\Delta\omega} \le c_{\rm ext}(\abs\omega+\abs{\nabla\omega}) \] and terminal value $\omega(\cdot,0)=0$. \paragraph{Backward uniqueness outside a ball.} Fix a unit vector $e$ and choose $O\in SO(3)$ with $Oe_3=e$. For $(y,s)\in\R^3_+\times[0,1)$ set \[ x=R_3e+\sqrt{2}\,Oy,\qquad t=-2s,\qquad w(y,s)=A_\omega^{-1}\omega(R_3e+\sqrt{2}\,Oy,-2s), \] where $A_\omega=\max\{1,\|\omega\|_{L^\infty(\{|x|>R_3\}\times(-2,0))}\}$. Then $x\cdot e=R_3+\sqrt2 y_3>R_3$ and the map sends $\{x\cdot e>R_3\}\times(-2,0]$ onto $\R^3_+\times[0,1)$. The weak chain rule gives \[ (\partial_s+\Delta_y)w =-2A_\omega^{-1}(\dt\omega-\Delta_x\omega) (R_3e+\sqrt2 Oy,-2s), \qquad \abs{\nabla_x\omega}/A_\omega=2^{-1/2}\abs{\nabla_y w}. \] Thus $\abs{\partial_s w+\Delta_yw}\le c_1(\abs w+\abs{\nabla_yw})$ with $c_1=2c_{\rm ext}$, after increasing it if necessary. The zero terminal trace gives $w(y,0)=0$; the exterior bound gives $\abs w\le1\le e^0$; and the local $W^{2,1}_2$ estimates give all required $L^2$ bounds on bounded subsets. Theorem~\ref{thm:bu} yields $w=0$ on $\R^3_+\times(0,1)$. As $e$ was arbitrary, choosing $e=x/\abs x$ shows \begin{equation}\label{eq:omega-exterior-zero} \omega=0\quad\text{on }(\R^3\setminus\overline{B_{R_3}})\times(-2,0). \end{equation} \paragraph{Regular time slices and unique continuation.} By Proposition~\ref{prop:blowup-limit}(d), the limit is suitable on every bounded subcylinder of $\R^3\times(-2,0)$. Every compactly supported test in the full open domain $\R^3\times(-2,0)$ is supported in one such bounded cylinder, so these local suitability statements give the global CKN suitability predicate. Clause (c) gives $U\in L^\infty((-2,0);L^3(\R^3))$, and clause (d) gives the global pressure identity $q=P[U\otimes U]$. Boundedness of the double Riesz transforms on $L^{3/2}(\R^3)$ therefore gives $q\in L^\infty((-2,0);L^{3/2}(\R^3))$, the required global pressure integrability; the local gradient and energy conditions are those in (d). Thus the limit lies in the exponent-three CKN class on the full open domain. CKN Theorem~C therefore says that the singular set $S$ there has zero one-dimensional parabolic Hausdorff measure. By Lemma~\ref{lem:time-projection}, for almost every $t_\ast\in(-2,0)$ there is an interval $I$ about $t_\ast$ such that $\overline{B_{4R_3}}\times I$ consists of regular points. Shrink $I$ so that its closure is compactly contained in $(-2,0)$. We record the small-scale input at each regular point. If $z$ is regular, choose $r_0>0$ so that $\overline{Q_{r_0}^-(z)}$ lies in its regularity neighborhood and in the open solution domain. By Proposition~\ref{prop:blowup-limit}(d), the limit is suitable with exponent $3$ there; by clauses (c)--(d), its pressure is the global associated pressure $q=P[U\otimes U]$. Lemma~\ref{lem:pressure-smallness-at-regular-point} applies directly and gives \[ r^{-2}\iint_{Q_r^-(z)}(\abs U^3+\abs q^{3/2})\longrightarrow0. \] For small $r$, Lemma~\ref{lem:thmA-rescaled} also gives a Hölder representative and an explicit bound on a smaller cylinder. Since the closed cylinder is inside the open domain, Theorem \ref{thm:vorticity-regularity} applies there after parabolic rescaling. Cover the compact cylinder $\overline{B_{4R_3}}\times\overline I$ by finitely many of these smaller regular neighborhoods. The resulting vorticity representatives glue; on every smaller cylinder they satisfy $W^{2,1}_2$ bounds, are continuous, and obey \begin{equation}\label{eq:omega-local-uc-inequality} \abs{\dt\omega-\Delta\omega} \le c_I(\abs\omega+\abs{\nabla\omega}). \end{equation} The constants and the bound $\abs\omega\le M_I$ are uniform on this compact cylinder. Define \[ V=\{x\in B_{4R_3}:\text{ for some }r>0, \ \omega=0\text{ on }B_r(x)\times I\}. \] This set is relatively open. It is nonempty by \eqref{eq:omega-exterior-zero}, since the exterior zero region intersects $B_{4R_3}$. We show it is relatively closed. Let $x\in\overline V\cap B_{4R_3}$. Choose $R'>0$ with $\overline B_{2R'}(x)\subset B_{4R_3}$, and choose $x_\ast\in V$ so close to $x$ that $\abs{x-x_\ast}0$ such that $\omega=0$ on $B_r(x_\ast)\times I$. Fix any $t_\ast\in I$. Choose $T'>0$ with $T'0$ so small that $r_0<\min\{r,R'\}$ and $r_0^21$, says those same integrals tend to zero. The contradiction proves that $u$ has a parabolically H\"older representative on $K$, which is the conclusion of Theorem~\ref{thm:ess-local}. \end{proof} \begin{remark}[Departure from \cite{ESS2003}]\label{rem:thm-local-departure} The proof follows the blow-up, exterior backward-uniqueness, and interior unique-continuation structure of Theorem~1.4 in~\cite[pp.~222--229]{ESS2003}. The endpoint good-point criterion is stated on open-top past cylinders, and the pressure split is the whole-space Riesz split of Lemma~\ref{lem:pressure-split}; the endpoint lower bound and pressure convergence are supplied by Proposition~\ref{prop:blowup-limit}. We give the terminal vorticity extension and the infinite-order vanishing step explicitly, and use CKN Theorem~C for the regular-time projection. These choices make the top-face trace and both applications of unique continuation explicit in the regularity classes used here. \end{remark} \section{Proof of the global theorem}\label{sec:thm-global} The pressure needed for the local theorem is the canonical whole-space pressure of the velocity tensor. Its mixed-norm bound follows directly from the slice-wise Riesz estimate. \begin{lemma}[Pressure bound under the critical velocity hypothesis] \label{lem:assoc-pressure-L3} Suppose $(u,Du)$ satisfies the hypotheses of Theorem~\ref{thm:ess-global}. Let $p_*=P[u\otimes u]$ be the pressure witness selected in the proof of Theorem~\ref{thm:assoc-pressure}. Then $p_*\in L^{5/3}(\R^3\times(0,T))$, the identity (S3) of Definition~\ref{def:sws} holds with $p=p_*$ and $f=0$, and $p_*\in L^\infty((0,T);L^{3/2}(\R^3))$. There is a constant $C$ depending only on the dimension and the $L^{3/2}$ double-Riesz-transform bound such that \[ \esssup_{t\in(0,T)}\norm{p_*(\cdot,t)}_{L^{3/2}(\R^3)} \le C\left(\esssup_{t\in(0,T)} \norm{u(\cdot,t)}_{L^3(\R^3)}\right)^2. \] \end{lemma} \begin{proof} The proof of Theorem~\ref{thm:assoc-pressure} takes $F_{ij}=u_i u_j$ and sets $p_*=P[F]$. \cite[Lemma~8.4]{CKNLean} gives $F\in L^{5/3}(\R^3\times(0,T))$; the proof there uses \cite[Lemma~8.2]{CKNLean} to verify the gradient part of (S3). Thus the existential theorem's chosen witness is exactly $P[u\otimes u]$, with the stated global integrability and (S3) identity. For almost every $t$, the hypothesis of Theorem~\ref{thm:ess-global} gives $u(\cdot,t)\in L^3(\R^3)$. The slice estimate in Definition~\ref{def:riesz-pressure}, applied with $r=3/2$, yields \[ \norm{p_*(\cdot,t)}_{L^{3/2}} \le C_{3/2}\sum_{i,j=1}^3 \norm{u_i(\cdot,t)u_j(\cdot,t)}_{L^{3/2}} \le 9C_{3/2}\norm{u(\cdot,t)}_{L^3}^2. \] Taking the essential supremum proves the estimate. The pressure representative in Definition~\ref{def:riesz-pressure} is jointly measurable, so it also has the measurability required below. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:ess-global}] Fix $z_0=(x_0,t_0)\in\R^3\times(0,T)$. Choose $R>0$ so small that \[ R^2<\min\left\{\frac{8t_0}{7},\,8(T-t_0)\right\}, \qquad t_1=t_0+\frac{R^2}{8}. \] Then $00$, \begin{equation}\label{eq:pv-heat-l5} \|S(\cdot)b\|_{L^5(Q_\tau)}\le C\|b\|_{L^3}\,, \end{equation} where $C$ depends only on the dimension. If in addition $b\in L^2$, then \begin{equation}\label{eq:pv-heat-l4} \|S(\cdot)b\|_{L^4(Q_\tau)}\le C\bigl(\|b\|_{L^2}+\|b\|_{L^3}\bigr)\,. \end{equation} The constants do not depend on $\tau$. \end{lemma} \begin{proof} First take $b\in C_c^\infty(\mathbb R^3;\mathbb R^3)$. The Gaussian heat kernel gives a smooth solution $h=S(t)b$ with enough decay to integrate by parts in space. Testing $\partial_t h-\Delta h=0$ against $|h|h$ yields \begin{equation*} \frac13\frac{d}{dt}\|h(t)\|_3^3+c\int_{\mathbb R^3}|h|\,|\nabla h|^2\,dx+c\int_{\mathbb R^3}|\nabla(|h|^{3/2})|^2\,dx\le0\,, \end{equation*} with a numerical $c>0$. At zeros of $h$, this identity follows first with $|h|$ replaced by $(|h|^2+\eta^2)^{1/2}$ and then by letting $\eta\downarrow0$; the nonnegative terms pass by Fatou's lemma. There is no boundary contribution, since one may first insert a cutoff equal to one on $B_R$ and use the Gaussian decay as $R\to\infty$. With $g=|h|^{3/2}$, integration in time gives \begin{equation*} \operatorname*{ess\,sup}_{00$, \begin{equation*} \|S(t)a_\rho\|_5\le Ct^{-3/40}\|a_\rho\|_4\,,\qquad \|S(t)a_\rho\|_4\le Ct^{-1/8}\|a_\rho\|_3\,. \end{equation*} Integrating the fifth and fourth powers in time gives \begin{equation*} \|S(\cdot)a_\rho\|_{L^5(Q_\tau)}+\|S(\cdot)a_\rho\|_{L^4(Q_\tau)} \le C\tau^{1/8}(\|a_\rho\|_4+\|a_\rho\|_3)\,. \end{equation*} Given $\epsilon>0$, choose $\rho$ so that the approximation error is below $\epsilon/2$, then choose $\tau$ so that the smooth-data term is below $\epsilon/2$. This proves the limit. The pointwise heat bounds are the cases $(s,s_1)=(5,4)$ and $(4,3)$ of (7.3) in~\cite[printed p.~243]{ESS2003}; their time integrals above are computed directly, without using the general mixed-norm assertion (7.4). \end{proof} \section{The forced Stokes response}\label{sec:pv-stokes} The fixed point needs an estimate that controls both the critical $L^5$ norm and an $L^4$ norm used to keep the quadratic forcing in $L^2$. The pressure in this linear equation must be included in the endpoint energy test; it is controlled by the same Riesz bounds as the force. \begin{lemma}[Zero-data Stokes estimate]\label{lem:pv-stokes} Let $\tau>0$ and let $F\in L^{5/2}(Q_\tau;\mathbb R^{3\times3})\cap L^2(Q_\tau;\mathbb R^{3\times3})$. There is a unique energy-class pair $(z,q)$ with zero initial datum satisfying, in distributions, \begin{equation*} \partial_tz-\Delta z=\operatorname{div}F-\nabla q\,,\qquad \operatorname{div}z=0\,,\qquad z(0)=0\,, \end{equation*} where $(\operatorname{div}F)_j=\sum_i\partial_iF_{ij}$ and $q=-P[F]$. It has \begin{equation*} z\in C([0,\tau];L^2)\cap L^2(0,\tau;H^1)\cap C([0,\tau];L^3)\cap L^5(Q_\tau)\cap L^4(Q_\tau),\qquad q\in L^2(Q_\tau)\cap L^{5/2}(Q_\tau), \end{equation*} and $\partial_tz\in L^2(0,\tau;V')$, where $V=\{\phi\in H^1(\mathbb R^3;\mathbb R^3):\operatorname{div}\phi=0\}$ and $V'$ is its dual on the divergence-free $L^2$ space. Moreover \begin{align} \|z\|_{L^\infty(0,\tau;L^3)}+\|z\|_{L^5(Q_\tau)}&\le C\|F\|_{L^{5/2}(Q_\tau)}\,,\label{eq:pv-stokes-l5}\\ \|z\|_{L^4(Q_\tau)}&\le C\bigl(\|F\|_{L^{5/2}(Q_\tau)}+\|F\|_{L^2(Q_\tau)}\bigr)\,,\label{eq:pv-stokes-l4}\\ \|z\|_{L^\infty(0,\tau;L^2)}+\|\nabla z\|_{L^2(Q_\tau)}&\le C\|F\|_{L^2(Q_\tau)}\,.\label{eq:pv-stokes-energy} \end{align} The constants depend only on dimension and the pressure-operator bounds at exponents $2$ and $5/2$, and are independent of $\tau$. \end{lemma} \begin{proof} For smooth tensor data, the Fourier formula in \cite[(168)]{CKNLean} gives the divergence-free evolution $z(t)=\int_0^t K(t-s)F(s)\,ds$. The pressure is $q=-P[F]$: the identity $-\Delta P[F]=\operatorname{div}\operatorname{div}F$ gives $\Delta q=\operatorname{div}\operatorname{div}F$, which is the divergence of the displayed Stokes equation. The $L^2$ energy estimate follows by pairing the projected equation with $z$ in the divergence-free energy space, where the Leray projection is orthogonal and the pressure is absent. Fourier truncation justifies this pairing for smooth data. Cauchy--Schwarz and Young's inequality give \eqref{eq:pv-stokes-energy}. The equation then gives the asserted $L^2(0,\tau;V')$ bound for $\partial_tz$. For the critical estimate, test instead with $|z|z$. First take $F$ smooth and compactly supported in space and time. The Riesz kernel gives $q(x,t)=O(|x|^{-3})$ and $\nabla q(x,t)=O(|x|^{-4})$ outside a fixed ball; the heat representation then gives $z(x,t)=O(|x|^{-4})$ there on a finite time interval. These bounds make all cutoff errors tend to zero as the spatial cutoff radius tends to infinity. A smooth regularization at $z=0$ justifies the chain rule. The diffusion term controls $|z||\nabla z|^2$ and $|\nabla(|z|^{3/2})|^2$. Integration by parts in the force and pressure terms gives \begin{equation*} \frac13\frac{d}{dt}\|z(t)\|_3^3+c\int_{\mathbb R^3}|z|\,|\nabla z|^2+c\int_{\mathbb R^3}|\nabla(|z|^{3/2})|^2 \le C\int_{\mathbb R^3}(|F|^2+|q|^2)|z|\,dx\,, \end{equation*} with numerical constants $c,C>0$. Indeed, $|\nabla(|z|z)|\le2|z||\nabla z|$ and $|\operatorname{div}(|z|z)|\le2|z||\nabla z|$; Cauchy--Schwarz absorbs half of the weighted diffusion term and leaves the right side displayed. Cutoff errors vanish for smooth approximants by their spatial decay. Definition~\ref{def:riesz-pressure} gives $\|q\|_{L^{5/2}(Q_\tau)}\le C\|F\|_{L^{5/2}(Q_\tau)}$. Let $Z=\|z\|_{L^5(Q_\tau)}$, $A=\operatorname*{ess\,sup}_{00$ and a Leray--Hopf solution $(U,DU)$ on $\mathbb R^3\times[0,\tau]$ with datum $a$ such that $U\in L^5(Q_\tau)\cap L^4(Q_\tau)$. Moreover $U(t)\to a$ strongly in $L^3$ as $t\downarrow0$, and the associated pressure $P[U\otimes U]$ belongs to $L^2(Q_\tau)\cap L^{5/2}(Q_\tau)$. \end{proposition} \begin{proof} Let $h(t)=S(t)a$. Lemma~\ref{lem:pv-small-heat} permits a choice of $\tau>0$ with $H:=\|h\|_{X_\tau}<1/(8K)$. If $H=0$, take $U=0$; otherwise the map $\Phi(U)=h+\mathcal B(U,U)$ maps the closed ball of radius $2H$ in $X_\tau$ into itself and contracts it. In fact, \eqref{eq:pv-bilinear} gives $\|\Phi(U)-h\|_{X_\tau}\le4KH^2\le H/2$ on that ball and Lipschitz constant at most $4KH<1/2$. The contraction theorem gives a unique fixed point $U$ in that ball. The heat flow and every Stokes response are divergence free, so $\operatorname{div}U=0$ in distributions. Set $U_0=h$ and $U_{n+1}=h+\mathcal B(U_n,U_n)$. The contraction estimate gives convergence to $U$ in $X_\tau$. Since $U_n\to U$ in $L^4$, the tensors $U_n\otimes U_n$ converge to $U\otimes U$ in $L^2$; the $L^5$ convergence also gives convergence in $L^{5/2}$. Thus the forcing tensors converge in both data spaces of Lemma~\ref{lem:pv-stokes}. The associated Stokes responses converge in $C_tL^2_x\cap L^2_tH^1_x$, so $U\in C([0,\tau];L^2)\cap L^2(0,\tau;H^1)$ with $U(0)=a$ strongly in $L^2$. Their $L^\infty_tL^3_x$ difference estimate gives convergence in $C_tL^3_x$ as well. The heat flow is strongly continuous in $L^3$, and the nonlinear response tends to zero in $L^3$ at time zero; hence $U(t)\to a$ strongly in $L^3$. The pressure convergence from Lemma~\ref{lem:pv-stokes} gives $P[U\otimes U]\in L^2\cap L^{5/2}$, and passing to the limit in the mild equation gives \begin{equation*} \partial_tU-\Delta U+\operatorname{div}(U\otimes U)+\nabla P[U\otimes U]=0 \end{equation*} in distributions. This is the unregularized Navier--Stokes equation; the Fourier multiplier supplies only the linear Stokes response, and the tensor in the fixed point is the actual $U\otimes U$. It remains to verify the Leray--Hopf clauses, including the energy inequality. The response energy estimate and $U\otimes U\in L^2$ give $\partial_tU\in L^2(0,\tau;V')$. The Hilbert-space energy identity for $L^2(0,\tau;H^1)$ fields with time derivative in $L^2(0,\tau;V')$ follows by time convolution: on each compact subinterval convolve in time, pair the equation with the convolved field, and pass in the $L^2H^1$ and $L^2V'$ norms; continuity of $U$ in $L^2$ supplies the endpoint values. The nonlinear pairing is integrable since $U\otimes U\in L^2$ and $DU\in L^2$. Its spatial integral vanishes: for a cutoff $\chi_R$ equal to one on $B_R$, supported in $B_{2R}$, and with $|\nabla\chi_R|\le C/R$, integration by parts gives a convection error supported on the annulus $A_R=B_{2R}\setminus B_R$ and bounded by $C R^{-1}\int_{Q_\tau\cap A_R}|U|^3$. Hölder's inequality and $|Q_\tau\cap A_R|\le C\tau R^3$ bound this by $C\tau^{1/4}R^{-1/4}\|U\|_{L^4(Q_\tau)}^3$, which tends to zero. The diffusion cutoff error is bounded by $C R^{-1}\|U\|_{L^2(Q_\tau)}\|DU\|_{L^2(Q_\tau)}$ and also tends to zero by the energy bounds. There is no pressure boundary term in this divergence-free pairing. Thus $\int (U\otimes U):\nabla U=0$. We obtain the energy equality \begin{equation*} \frac12\|U(t)\|_2^2+\int_0^t\|DU(s)\|_2^2\,ds=\frac12\|a\|_2^2\qquad(0\le t\le\tau)\,, \end{equation*} first for almost every $t$ and then for every $t$ by $L^2$ continuity and absolute continuity of the dissipation integral. This implies (LH1), (LH2), and the stronger equality version of (LH4); the distributional equation gives (LH3), and the strong $L^2$ trace gives (LH5). The pressure term does not enter the energy pairing because we work in the divergence-free space $V$. This proves the proposition. \end{proof} \section{Weak--strong comparison}\label{sec:pv-serrin} The short-time solution is a Leray--Hopf solution, so it can be compared with the given weak solution using only the latter's energy inequality. The comparison identity is obtained by mollifying the time slices of both solutions in space and removing the mollification; the $L^5$ integrability of one solution makes every nonlinear term integrable down to time zero, and (LH4) is used for the other solution only from zero. \begin{lemma}[Weak--strong uniqueness for $L^5$ solutions]\label{lem:pv-serrin-uniqueness} Let $T>0$, and let $(u,Du)$ and $(v,Dv)$ be Leray--Hopf solutions on $\mathbb R^3\times[0,T]$ with the same datum in $\J$. If $u\in L^5(\mathbb R^3\times(0,T))$, then $u=v$ almost everywhere on $\mathbb R^3\times(0,T)$. \end{lemma} \begin{proof} By Theorem~\ref{thm:assoc-pressure}, each of $u$ and $v$ has an associated pressure in $L^{5/3}(Q_T)$ with which it satisfies the momentum equation against compactly supported smooth vector tests; the pairing of its slices with a fixed test field is continuous on $[0,T]$ and equals the pairing with $a$ at $t=0$. The spatial and temporal exponents of $u$ are both $5$, and $2/5+3/5=1$. The only nontrivial nonlinear integrability check is \begin{align*} \int_0^T\|v(t)\|_{10/3}\|Dv(t)\|_2\|u(t)\|_5\,dt &\le C\int_0^T\|v(t)\|_2^{2/5}\|Dv(t)\|_2^{8/5}\|u(t)\|_5\,dt\\ &\le \epsilon\|Dv\|_{L^2(Q_T)}^2+C_\epsilon\|v\|_{L^\infty_tL^2_x}^2\|u\|_{L^5_tL^5_x}^5<\infty\,, \end{align*} where the first inequality is spatial interpolation between $L^2$ and $L^6$, and the second is Young's inequality with exponents $5/4$ and $5$. It makes the convection terms $u\cdot(v\cdot\nabla)v$ and $v\cdot(u\cdot\nabla)u$ integrable on $Q_T$, down to time zero. The gradient pairing is integrable by the energy bounds, and the pressures pair integrably with both velocities, which lie in $L^{5/2}(Q_T)$ by interpolation. We derive the cross-testing identity without any regularity of $u$ beyond $L^5$. Let $u_n(\cdot,t)$ and $v_n(\cdot,t)$ be the spatial mollifications of the slices at radius $1/(n+1)$, and let $\chi_n$ be a smooth cutoff equal to one on $B_{n+1}$ with $|\nabla\chi_n|\le C/(n+1)$. At a fixed point $y$, testing the momentum equation of $u$ with the kernel $\rho_n(y-\cdot)e_k$ and a time cutoff shows that $t\mapsto u_n(y,t)_k$ is the primitive in time of an integrable tested flux, and likewise for $v_n$. The product rule for primitives then expresses $\int\chi_n\,v_n(t)\cdot u_n(t)\,dy-\int\chi_n\,v_n(0)\cdot u_n(0)\,dy$ as a space-time integral of mollified velocities, gradients, convection terms and pressures. The mollified slices are divergence free, so the pressure contributions only pair the mollified pressures with $\nabla\chi_n$. As $n\to\infty$, spatial mollification converges in the specific space-time $L^p$ classes used for these terms, and the product estimates above and the energy bounds give convergence of the cross integrals; the terms carrying $\nabla\chi_n$ vanish by the cutoff derivative bound and the same integrability estimates. The initial pairing converges to $\|a\|_2^2$. For almost every $t\in(0,T)$ this gives \begin{equation*} \int_{\mathbb R^3}v(t)\cdot u(t)\,dx-\|a\|_2^2 =-\int_0^t\!\!\int_{\mathbb R^3}\bigl(u\cdot(v\cdot\nabla)v+\nabla u:\nabla v\bigr) -\int_0^t\!\!\int_{\mathbb R^3}\bigl(v\cdot(u\cdot\nabla)u+\nabla v:\nabla u\bigr)\,. \end{equation*} No test by an arbitrary weak solution against itself is used: the identity pairs mollified slices, and the limits use only the integrability above. With $v=u$, the transport term $\int u\cdot(u\cdot\nabla)u\,dx$ vanishes for almost every time by the divergence-free transport identity: the Leray--Hopf slice bounds and $u\in L^5(Q_T)$ provide the integrability needed to justify it by cutoff and mollification. Thus the identity is the energy equality for $u$ at almost every time. We use only (LH4) for $v$ on $[0,t]$. Combining these gives, for almost every $t\in(0,T)$, \begin{equation*} \frac12\|w(t)\|_2^2+\int_0^t\|Dw(s)\|_2^2\,ds \le\int_0^t\int_{\mathbb R^3}u\cdot(w\cdot\nabla)w\,dx\,ds\,,\qquad w=v-u\,. \end{equation*} The transport identities $\int b\cdot\nabla(f\cdot g)\,dx=0$ for divergence-free $b$, applied at almost every time with the integrability above, reduce the combination to this form. This is the zero-start energy comparison of~\cite[(8.12)]{RobinsonRodrigoSadowski2016}, printed p.~174, where it is derived from Serrin regularity and their testing lemma (Theorem~8.17 and Lemma~8.18, printed pp.~173--174); the mollified identity above replaces both. The weak solution contributes only its energy inequality from zero. For almost every $s$, interpolation and H\"older's inequality give \begin{align*} \left|\int_{\mathbb R^3}u\cdot(w\cdot\nabla)w\,dx\right| &\le\|u(s)\|_5\|w(s)\|_{10/3}\|Dw(s)\|_2\\ &\le C\|u(s)\|_5\|w(s)\|_2^{2/5}\|Dw(s)\|_2^{8/5}\\ &\le\frac12\|Dw(s)\|_2^2+C\|u(s)\|_5^5\|w(s)\|_2^2\,. \end{align*} The coefficient $s\mapsto\|u(s)\|_5^5$ is integrable on $(0,T)$. Substitution and Gronwall's inequality yield \begin{equation*} \|w(t)\|_2^2\le \|w(0)\|_2^2\exp\!\left(C\int_0^t\|u(s)\|_5^5\,ds\right)=0\,. \end{equation*} The strong $L^2$ initial traces give $w(0)=0$. Gronwall therefore proves $w(t)=0$ in $L^2$ for almost every $t$, hence $u=v$ almost everywhere on the open cylinder. This proof does not use energy equality for $v$, and it does not use the positive-time regularity of $u$. \end{proof} \section{Positive times and the global conclusion}\label{sec:pv-assembly} We now prove the proposed successor of Theorem~\ref{thm:ess-global}. The only estimate still needed away from $t=0$ is a velocity bound uniform in space. Local regularity gives such a bound on bounded spatial sets; at spatial infinity, the tails of the velocity and its canonical pressure make the rescaled local energy arbitrarily small at a fixed radius. \begin{lemma}[A global bound on positive-time slabs]\label{lem:pv-positive-bound} Under the hypotheses of Theorem~\ref{thm:ess-global}, put $M^3=\operatorname*{ess\,sup}_{t\in(0,T)}\int_{\mathbb R^3}|u(x,t)|^3\,dx$. For every $0<\delta0$ with $t_0-\rho^2>0$. Define the translated and rescaled fields on $B_1\times(-1,0)$ by \begin{align*} u_\rho(y,s)&=\rho\,u(x_0+\rho y,t_0+\rho^2s),\\ (Du)_\rho(y,s)&=\rho^2Du(x_0+\rho y,t_0+\rho^2s),\\ (p_*)_\rho(y,s)&=\rho^2p_*(x_0+\rho y,t_0+\rho^2s). \end{align*} By (LH1), these fields have the local energy bounds, weak gradient, and divergence condition required by Theorem~\ref{thm:ess-local}. Lemma~\ref{lem:assoc-pressure-L3} supplies the pressure identity (S3) with this same canonical pressure, and the global $L^\infty_tL^3_x$ hypothesis supplies the remaining norm bound. These properties are preserved by the displayed change of variables, so the hypotheses of Theorem~\ref{thm:ess-local} hold for the rescaled fields. Applying Lemma~\ref{lem:lei-L4} with inner radius $3/4$ proves that $(u_\rho,(Du)_\rho,(p_*)_\rho,0)$ is suitable on $B_{3/4}\times(-1,0)$. Scaling back, $(u,Du,p_*,0)$ is suitable on $B_{3\rho/4}(x_0)\times(t_0-\rho^2,t_0)$. The function $|u|^3+|p_*|^{3/2}$ is integrable on every finite slab, and its integral outside $B_R$ tends to zero as $R\to\infty$. Fix $r>0$ with $r^2<\delta/4$. Since $u\in L^\infty((0,T);L^3)$ and $p_*\in L^\infty((0,T);L^{3/2})$, the function $|u|^3+|p_*|^{3/2}$ has an integrable spatial tail uniformly after integration over $(\delta/2,T)$. Choose $R$ so large that for every $x_0$ with $|x_0|>R$, \begin{equation*} r^{-2}\iint_{B_r(x_0)\times(t_0-r^2,t_0)}(|u|^3+|p_*|^{3/2})\,dx\,dt<\varepsilon_0/8 \end{equation*} whenever $\delta\le t_0\le T$. Indeed all these cylinders lie in $(\delta/2,T)$, and their spatial sections lie outside $B_{R-r}$; the integral is bounded by the tail integral over $(\delta/2,T)\times(\mathbb R^3\setminus B_{R-r})$, which tends to zero. For each such $t_0$, the preceding rescaled argument with outer scale $2r$ gives suitability on $B_{3r/2}(x_0)\times(t_0-4r^2,t_0)$, since $4r^2<\delta$. This contains the smallness cylinder displayed above. Lemma~\ref{lem:thmA-top} then bounds a representative by $2C_4/r$ on $B_{r/2}(x_0)\times(t_0-r^2/4,t_0)$, with constants independent of $x_0$ and $t_0$. These cylinders cover the part of $\{|x|>R+r\}\times(\delta,T)$ with $t\le T-r^2/8$ by taking $t_0=t+r^2/8$; the remaining terminal strip is covered by the same estimate with $t_0=T$. Hence the essential supremum on the unbounded spatial part is at most $2C_4/r$. On the bounded spatial part, Theorem~\ref{thm:ess-global} gives a regular point at every $(x,t)$ with $\delta\le t0$. \end{lemma} \begin{proof} Fix $\delta>0$ and take $B_{\delta/2}$ from Lemma~\ref{lem:pv-positive-bound}. Set $w=|u|^{3/2}$ on $Q_{\delta,T}$. The critical hypothesis gives $\operatorname*{ess\,sup}_{t\in(\delta,T)}\|w(t)\|_2^2\le M^3$. The weak chain rule, obtained by applying the Sobolev chain rule to smooth truncations of $z\mapsto|z|^{3/2}$ and passing by monotone convergence, gives \begin{equation*} \int_{Q_{\delta,T}}|\nabla w|^2\,dx\,dt\le C B_\delta\int_{Q_{\delta,T}}|Du|^2\,dx\,dt<\infty\,. \end{equation*} Indeed $|\nabla w|\le(3/2)|u|^{1/2}|Du|$ almost everywhere and $|u|\le B_\delta$ there. The multiplicative inequality proved in \cite[Lemma~8.4]{CKNLean}, applied to $w$ on this slab, now yields \begin{equation*} \int_{Q_{\delta,T}}|u|^5\,dx\,dt=\int_{Q_{\delta,T}}|w|^{10/3}\,dx\,dt \le C\left(\operatorname*{ess\,sup}_{t\in(\delta,T)}\|w(t)\|_2\right)^{4/3}\int_{Q_{\delta,T}}|\nabla w|^2\,dx\,dt<\infty\,. \end{equation*} All constants are finite because the Leray--Hopf energy inequality controls the last integral. This proves the assertion. \end{proof} \begin{theorem}[The $L^5$ bound and uniqueness]\label{thm:ess-l5-unique} Let $00$ put $\kappa=\rho_\varepsilon-\rho_{\varepsilon'}$; convolutions act in $x$ at almost every time. The field $d=\kappa*g=(\partial_j\kappa)*w$ lies in $L^2(Q)$, because $\|\kappa*g(t)\|_2\le\|\kappa\|_1\|g(t)\|_2$, and $\dt d=(\partial_j\kappa)*r=\partial_j(\kappa*r)\in L^2(Q)$. Part~(i) applies to $d$, with representative $\bar d$: for $s\le t$, \[ \|\bar d(t)\|_2^2-\|\bar d(s)\|_2^2 =2\int_s^t\!\!\int\partial_j(\kappa*r)\,(\kappa*g)\,dx\,d\tau =-2\int_s^t\!\!\int(\kappa*r)\,(\kappa*h)\,dx\,d\tau , \] where $\kappa*g\in H^1$ has $\partial_j(\kappa*g)=\kappa*h$ at almost every time, and the derivative was moved onto this factor. Hence $|\|\bar d(t)\|_2^2-\|\bar d(s)\|_2^2|\le2\|\kappa*r\|_{L^2(Q)}\|\kappa*h\|_{L^2(Q)}$. Averaging over $s\in(a,b)$ gives for every $t\in[a,b]$ \[ \|\bar d(t)\|_2^2\le(b-a)^{-1}\|d\|_{L^2(Q)}^2 +2\|\kappa*r\|_{L^2(Q)}\|\kappa*h\|_{L^2(Q)} . \] As $\varepsilon,\varepsilon'\to0$ the fields $\rho_\varepsilon*g$, $\rho_\varepsilon*r$ and $\rho_\varepsilon*h$ converge in $L^2(Q)$ to $g$, $r$ and $h$, so the right side tends to zero. The curves $\bar g_\varepsilon$ representing $\rho_\varepsilon*g$ are therefore uniformly Cauchy in $C([a,b];L^2(\R^3))$, and their limit $\bar g$ is a representative of $g$. The same computation with $\kappa$ replaced by $\rho_\varepsilon$ gives the identity for $\bar g_\varepsilon$, and it passes to the limit $\varepsilon\to0$. \end{proof} \begin{lemma}[Energy equality at almost every time]\label{lem:lps-energy-equality} Under the hypotheses of Theorem~\ref{thm:lps}, there is a full-measure set $E\subset(0,T)$ of times at which the slice $u(\cdot,t)$ is a measurable $L^2$ field such that for every $t\in E$, \begin{equation}\label{eq:lps-energy-equality} \frac12\|u(t)\|_2^2+ \int_0^t\|Du(\tau)\|_2^2\,d\tau =\frac12\|a\|_2^2\,. \end{equation} Consequently, for $t_0,t\in E$ with $t_00$, let $U_\varepsilon$ and $p_\varepsilon$ be the solution and pressure of \cite[Theorem~9.4]{CKNLean} with datum $b$, and put $V_\varepsilon=\Jeps U_\varepsilon$. For every finite $S>0$: \begin{enumerate}[label=(\alph*)] \item for every integer $k\ge0$, $U_\varepsilon\in C([0,S];H^k)$ and $U_\varepsilon(0)=\Jeps b$ in $H^k$; \item for $t>0$ the time derivative of $U_\varepsilon$ exists classically and $\dt U_\varepsilon=R_\varepsilon-\nabla p_\varepsilon$, where $R_\varepsilon=\Delta U_\varepsilon-(V_\varepsilon\cdot\nabla)U_\varepsilon$ is a continuous map from $(0,S]$ into $L^2(\R^3;\R^3)$; \item the function $y_\varepsilon(t)=\|\nabla U_\varepsilon(t)\|_2^2$ is continuous on $[0,S]$ and differentiable on $(0,S)$, and for $00$, depending only on $\|b\|_{H^1}$, a solenoidal solution $U$ of the unregularized equation on $[0,\tau_b]$, and a pressure $p\in L^2(\R^3\times(0,\tau_b))$ for which the momentum equation with pressure holds in the weak sense against compactly supported smooth vector tests, such that \begin{equation}\label{eq:lps-strong-class} U\in C([0,\tau_b];H^1)\cap L^2(0,\tau_b;H^2), \qquad \dt U\in L^2(0,\tau_b;L^2),\qquad U(0)=b\,. \end{equation} It has energy equality from every time and defines a Leray--Hopf solution on this interval. A choice of $\tau_b$ bounded below by $c(1+\|b\|_{H^1})^{-4}$ is possible, with absolute $c>0$. \end{proposition} \begin{proof} Apply \cite[Theorem~9.4]{CKNLean} with datum $b$ and mollifier radius $\varepsilon\downarrow0$. Lemma~\ref{lem:lps-regularized-Hk-start} adds the $H^1$ continuity at zero and the differentiated gradient-energy identity \eqref{eq:lps-regularized-Hm-identity} needed here; these are not clauses (R1)--(R5) of that theorem. Write $y=\|\nabla U_\varepsilon\|_2^2$ and $d=\|\nabla^2U_\varepsilon\|_2^2=\|\Delta U_\varepsilon\|_2^2$. The transport velocity $\Jeps U_\varepsilon$ is divergence free, and Young's inequality for convolution and the homogeneous $H^1\to L^6$ inequality give $\|\Jeps U_\varepsilon\|_6\le C\|\nabla U_\varepsilon\|_2$. By H\"older's inequality and the interpolation of $\nabla U_\varepsilon$ between $L^2$ and $L^6$, whose $L^6$ norm is controlled by $\|\nabla^2U_\varepsilon\|_2$, \[ |\langle(\Jeps U_\varepsilon\cdot\nabla)U_\varepsilon,\Delta U_\varepsilon\rangle| \le\|\Jeps U_\varepsilon\|_6\|\nabla U_\varepsilon\|_3\|\Delta U_\varepsilon\|_2 \le C\,y^{3/4}d^{3/4} \] with an absolute constant, independent of $\varepsilon$ and of the mollifier profile. Inserting this in \eqref{eq:lps-regularized-Hm-identity} and using Young's inequality with exponents $4/3$ and $4$ gives, for $t>0$, \begin{equation}\label{eq:lps-uniform-H1} \frac{d}{dt}\|\nabla U_\varepsilon\|_2^2 +\|\Delta U_\varepsilon\|_2^2 \le C\|\nabla U_\varepsilon\|_2^6\,. \end{equation} The initial field is $\Jeps b$, so its $L^2$ and $H^1$ norms do not exceed those of $b$. Set $X_0=1+\|b\|_{H^1}^2$. Since $y$ is continuous on $[0,S]$ and satisfies $y'\le Cy^3$ on $(0,S)$ for every $S$, comparison of $X'=CX^3$ with $X(0)=X_0$ (applied on $[s,\tau]$ and letting $s\downarrow0$) shows that for $\tau_b=cX_0^{-2}$, with $c$ chosen so that $2cC<1/2$, $\sup_{0\le t\le\tau_b}\|\nabla U_\varepsilon(t)\|_2^2 \le2X_0$. Integrating \eqref{eq:lps-uniform-H1} over $[s,\tau_b]$, using this bound and letting $s\downarrow0$ controls $\int_0^{\tau_b}\|\Delta U_\varepsilon\|_2^2$ independently of $\varepsilon$. The global $L^2$ energy identity adds $\sup_t\|U_\varepsilon(t)\|_2^2\le\|b\|_2^2$. The nonlinear term belongs uniformly to $L^2_tL^2_x$: $\|(\Jeps U_\varepsilon\cdot\nabla)U_\varepsilon\|_2 \le\|\Jeps U_\varepsilon\|_\infty\|\nabla U_\varepsilon\|_2 \le C\|U_\varepsilon\|_{H^2}\|U_\varepsilon\|_{H^1}$. The projected equation therefore bounds $\dt U_\varepsilon$ in $L^2_tL^2_x$; the Leray projection has operator norm one on $L^2$. \cite[Proposition~12.1]{CKNLean}, applied to this family with datum $b$, supplies a subsequence converging strongly in $L^2$ on bounded space-time cylinders and weakly in the energy gradients. Its compactness input is \cite[Lemma~10.6]{CKNLean}, which gives the required time-pairing modulus; the present $L^2_tL^2_x$ time-derivative bound is an additional estimate, not a replacement for that input. The uniform $H^2$ bound improves local convergence to strong $L^2_tH^1_x$: for a cutoff $\chi$ and $W=\chi(U_\varepsilon-U)$, integration by parts gives $\|\nabla W\|_2^2\le\|W\|_2\|\Delta W\|_2$; the first factor tends to zero and the second is bounded. Weak compactness retains $U\in L^\infty_tH^1_x\cap L^2_tH^2_x$ and $\dt U\in L^2_tL^2_x$. The convergence $\Jeps U_\varepsilon-U\to0$ locally in $L^2_tH^1_x$ follows by writing it as $\Jeps(U_\varepsilon-U)+(\Jeps U-U)$ and using the uniform derivative bound and approximate-identity convergence. The quadratic transport term therefore converges in distributions to $(U\cdot\nabla)U$; pressure is recovered by the fixed double-Riesz formula of \cite[Section~8]{CKNLean}, and lies in $L^2(\R^3\times(0,\tau_b))$ because $U\otimes U\in L^\infty_tL^2_x$ and the Riesz projection is bounded on $L^2$. The initial value follows from the uniform time-pairing modulus and $\Jeps b\to b$ in $L^2$. For the time continuity, apply Lemma~\ref{lem:lps-Hilbert-triple} componentwise. Here $U\in L^2_tH^2_x$, so the weak derivatives $\partial_jU$ and $\partial_j\partial_jU$ lie in $L^2$, and $\dt U\in L^2_tL^2_x$. Part~(i) of the lemma gives the $L^2$ continuity of $U$ together with $\|U(t)\|_2^2-\|U(s)\|_2^2=2\int_s^t\!\int U\cdot\dt U$. Part~(ii), applied to each derivative $\partial_jU_i$ and summed over $i,j$, gives the $L^2$ continuity of $\nabla U$ together with $\|\nabla U(t)\|_2^2-\|\nabla U(s)\|_2^2 =-2\int_s^t\!\int\Delta U\cdot\dt U$. Thus $U\in C_tH^1_x$. Choose this continuous representative for the strong solution; it agrees almost everywhere with the limit field of \cite[Proposition~12.1]{CKNLean}. The resulting energy identity gives the $L^2$ trace and, after pairing the projected equation with $-\Delta U$, the $H^1$ trace. The nonlinearity is in $L^2_tL^2_x$, so the equation and spatial cutoffs justify the energy equality for every subinterval. These properties, weak continuity, and the weak equation verify (LH1)--(LH5) with datum $b$; in particular no positive-time inequality is inserted as an extra hypothesis. \end{proof} \begin{lemma}[Serrin control of the $H^1$ norm]\label{lem:lps-H1-estimate} Let $U$ be a strong solution on $[t_0,t_1]$, that is, a solenoidal field in the class \eqref{eq:lps-strong-class} with an $L^2$ pressure as in Proposition~\ref{prop:lps-local-strong}, with $t_0t_0$ for which there is a strong solution $U$ on $[t_0,t']$ with $U(t_0)=u(t_0)$ in $L^2$ and $U=u$ almost everywhere on $\R^3\times(t_0,\min\{t',T\})$. Then $S$ contains a time larger than $T$. \end{lemma} \begin{proof} \emph{Step 1: comparison after a restart by concatenation.} Let $t'>t_0$ and let $W$ be a strong solution on $[t_0,t']$ with $W(t_0)=u(t_0)$ in $L^2$. We claim that $W=u$ almost everywhere on $\R^3\times(t_0,\sigma)$, $\sigma=\min\{t',T\}$. Let $V$ be the field equal to $u$ at times $t0$ bounded below in terms of $M$ alone. Restart from $U(t')\in H^1\cap\J$ and glue the two strong fields. Continuity in $H^1$ at the junction eliminates a distributional time jump; the two $L^2$ time derivatives remain integrable on the union, and the pressures are glued in the same way. The glued field is a strong solution on $[t_0,t'+\tau_*]$ with the same value $u(t_0)$ at $t_0$, so Step~1 shows that it agrees with $u$ almost everywhere on $\R^3\times(t_0,\min\{t'+\tau_*,T\})$. Thus $t'+\tau_*\in S$. If $\sup S\le T$, choose $t'\in S$ with $t'>\sup S-\tau_*/2$; the preceding extension exceeds $\sup S$, a contradiction. Hence some member of $S$ exceeds $T$. This uses the continuation remark after \cite[Corollary~6.9, printed p.~137]{RobinsonRodrigoSadowski2016}; on $\R^3$ the local construction is the one underlying \cite[Theorem~6.15, printed pp.~144--145]{RobinsonRodrigoSadowski2016}. \end{proof} \section{Smooth representatives}\label{sec:lps-smooth} The strong solution identifies the weak field on every interval after a good time. On a positive-time slab it then gains spatial regularity by a direct linear ladder: a bound for $\dt U$ in $L^2_x$ gives the $H^2$ regularity of every slice, and the whole-space heat gain, applied to the projected convection term as a source, gives one further order at a time. The projected equation supplies all time derivatives. The endpoint $t_1$ is included by working on closed time intervals throughout. On $\R^3$ there is no boundary compatibility condition. \begin{lemma}[Joint smoothness from Sobolev curves]\label{lem:lps-Bochner-joint-smooth} Let $t_0k+3/2$, turns each curve $Z^j$ into a continuous curve in $C_b^k$ for every $k$. Since $Z^{j+1}$ is continuous with values in $H^m$, the differentiability of the pairings with all test functions gives $Z^j(t)-Z^j(s)=\int_s^tZ^{j+1}(\tau)\,d\tau$ as a Bochner integral in $H^m$, hence in $C_b^k$, so $Z^j$ is a $C^1$ curve into $C_b^k$ with derivative $Z^{j+1}$ (one-sided at the endpoints). Evaluation of a $C_b^k$ function at $x$ is jointly $C^k$ in the function and the $x$ variable. Composing the evaluation with the curves shows that $(x,t)\mapsto Z^0(t)(x)$ has continuous mixed partial derivatives $\partial_t^j\partial_x^\alpha=\partial_x^\alpha Z^j(t)(x)$ of every order $j+|\alpha|$, jointly in $(x,t)$ on $\R^3\times[t_0+\delta,t_1]$; it is therefore $C^\infty$ there, with one-sided derivatives at the ends of the interval. It equals $V$ almost everywhere. For $\delta'<\delta$ the two smooth functions agree almost everywhere on the common set, hence everywhere by continuity, so they patch as $\delta\downarrow0$ to a representative on $\R^3\times(t_0,t_1]$. For formalization, the embedding into $C_b^k$ can be constructed by Fourier inversion and differentiation under the integral, as in the $H^4\hookrightarrow C_b^2$ argument of the CKN paper~\cite{CKNLean}, iterated to higher orders. \end{proof} \begin{proposition}[Smoothing of strong solutions]\label{prop:lps-smoothing} If $U$ is a strong solution on $[t_0,t_1]$ with $t_00$ with $t_0+\delta0$ put $v_h(t)=h^{-1}(U(t+h)-U(t))$ on $[t_0,t_1-h]$. Since $U(t+h)\otimes U(t+h)-U(t)\otimes U(t)$ equals $U(t+h)\otimes(U(t+h)-U(t))+(U(t+h)-U(t))\otimes U(t)$, the difference of the equations at times $t+h$ and $t$, divided by $h$, states that $v_h$ is solenoidal and solves $\dt v_h-\Delta v_h+\nabla\cdot F_h+\nabla q_h=0$, with flux $|F_h|\le(|U(t+h)|+|U(t)|)|v_h|$. We have $v_h\in L^2H^2$ and $\dt v_h\in L^2L^2$, so part~(i) of Lemma~\ref{lem:lps-Hilbert-triple} gives the energy identity for $E_h=\|v_h\|_2^2$; the pressure term vanishes against the solenoidal $v_h$, and \[ E_h'\le-2\|\nabla v_h\|_2^2 +(\|U(t+h)\|_\infty+\|U(t)\|_\infty)\,2\|v_h\|_2\|\nabla v_h\|_2 \le-\|\nabla v_h\|_2^2+\kappa_hE_h, \] where $\kappa_h(t)=2K^2\bigl(M(t)+M(t+h)\bigr)$ has integral at most $4K^2\|M\|_{L^1}$, uniformly in $h$. Moreover $\|v_h(s)\|_2^2\le h^{-1}\int_0^h\|\dt U(s+\theta)\|_2^2\,d\theta$, so $\int_{t_0}^{t_0+\delta}E_h\le\|\dt U\|_{L^2(Q)}^2$. Averaging the Gronwall inequality $E_h(t)\le E_h(s)\exp\int\kappa_h$ over $s\in(t_0,t_0+\delta)$ gives, for $t\in[t_0+\delta,t_1-h]$, \[ E_h(t)\le B_\delta:=\delta^{-1}e^{4K^2\|M\|_{L^1}}\|\dt U\|_{L^2(Q)}^2 , \] independently of $h$. Since $v_h\to\dt U$ in $L^2(Q)$, weak lower semicontinuity gives $\|\dt U(t)\|_2^2\le B_\delta$ for almost every $t\in(t_0+\delta,t_1)$. \emph{Step 2: $H^2$ bound of the slices and the base of the ladder.} At almost every time $U(t)\in H^2$ solves the stationary system $\Delta U=\dt U+(U\cdot\nabla)U+\nabla p$. Test it with $\Delta U$: the pressure drops because $\Delta U$ is solenoidal, and $|\langle(U\cdot\nabla)U,\Delta U\rangle|\le\|U\|_6\|\nabla U\|_3\|\Delta U\|_2 \le C\|\nabla U\|_2^{3/2}\|\nabla^2U\|_2^{3/2}$ by Sobolev interpolation, with $\|\nabla^2U\|_2=\|\Delta U\|_2$. Young's inequality gives $\|\nabla^2U(t)\|_2\le2\|\dt U(t)\|_2+C\|\nabla U(t)\|_2^3$. By Step~1 and $U\in C([t_0,t_1];H^1)$, $\sup\|U(t)\|_{H^2}\le C_\delta$ for almost every $t\in(t_0+\delta,t_1)$. Consequently $B=(U\cdot\nabla)U$ is bounded in $H^1$ there: $|\nabla B|\le|\nabla U|^2+|U||\nabla^2U|$, with $\|\nabla U\|_4^2\le C\|U\|_{H^2}^2$ and $\|U\|_\infty\|\nabla^2U\|_2\le C\|U\|_{H^2}^2$. We use the whole-space heat gain. \emph{Let $m\ge1$, $a0$ with $t_0+\delta3/2$), $B=\sum_jU_j\partial_jU\in L^2H^m$ there, and $\Leray B\in L^2H^m$. Now $U\in L^2H^{m+1}$ solves \eqref{eq:lps-projected} with source $-\Leray B\in L^2H^m$. The heat gain of order $m+1$ with $(a,a+\beta)=(t_0+\delta/2,t_1)$ and $\sigma=\delta/2$ gives $U\in C([t_0+\delta,t_1];H^{m+1})\cap L^2(t_0+\delta,t_1;H^{m+2})$. This proves all the spatial assertions on the closed interval $[t_0+\delta,t_1]$. \emph{Step 4: the time chain.} Fix $\delta$ and put $I=[t_0+\delta,t_1]$. Define $Z^0=U$ and, recursively, \[ Z^{j+1}=\Delta Z^j-\Leray\sum_{i=0}^{j}\binom ji \,(Z^i\cdot\nabla)Z^{j-i}. \] If $Z^0,\dots,Z^j\in C(I;H^m)$ for every $m$, then $Z^{j+1}\in C(I;H^m)$ for every $m$, by the algebra property and the boundedness of $\Leray$ on $H^m$; so by Step~3 and induction all $Z^j$ belong to $C(I;H^m)$ for every $m$. By \eqref{eq:lps-projected}, $\dt Z^0=Z^1$ in the sense of distributions on $(t_0+\delta,t_1)$. If $\dt Z^i=Z^{i+1}$ for $i\le j$, then the product rule for products of curves that are continuous into $H^m$ and have continuous distributional time derivatives (by time mollification), together with Pascal's rule for the binomial coefficients, gives $\dt Z^{j+1}=Z^{j+2}$. For each $\psi\in C_c^\infty(\R^3)$ the pairing $\langle Z^j(t),\psi\rangle$ is thus a $C^1$ function on the open interval with the continuous derivative $\langle Z^{j+1}(t),\psi\rangle$, and therefore is differentiable on the closed interval $I$, one-sidedly at the ends, with this derivative. \emph{Step 5: joint smoothness.} Step 4 gives exactly the hypotheses of Lemma~\ref{lem:lps-Bochner-joint-smooth}, for every $\delta$, with $V=U$; the identification $Z^0(t)=U(t)$ for almost every $t$ was established in Step~3. The lemma produces a representative smooth jointly on $\R^3\times(t_0,t_1]$ with one-sided derivatives at $t_1$. No continuation beyond $t_1$ is used for the endpoint: all estimates hold on the closed intervals $[t_0+\delta,t_1]$. This is the whole-space version of \cite[Theorems~7.1, 7.3 and 7.5, printed pp.~149--154]{RobinsonRodrigoSadowski2016}; the argument above uses a direct heat-gain ladder and the Fourier multipliers of the CKN paper~\cite{CKNLean} in place of the periodic Galerkin basis used for the detailed source proof. \end{proof} \begin{remark}[An a posteriori energy inequality]\label{rem:lps-Hm-energy} Once Proposition~\ref{prop:lps-smoothing} is proved, $U\in C^1(I;H^m)$, $I=[t_0+\delta,t_1]$, for every $m$ and $\delta$ by Step~4, and for $m\ge2$ with $\|U\|_{H^m}^2=\sum_{|\alpha|\le m}\|\partial^\alpha U\|_2^2$ pairing the projected equation with $\partial^\alpha U$ and integrating one derivative by parts in the divergence form of the convection term gives \begin{equation}\label{eq:lps-Hm-energy} \frac{d}{dt}\|U\|_{H^m}^2+\|\nabla U\|_{H^m}^2\le C_m\|U\|_{H^m}^4 . \end{equation} Indeed the heat pairing gives $-\|\nabla U\|_{H^m}^2$, the projection is orthogonal and drops out against the solenoidal $\partial^\alpha U$, and the tensor-gradient pairing is bounded by $C_m\|U\|_{H^m}^2\|\nabla U\|_{H^m}$ by the algebra property, after which Young's inequality is used. This inequality is not used in the proof of the proposition, which does not pass through regularized solutions at higher order. \end{remark} \begin{proof}[Proof of Theorem~\ref{thm:lps}] Lemmas~\ref{lem:lps-mixed-interpolation} and \ref{lem:lps-energy-equality} give energy equality for $u$ at almost every measurable time. At $t_0=0$, the strong initial trace (LH5) gives the common $L^2$ trace $a$ through those times for $u$ and for any rival Leray--Hopf solution. Theorem~\ref{thm:assoc-pressure-forced} and the $L^4$ estimate give the $L^2$ canonical pressure for $u$; Theorem~\ref{thm:assoc-pressure} gives an $L^{5/3}$ pressure for the rival. Lemma~\ref{lem:lps-comparison} with $t_0=0$ identifies them almost everywhere. The rival's (LH4) supplies its energy inequality. Choose $t_0$ in the intersection of the full-measure good-time and energy-equality sets of Lemmas~\ref{lem:lps-good-times} and \ref{lem:lps-energy-equality}. Subtracting two instances of \eqref{eq:lps-energy-equality} gives the equality from $t_0$ for almost every later time. The trace assertion of the energy lemma identifies the strong $L^2$ trace there with $u(t_0)$; no positive-time energy clause is added to the stated Leray--Hopf definition. Proposition~\ref{prop:lps-local-strong} constructs a strong solution from this datum. It belongs to $L^2_tL^\infty_x$ by $H^2\hookrightarrow L^\infty$ and its $L^2_tH^2$ bound. Lemma~\ref{lem:lps-comparison}, with that strong field in the Serrin role and the original weak field in the energy role, identifies them on their common interval. Lemma~\ref{lem:lps-continuation} repeats this identification at each restart and extends the strong field beyond $T$. For any $\delta\in(0,T)$ choose such a $t_0<\delta/2$. Proposition~\ref{prop:lps-smoothing} gives a smooth representative on $\R^3\times[\delta,T]$. Representatives from two choices of good time agree almost everywhere on the overlap. Since both are continuous, they agree everywhere there: a nonempty open set on which they differ would have positive measure. The representatives thus patch as $\delta\downarrow0$. The strong continuation past $T$ supplies the one-sided space-time derivatives there. \end{proof} \begin{proof}[Proof of Corollary~\ref{cor:ess-smooth}] Theorem~\ref{thm:ess-l5-unique} gives $u\in L^5(Q_T)$ and its uniqueness conclusion. Fubini identifies this spacetime norm with $L^5(0,T;L^5(\R^3))$, and $3/5+2/5=1$. Apply Theorem~\ref{thm:lps} with $(s,\ell)=(5,5)$ to obtain the smooth representative through the top time. \end{proof} \appendix \section{Corrections to the sources}\label{app:errata} \subsection*{Escauriaza--Seregin--Šverák (2003)} \begin{enumerate} \item \textbf{Cylinder-sign typo.} Theorem~1.4, printed p.~214, writes the ambient cylinder as \(B_1\times(0,1)\), while its Sobolev hypothesis and target backward cylinder use negative times. The coherent cylinder is \(B_1\times(-1,0)\). The same inconsistency occurs in the IMA preprint. The manuscript states the negative-time domain in Theorem~\ref{thm:ess-local} and proves the shifted-cylinder argument in Lemma~\ref{lem:regular-point-shift}. \item \textbf{Test-space mismatch.} Definition~2.1, printed p.~215, names spatial tests \(\phi\in C^\infty_0(\mathbb R^3)\), but its local energy formula uses \(\phi(x,t)\) and \(\partial_t\phi\). The IMA preprint repeats the mismatch. The precise space-time test class is not fixed there. The manuscript fixes its suitable-solution test clauses in Definition~\ref{def:sws}. \item \textbf{Higher-order regularity delegated.} Lemma~2.2, printed p.~216, asserts bounds for every spatial derivative order but refers \(k>1\) to an external Stokes regularity result instead of proving the induction. Its proof ends on printed p.~220; §3 begins on p.~221. Constants are indexed by \(k\), but their dependence is not developed. Theorem~\ref{thm:vorticity-regularity} supplies the finite weak-to-regular bootstrap needed here, but does not reproduce the source's all-orders theorem or supply the all-orders exterior estimates in (3.31), (3.38), and (3.42)--(3.46). \item \textbf{Compressed Campanato compactness.} Propositions~2.4--2.6 and the proof of Lemma~2.2, printed pp.~216--220, appeal to “well-known compactness” and leave the relevant function-space operators and pressure-limit details compressed. No result of the manuscript relies on this argument; the manuscript uses the CKN regularity criterion~\cite{CKNLean} and proves the compactness it needs in \cite[Lemma~11.1]{CKNLean}. \item \textbf{Time representative and all-time \(L^3\) bound.} The assertion following (3.1), printed p.~221, extends weak continuity from the Leray--Hopf representative and the essential \(L^\infty_tL^3_x\) bound to every time, but does not spell out the density and lower-semicontinuity argument. The manuscript writes that representative and modulus in Lemma~\ref{lem:weak-cont-L3}. \item \textbf{Strong regularity estimates are used without a detailed proof.} The local estimate (3.10), printed p.~222, is cited through mixed-norm coercive estimates and supplies much more than the weak continuity and \(L^3\)-slice bound needed there. The manuscript replaces those conclusions by the momentum-pairing modulus in Lemma~\ref{lem:weak-cont-L3}; this does not claim the full derivative package in (3.10). \item \textbf{Singular-point lower bound is externally imported.} The lower bound (3.13), printed p.~223, is attributed to Seregin--Šverák [39], a result not used here. The contrapositive of CKN Theorem~A does not replace it directly: Theorem~A gives full-neighborhood regularity below the top face but only one-sided Hölder control at the top, so its contrapositive does not yield ESS's lower bound at a full-neighborhood singular point. The manuscript avoids the singular-point formulation. Lemma~\ref{lem:thmA-top} proves top-boundary \(\varepsilon\)-regularity with threshold \(\varepsilon_0/8\), and Definition~\ref{def:good-point} defines good points by that smallness. Lemma~\ref{lem:good-open-glue} shows that the solution is Hölder on a compact set of good points, and that a point which is not good satisfies the lower bound \eqref{eq:bad-point-lower-bound} at every admissible scale. The blow-up argument of Proposition~\ref{prop:blowup-limit} uses only this lower bound, so no input from [39] is needed. \item \textbf{Blow-up compactness conclusion is stronger than the replacement lemma.} The estimates (3.21)--(3.23), printed pp.~224--225, invoke nonstationary Stokes coercivity to obtain derivative bounds, strong \(L^3\) convergence, and \(C([a,b];L^2_{\rm loc})\) convergence. The cited nonstationary Stokes coercivity is not proved in detail. The manuscript compactness lemma \cite[Lemma~11.1]{CKNLean} gives strong local space-time \(L^2\), weak gradients, and weak slice convergence at every time; it does not claim the stronger \(C_tL^2_{\rm loc}\) topology. The blow-up proof here instead handles its endpoint and strong-\(L^3\) passages using pairing continuity, weak slice convergence, lower semicontinuity, and time-strip estimates. \item \textbf{Pressure-splitting normalization matters.} Equations (3.11)--(3.19) begin on printed p.~223 and end on p.~224. In these equations, \(p_1\) is defined by a local Dirichlet problem and \(p_2\) is that split's harmonic remainder. The decay in (3.19) belongs to this fixed remainder; it cannot be transferred to a different harmonic remainder chosen after rescaling. The manuscript's Lemma~\ref{lem:pressure-split} instead fixes a whole-space Riesz pressure of the zero-extended tensor and its harmonic difference, then proves the interior bound and rescaled vanishing directly. It does not transfer (3.19) to that changed split or redefine the remainder after rescaling. \item \textbf{Time reversal in the vorticity equation.} The §3 vorticity equation and (3.32), printed pp.~226--227, use \(\partial_t-\Delta\); §5 is written for \(\partial_t+\Delta\) in its forward variable. Reversing time reconciles them. This is a convention to state when applying Theorem~\ref{thm:bu}, not a sign correction to either displayed equation. \item \textbf{Exterior continuation proof is abbreviated.} Equations (3.38)--(3.46) are printed on pp.~227--228, and the closing text continues on p.~229. They use all-order velocity, time, and pressure bounds, a divergence correction via stationary Stokes, local strong solvability, and linear regularity; the closing harmonic \(L^3\) Liouville step is only described. The manuscript does not follow this route. Theorem~\ref{thm:vorticity-regularity} and Lemma~\ref{lem:vorticity-top-extension} give a bounded, continuous vorticity of class \(W^{2,1}_2\) with zero trace, which is the class of Theorem~\ref{thm:uc} and Theorem~\ref{thm:bu}. The infinite-order vanishing hypothesis of Theorem~\ref{thm:uc} follows from Lemma~\ref{lem:flat-from-vanishing}, and the Liouville step is Lemma~\ref{lem:liouville-L3}. No higher-order regularity is used. \item \textbf{Initial-time \(L^5\) estimate.} Lemma~7.1, printed pp.~243--244, explicitly cites an external theorem for its general mixed-norm estimate (7.4); Theorem~7.4, printed pp.~246--248, abbreviates the contraction and limit details. These are inputs to the \(L^5\) and uniqueness conclusions of the source's Theorem~1.3, not to its local Theorem~1.4. The manuscript proves those conclusions in Theorem~\ref{thm:ess-l5-unique} without the general estimate (7.4): Lemma~\ref{lem:pv-small-heat} proves the special heat estimate that is needed, and Proposition~\ref{prop:pv-local-solution} carries out the fixed-point construction and its limit (see Remark~\ref{rem:pv-departure}). \item \textbf{Pointwise boundary trace and dimension conventions.} Theorem~4.1, equations (4.1)--(4.3), printed p.~230, evaluates \(u(x,0)\) although the cylinder is open and does not specify a trace or representative. It states a \(W^{2,1}_2\) hypothesis and a differential inequality but concludes pointwise vanishing. The source uses a spatial \(\mathbb R^3\) cylinder, writes an \(\mathbb R^n\)-valued field, and uses a \(\rho^{n-1}\) factor in the proof; it also concludes on the whole ball although Lemma~4.3 estimates only an inner ball before saying the extension is immediate. The notation \(A_0(R,T)\) in §4 and the small absolute \(A_0\) in §5 are unrelated. Theorem~\ref{thm:uc} fixes the continuous trace convention and dimension three; its proof uses the Gaussian-box estimate in Lemma~\ref{lem:uc-gaussian} and gives the explicit radius iteration that enlarges the inner-ball estimate to the whole ball. \item \textbf{Gaussian propagation omits closure and radius steps.} Lemma~4.3, printed pp.~230--233, applies Proposition~6.1 to a cutoff field in \(W^{2,1}_2\), while that proposition is stated for smooth compactly supported fields; the density passage and the singular \(\varepsilon\downarrow0\) bound are not supplied. It also moves from an inner-ball estimate to the whole target ball without the radius iteration and cites the local \(L^2\)-to-pointwise estimate without proof. The manuscript supplies the smooth-to-\(W^{2,1}_2\) passage in Lemma~\ref{lem:carleman-sobolev}, proves the Gaussian-box estimate in Lemma~\ref{lem:uc-gaussian}, and carries out the explicit radius iteration in Theorem~\ref{thm:uc}. \item \textbf{Half-space trace and weak-gradient hypotheses.} Theorem~5.1, equations (5.2)--(5.4), printed p.~234, does not define the zero initial trace needed for extension across \(t=0\); without it, the extension can create a boundary distribution. Condition (5.4) also omits the weak spatial gradient assumed explicitly in Seregin's corresponding class. The source sets the spatial cylinder in \(\mathbb R^n_+\) and the field target in \(\mathbb R^n\), then states that the result also holds for every finite target dimension \(m\). The manuscript and the formalization fix \(n=3\). The source's repeated symbol \(A_0\) denotes unrelated quantities in §§4 and 5. Theorem~\ref{thm:bu} makes the continuous trace and weak-gradient clauses explicit. \item \textbf{Omitted quantitative choice and noncompact test field.} Lemma~5.2, printed pp.~234--236, existentially chooses \(A_0<1/32\); it does not display a numerical choice small enough for the later bounds, although the stated existential choice permits one. This is an omitted explicit quantitative choice, not a threshold contradiction. Lemma~5.3, printed pp.~236--238, applies Proposition~6.2 to a noncompact Sobolev cutoff without a cutoff/tail limit or density argument and compresses the propagation step from Theorem~4.1. Its general-dimensional notation writes \(\mathbb R^3_+\) where \(\mathbb R^n_+\) is required. The manuscript computes the absorption constants in Proposition~\ref{prop:carleman-halfspace}, extends the estimate by density in Lemma~\ref{lem:carleman-sobolev}, and carries out the needed cutoff and iteration in Theorem~\ref{thm:bu}. \item \textbf{Final large-growth iteration is compressed.} The proof of Lemma~5.4 and Theorem~5.1, printed p.~238, invokes the small-growth lemma with a parameter \(M>A_0\), outside that lemma's stated hypothesis; the trace and rescaling steps are also skipped. A valid repetition requires translating and rescaling each slab and justifying its trace. The manuscript gives the corresponding explicit iteration in Theorem~\ref{thm:bu}; the source's compressed step is not a counterexample to Theorem~5.1. \item \textbf{Smooth Carleman estimates applied beyond their stated class.} Propositions~6.1--6.2, printed pp.~238--243, are stated for smooth compactly supported fields, but the preceding applications require Sobolev cutoff fields. Proposition~6.1 asserts an absolute positive constant \(c_0\) but gives no numerical value; Proposition~6.2 compresses the final Cauchy--Schwarz absorption without recording the uniform constants. The manuscript computes explicit constants in Proposition~\ref{prop:carleman-gauss} and Proposition~\ref{prop:carleman-halfspace}, and supplies the density extension in Lemma~\ref{lem:carleman-sobolev}. \end{enumerate} \subsection*{Sources of the existence and pressure results} The corrections to Ożański--Pooley, Tsai, Robinson--Rodrigo--Sadowski and Lemarié--Rieusset, and the corrections to Theorem~1.1 and equations (3.2)--(3.4) of Escauriaza--Seregin--\v{S}ver\'ak, concern the existence and associated-pressure results of Section~\ref{sec:existence-in-ckn}. They are recorded in the appendix of the CKN paper~\cite{CKNLean}. \subsection*{Seregin lecture notes} \begin{enumerate} \item Appendix~A.1, Proposition~1.1 and Proposition~1.2, printed pp.~229--234, give the matching Carleman calculations but also compress the final \(L^2\) absorption/Cauchy--Schwarz step after (A.1.26); in particular the uniform constant is not derived there. The manuscript computes the constants in Proposition~\ref{prop:carleman-gauss} and Proposition~\ref{prop:carleman-halfspace}. \item Appendix~A.2, Theorem~2.1 and Lemma~2.2, repeats the pointwise Gaussian route and compressed inner-to-whole-ball continuation; Appendix~A.3 repeats the half-space proof's final iteration without fully stating its trace/rescaling details. The manuscript instead supplies an integral-box estimate and explicit radius/large-growth iterations in Lemma~\ref{lem:uc-gaussian}, Theorem~\ref{thm:uc}, and Theorem~\ref{thm:bu}. \item Appendix~A.3, condition (A.3.4), explicitly includes weak \(\nabla u\in L^2_{\rm loc}\), which ESS's (5.4) omits. The appendix's initial trace convention is still unstated. Its \(Q_1^+\) notation uses \(\mathbb R^3_++e_n\) in an \(n\)-dimensional argument; it should be \(\mathbb R^n_+\). The manuscript fixes spatial and target dimension three and makes the weak-gradient and continuous trace clauses explicit in Theorem~\ref{thm:bu}. The notes also describe the first Carleman cutoff with support outside a larger parabolic cylinder; after the separate cutoff at the initial time, the estimates agree with the published proof. Appendix~A.3 has the ordinary editorial typos “result result” and “fist integral,” which do not change the argument. \end{enumerate} \bibliographystyle{amsplain} \bibliography{refs} \end{document}