{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Nonlinear Schrödinger as a Dynamical System\n", "\n", "#### [Ascona Winter School 2016](http://www.math.uzh.ch/pde16/index-Ascona2016.html), [(alternate link)](http://www.monteverita.org/en/90/default.aspx?idEvent=295&archive=)\n", "\n", "\n", "#### [J. Colliander](http://colliand.com) ([UBC](http://www.math.ubc.ca))" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Lectures\n", "\n", "1. [Introduction](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/colliander-lecture1.ipynb)\n", "2. **[Conservation](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/colliander-lecture2.ipynb)**\n", "3. [Monotonicity](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/colliander-lecture3.ipynb)\n", "4. [Research Frontier](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/colliander-lecture4.ipynb)\n", "\n", "### https://github.com/colliand/ascona2016\n", "\n", "***" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Overview of Lecture 2\n", "\n", "* Conserved Quantities\n", "* Bourgain's High/Low Frequency Decomposition\n", "* $I$-Method\n", "* Multilinear Correction Terms\n", "* Applications" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Lecture Notes\n", "\n", "### [https://github.com/colliand/ascona2016](https://github.com/colliand/ascona2016)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Conserved Quantities\n", "\n", "$$\n", "\\begin{align*}\n", "{\\mbox{Mass}}& = \\| u \\|_{L^2_x}^2 = \\int_{\\mathbb{R}^d} |u(t,x)|^2 dx. \\\\\n", "{\\mbox{Momentum}}& = {\\textbf{p}}(u) = 2 \\Im \\int_{\\mathbb{R}^2} {\\overline{u}(t)} \\nabla u (t)\n", "dx. \\\\\n", "{\\mbox{Energy}} & = H[u(t)] = \\frac{1}{2} \\int_{\\mathbb{R}^2} |\\nabla u(t) |^2 dx {\\pm} \\frac{2}{p+1} |u(t)|^{p+1} dx .\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Conserved ##\n", "$$ \\partial_t Q[u] = 0.$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ " \n", "## Almost Conserved##\n", "$$\\big| \\partial_t Q[u] \\big| ~\\mbox{is small}.$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "$$ \\sup_{t \\in T_{lwp}} Q[u(t)] - \\inf_{t \\in T_{lwp}} Q[u(t)] < \\epsilon$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "$$ \\int_0^{T_{lwp}} (\\partial_t Q)[u(\\tau)] d\\tau < \\epsilon $$ " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Conservation of Mass" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "$$ \\partial_t |u(t)|^2 = \\partial_t ( u \\overline{u} ) = u_t \\overline{u} + u \\overline{u}_t$$\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "From the equation $ (i \\partial_t + \\Delta) u = \\pm |u|^{p-1} u $, we have:\n", "$$ u_t = i \\Delta u \\mp i |u|^{p-1} u$$\n", "$$ {\\overline{u}}_t = -i \\Delta {\\overline{u}} \\pm i |u|^{p-1} {\\overline{u}}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Thus,\n", "$$ \\partial_t |u(t)|^2 = [ i \\Delta u \\mp i |u|^{p-1} u] \\overline{u} + u [ -i \\Delta {\\overline{u}} \\pm i |u|^{p-1} {\\overline{u}}]$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "$$ \\partial_t |u(t)|^2 = i[ \\overline{u} \\Delta u - u\\Delta {\\overline{u}} ] \\pm i [ |u|^{p+1} - |u|^{p+1} ] $$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "$$ \\partial_t |u(t)|^2 = \\nabla \\cdot \\Im [ \\overline{u} \\nabla u ]$$ " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Bourgain's High/Low Method" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "![bourgain-front-page](https://wwejubwfy.s3.amazonaws.com/1998_Bourgain_IMRN_FrontPage.pdf-2016-01-11-06-15-59.jpg)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", "\n", "Consider the Cauchy problem for defocusing cubic NLS on $\\mathbb{R}^2$: \n", "\\begin{equation*}\n", "\\tag{{$NLS^{+}_3 (\\mathbb{R}^2)$}}\n", " \\left\\{\n", " \\begin{matrix}\n", " (i \\partial_t + \\Delta) u = +|u|^{2} u \\\\\n", " u(0,x) = u_{hi_0} (x).\n", " \\end{matrix}\n", " \\right.\n", "\\end{equation*}\n", "We describe the first result to give global well-posedness below\n", "$H^1$.\n", "\n", "* $NLS_3^+ (\\mathbb{R}^2)$ is GWP in $H^s$ for $s > \\frac{2}{3}$.\n", "* First use of *Bilinear Strichartz*\n", " estimate was in this proof. \n", "* Proof cuts solution into low and high frequency parts.\n", "* For $u_0 \\in H^s,\n", " ~s>\\frac{2}{3},$ Proof gives (and *crucially exploits*),\n", "$$ u(t) - e^{it \\Delta } u_{hi_0} \\in H^1 (\\mathbb{R}^2_x).$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", "# Setting up; Decomposing Data\n", "\n", "* Fix a large target time $T$. \n", "* Let $N = N(T)$ be large to be determined.\n", "* Decompose the initial data:\n", "$$\n", "u_0 = u_{low} + u_{high}\n", "$$\n", "where\n", "$$\n", "u_{low} (x) = \\int_{{|\\xi| < N}} ~e^{i x \\cdot \\xi }\n", "\\widehat{u_0} (\\xi) d \\xi.\n", "$$ \n", "* Our plan is to evolve:\n", "$$\n", "u_0 = u_{low} + u_{high}\n", "$$\n", "to\n", "$$ \n", "u(t) = u_{{low}} (t) + u_{{high}} (t) .\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "# Sizes of the Data Components\n", "\n", " \n", " Low Frequency Data Size:\n", "\n", "* Kinetic Energy:\n", "\\begin{align*}\n", " \\| \\nabla u_{low} \\|^2_{L^2} &= \\int_{|\\xi| < N} |\\xi|^{2}\n", " | \\widehat{u_0} (\\xi)|^2 dx \\ \\\\\n", "&= \\int_{|\\xi| < N} |\\xi|^{2(1-s)}\n", " |\\xi|^{2s} |\\widehat{u_0} (\\xi)|^2 dx \\\\\n", "& \\leq N^{2(1-s)} \\| u_0 \\|^2_{H^s} \\leq C_0 N^{2(1-s)}.\n", "\\end{align*}\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* Potential Energy:\n", "$\n", "\\| u_{low} \\|_{L^4_x} \\leq \\| u_{low} \\|_{L^2}^{1/2} \\|\n", "\\nabla u_{low} \\|_{L^2}^{1/2}\n", "$\n", "$$\n", "\\implies H[ u_{low} ] \\leq C N^{2(1-s)}.\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "High Frequency Data Size:\n", "$$\\| u_{high} \\|_{L^2} \\leq C_0 N^{-s}, ~\\| u_{high} \\|_{H^s} \\leq C_0.$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", "# LWP of $u_{low}$ Frequency Evolution along NLS\n", "\n", "The NLS Cauchy Problem for the low frequency data\n", "\\begin{equation*}\n", "\\tag{{${{NLS}}$}}\n", "%\\tag{{$NLS^{+}_3 (\\mathbb{R}^2)$}}\n", " \\left\\{\n", " \\begin{matrix}\n", " (i \\partial_t + \\Delta) u_{{low}} = +|u_{{low}}|^{2} u_{{low}} \\\\\n", " u_{{low}}(0,x) = u_{low} (x)\n", " \\end{matrix}\n", " \\right.\n", "\\end{equation*}\n", "is well-posed on $[0, T_{lwp}]$ with $T_{lwp} \\thicksim \\|\n", "u_{low} \\|_{H^1}^{-2} \\thicksim N^{-2(1-s)}$.\n", "\n", "We obtain, as a consequence of the local theory, that\n", "$$\n", "\\| u_{{low}} \\|_{L^4_{[0,T_{lwp}], x}} \\leq \\frac{1}{100}.\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "# LWP of $u_{high}$ Evolution along DE\n", "\n", "The NLS Cauchy Problem for the high frequency data\n", "\\begin{equation*}\n", "%\\tag{{$NLS^{+}_3 (\\mathbb{R}^2)$}}\n", " \\left\\{\n", " \\begin{matrix}\n", " (i \\partial_t + \\Delta) u_{{high}} = +2 |u_{{low}}|^2 u_{{high}} +\n", " {\\mbox{similar}} + |u_{{high}}|^{2} u_{{high}} \\\\\n", " u_{{high}} (0,x) = u_{high} (x)\n", " \\end{matrix}\n", " \\right.\n", "\\end{equation*}\n", "is also well-posed on $[0, T_{lwp}]$. \n", "\n", "\n", "**Crucial Observation:** The LWP lifetime of $NLS$ evolution of $u_{{low}}$ AND\n", "the LWP lifetime of the $DE$ evolution of $u_{{high}}$ are controlled by\n", "$\\| u_{{low}}(0)\\|_{H^1}$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", "# Extra Smoothing of Nonlinear Duhamel Term\n", "\n", "The high frequency evolution may\n", "be written\n", "$$\n", "u_{{high}} (t) = e^{it \\Delta} u_{{high}} + w.\n", "$$\n", "The local theory gives $\\| w(t) \\|_{L^2} \\lesssim N^{-s}$. Moreover, \n", "due to smoothing (obtained via bilinear Strichartz), we have that\n", "\\begin{equation}\n", "\\tag{SMOOTH!}\n", "w \\in H^1, ~ \\| w(t) \\|_{H^1}\n", "\\lesssim N^{1-2s+}. \n", "\\end{equation}\n", "Let's postpone the proof of (SMOOTH!)." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", "# Nonlinear High Frequency Term Hiding Step!\n", "\n", "* $\\forall ~t \\in [0, T_{lwp}]$, we have \n", "$$\n", "u(t) = u_{{low}} (t) + e^{it \\Delta } u_{high} + w(t).\n", "$$\n", "* \n", "At time $T_{lwp}$, we define data for the progressive scheme:\n", "$$\n", "u(T_{lwp} ) = u_{{low}} (T_{lwp}) + w(T_{lwp} ) + e^{iT_{lwp} \\Delta}\n", "u_{high}.\n", "$$\n", "\n", "$$\n", "u(t) = u^{(2)}_{{low}} (t) + u^{(2)}_{{high}} (t)\n", "$$\n", "for $ t > T_{lwp}$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", "# Hamiltonian Increment: $u_{low} (0) \\longmapsto\n", " u^{(2)}_{{low}} (T_{lwp})$\n", "\n", "The Hamiltonian increment due to $w(T_{lwp})$ being added to low\n", "frequency evolution can be calcluated. Indeed, by Taylor expansion,\n", "using the bound (SMOOTH!) and energy conservation\n", "of $u_{{low}}$ evolution, we have\n", "using \n", "\\begin{align*}\n", "H[u^{(2)}_{{l}} (T_{lwp})] &= H[u_{{l}} (0)] + (H[ u_{{l}} (T_{lwp}) +\n", "w(T_{lwp}) ] - H[u_{{l}} (T_{lwp})]) \\\\\n", "& \\thicksim N^{2(1-s) } + N^{2 -3s+} \\thicksim N^{2(1-s)}.\n", "\\end{align*}\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "We can accumulate $N^s$ increments of size $N^{2-3s+}$\n", "before we double the size $N^{2(1-s)}$\n", "of the Hamiltonian. During the iteration, Hamiltonian of ``low\n", "frequency'' pieces remains of size $\\lesssim N^{2(1-s)}$ so the LWP\n", "steps are of uniform size $N^{-2(1-s)}$. We advance the solution on a\n", "time interval of size:\n", "$$\n", "N^s N^{-2(1-s)} = N^{-2 + 3s}.\n", "$$\n", "For $s>\\frac{2}{3}$, we can choose $N$ to go past target time $T. ~\\blacksquare$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "# How do we prove (SMOOTH!)?\n", "\n", "The proof follows from a **bilinear estimate**.\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "# Bilinear Strichartz Estimate\n", "\n", "\n", "\n", "* Recall the Strichartz estimate for $(i \\partial_t + \\Delta)$ on $\\mathbb{R}^2$:\n", "$$\n", "\\| e^{it \\Delta} u_0 \\|_{L^4 ( \\mathbb{R}_t \\times \\mathbb{R}^2_x)} \\leq C \n", "\\| u_0 \\|_{L^2 (\\mathbb{R}^2_x)}.\n", " $$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* We can view this trivially as a bilinear estimate by writing\n", "$$\n", " \\| e^{it \\Delta} u_0 ~ e^{it \\Delta} v_0 \\|_{L^2 ( \\mathbb{R}_t \\times \\mathbb{R}^2_x)} \\leq C \n", " \\| u_0 \\|_{L^2 (\\mathbb{R}^2_x)} \\| v_0 \\|_{L^2 (\\mathbb{R}^2_x)} .\n", "$$\n", "* Bourgain refined this trivial bilinear estimate for\n", "functions having certain Fourier support properties. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", "# Bilinear Strichartz Estimate\n", "\n", "For (dyadic) $N \\leq L$ and for $x \\in \\mathbb{R}^2$,\n", "$$\n", "\\| e^{it\\Delta} f_L e^{it\\Delta} g_N \\|_{L^2_{t,x}} \\leq\n", "\\frac{N^{\\frac{1}{2}}}{L^{\\frac{1}{2}}} \\| f_L \\|_{L^2_x} \\| g_N\n", "\\|_{L^2_x}.\n", "$$\n", "\n", "\n", "\n", "* Here $\\mbox{spt}~(\\widehat{f_L}) \\subset \\{ |\\xi | \\thicksim L\\},\n", " ~g_N$ similar.\n", "* Observe that $\\sqrt{\\frac{N}{L}} \\ll 1$ when $N \\ll L$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# $I$-Method" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "# The $I$-Method of Almost Conservation\n", "\n", "\n", "\n", "Let $H^s \\ni u_0 \\longmapsto u$ solve $NLS$ for $t \\in [0, T_{lwp}], T_{lwp} \\thicksim \\|u_0 \\|_{H^s}^{-2/s}.$\n", "\n", "\n", " Consider two ingredients (to be defined):\n", " \n", " * A **smoothing operator** $I = I_N: H^s \\longmapsto H^1$. The $NLS$ evolution $u_0 \\longmapsto u$ induces a **smooth reference evolution** $H^1 \\ni Iu_0 \\longmapsto Iu$ solving $I(NLS)$ equation on $[0,T_{lwp}]$.\n", " * A **modified energy** $\\widetilde{E}[Iu]$ built using the reference evolution." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "# First Version of the $I$-method: ${\\widetilde{E}}= H[Iu]$\n", "\n", "\n", " \n", "For $s<1, N \\gg 1$ define smooth monotone $m: \\mathbb{R}^2_\\xi \\rightarrow \\mathbb{R}^+$ s.t.\n", "$$\n", "m(\\xi) = \n", "\\left\\{\n", "\\begin{matrix}\n", "1 & {\\mbox{for}}~ |\\xi | 2N.\n", "\\end{matrix}\n", "\\right.\n", "$$\n", "\n", "\n", "The associated Fourier multiplier operator,\n", "${\\widehat{(Iu)}} (\\xi) = m(\\xi) \\widehat{u} (\\xi),$\n", "satisfies $I: H^s \\rightarrow H^1 $. Note that, pointwise in time, we have\n", "$$\n", "\\| u \\|_{H^s} \\lesssim \\| Iu \\|_{H^1} \\lesssim N^{1-s} \\|u \\|_{H^s}.\n", "$$\n", "\n", "\n", "Set $\\widetilde{E}[Iu(t)] = H[Iu(t)]$. Other choices of $\\widetilde E$\n", "are mentioned later." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", "# AC Law Decay and Sobolev GWP index\n", "\n", "\n", "* **Modified LWP.** Initial $v_0$ s.t. $\\| \\nabla I v_0\n", "\\|_{L^2} \\thicksim 1$ has $T_{lwp} \\thicksim 1$. \n", "\n", "* **Goal.** $\\forall ~u_0 \\in H^s, ~\\forall ~T > 0$, construct \n", "$u:[0,T] \\times \\mathbb{R}^2 \\rightarrow \\mathbb{C}.$\n", "\n", "* $\\iff$ **Dilated Goal.** Construct\n", "$\n", "u^\\lambda: [0, \\lambda^2 T] \\times \\mathbb{R}^2 \\rightarrow \\mathbb{C}.\n", "$\n", "\n", "* **Rescale Data.** $\\| I \\nabla u_0^\\lambda\n", " \\|_{L^2} \\lesssim N^{1-s} \\lambda^{-s} \\| u_0 \\|_{H^s} \\thicksim 1$\n", "provided we choose $\\lambda = \\lambda (N) \\thicksim\n", "N^{\\frac{1-s}{s}} \\iff N^{1-s} \\lambda^{-s} \\thicksim 1$.\n", "* **Almost Conservation Law.** $\\| I \\nabla u ( t ) \\|_{L^2}\n", " \\lesssim H[Iu(t)]$ and\n", "$$\n", "\\sup_{t \\in [0, T_{lwp}]} H[Iu(t) ] \\leq H [Iu(0)] + N^{-\\alpha}.\n", "$$\n", "* **Delay of Data Doubling.** Iterate modified LWP $N^\\alpha$ steps\n", " with $T_{lwp} \\thicksim 1$. We obtain rescaled solution for $t \\in\n", " [0, N^\\alpha]$.\n", "$$\n", "\\lambda^2(N) T < N^\\alpha \\iff T < N^{\\alpha + \\frac{2(s-1)}{s}}\n", "~{\\mbox{so}}~ s > \\frac{2}{2+\\alpha}~{\\mbox{suffices}}.\n", "$$ \n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "# Almost Conservation Law for $H[Iu]$\n", "\n", "\n", "\n", "Given $s > \\frac{4}{7}, N \\gg 1,$ and initial data \n", "$u_0 \\in C^{\\infty}_0(\\mathbb{R}^2)$ with $E(I_N u_0) \\leq 1$, then\n", "there exists a $ T_{lwp}\\thicksim 1$ so that the solution\n", "\\begin{align*}\n", "u(t,x) & \\in C([0,T_{lwp}], H^s(\\mathbb{R}^2))\n", "\\end{align*}\n", "of $NLS_3^+ (\\mathbb{R}^2)$ satisfies\n", "\\begin{equation*}\n", "\\label{increment}\n", "E(I_N u)(t) = E(I_N u)(0) + O(N^{- \\frac{3}{2}+}),\n", "\\end{equation*}\n", "for all $t \\in [0, T_{lwp}]$.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", " # Ideas in the Proof of Almost Conservation\n", " \n", " * Standard Energy Conservation Calculation:\n", "\\begin{align*}\n", " \\partial_t H(u) &= \\Re \\int_{\\mathbb{R}^2} \\overline{u_t} (|u|^2 u -\n", "\\Delta u) dx \\\\\n", "& = \\Re \\int_{\\mathbb{R}^2} \\overline{u_t} ( |u|^2 u -\n", "\\Delta u - i u_t) dx = 0.\n", "\\end{align*} \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "* For the smoothed reference evolution, we imitate....\n", " \\begin{align*}\n", " \\partial_t H(Iu) &= \\Re \\int_{\\mathbb{R}^2} \\overline{Iu_t} (|Iu|^2 Iu -\n", " \\Delta Iu - i I u_t)dx \\\\\n", " & = \\Re \\int_{\\mathbb{R}^2} \\overline{Iu_t} ( |Iu|^2 Iu - I (|u|^2 u)) dx \\neq 0.\n", " \\end{align*} \n", " * The increment in modified energy involves a commutator,\n", " $$H(Iu)(t) - H(Iu)(0) = \\Re \\int_0^t \\int_{\\mathbb{R}^2} \\overline{Iu_t} ( |Iu|^2 Iu - I (|u|^2 u)) dx dt.\n", " $$\n", " * Littlewood-Paley, Case-by-Case, **(Bi)linear Strichartz**, $X_{s,b}$...." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "# Remarks\n", "\n", "* The almost conservation property\n", "$$ \\sup_{t \\in [0, T_{lwp}]} \\widetilde{E}[Iu(t)] \\leq\n", "\\widetilde{E}[Iu_0] + N^{-\\alpha}$$\n", "leads to GWP for \n", "$$\n", "s > s_\\alpha = \\frac{2}{2+\\alpha}.\n", "$$\n", "* The $I$-method is a **subcritical method**. \n", "\n", "* The $I$-method **localizes the conserved density** in\n", " frequency}}. \n", "\n", "* There is a **multilinear corrections algorithm** for defining\n", " other choices of \n", " $\\widetilde{E}$ which yield a better AC property." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Multilinear Correction Terms" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true, "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", "## Multilinear Correction Terms\n", "\n", "* For $k \\in {\\mathbb{N}}$, define the **convolution hypersurface** \n", "$$ \\Sigma_k := \\{ (\\xi_1,\\ldots,\\xi_k) \\in (\\mathbb{R}^2)^k: \\xi_1 + \\ldots +\n", "\\xi_k = 0 \\} \\subset (\\mathbb{R}^2)^k.$$\n", " * For $M: \\Sigma_k \\to \\mathbb{C}$ and $u_1,\\ldots,u_k$ nice, define\n", " **$k$-linear functional** \n", " $$ \\Lambda_k( M; u_1,\\ldots,u_k ) :=\n", " c_k ~\\mathbb{R}e \\int\\limits_{\\Sigma_k} M(\\xi_1,\\ldots,\\xi_k) \\widehat{u_1}(\\xi_1) \\ldots \\widehat{u_k}(\\xi_k).$$\n", "* For $k \\in 2{\\mathbb{N}}$ abbreviate\n", "$\\Lambda_k (M; u) = \\Lambda_k (M; u, \\overline{u}, \\ldots, \\overline{u}).$\n", "* $\\Lambda_k (M;u)$ invariant under interchange of even/odd arguments,\n", "$$\n", "M (\\xi_1,\\xi_2,\\ldots,\\xi_{k-1},\\xi_k) \\mapsto \\overline{M}(\\xi_2,\\xi_1,\\ldots,\\xi_k,\\xi_{k-1}).$$\n", "* We can define a symmetrization rule via group orbit." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "## Examples\n", "\n", "\n", "* \n", "$$\n", "\\int\\limits_x u \\overline{u} u \\overline{u} dx = \\int (\\int e^{i x \\cdot\n", " \\xi_1} \\widehat{u} (\\xi_1) d\\xi_1) \\ldots (\\int e^{i x \\cdot\n", " \\xi_4} \\widehat{\\overline{u}} (\\xi_4) d\\xi_4) dx\n", "$$\n", "$$\n", "= \\int_{\\xi_1, \\dots, \\xi_4} \\left[\\int_x e^{i x \\cdot ( \\xi_1 + \\xi_2 +\n", " \\xi_3 + \\xi_4)} dx\\right] \n", " \\widehat{u} (\\xi_1) \\widehat{\\overline{u}} (\\xi_2) \\widehat{u} (\\xi_3) \\widehat{\\overline{u}} (\\xi_4) \n", " d \\xi_{1, \\ldots ,4}\n", "$$\n", "$$\n", "= \\int\\limits_{\\Sigma_4} \\widehat{u} (\\xi_1) \\widehat{\\overline{u}} (\\xi_2) \n", "\\widehat{u} (\\xi_3) \\widehat{\\overline{u}} (\\xi_4)\n", "= \\Lambda_4 (1; u).\n", "$$\n", "\n", "* \n", "$$\\Lambda_2 (-\\xi_1 \\cdot \\xi_2; u) = \\| \\nabla u \\|_{L^2}^2.$$\n", "\n", "\n", "\n", "\n", "Thus, $H[u] = \\Lambda_2 ( - \\xi_1 \\cdot \\xi_2; u) \\pm \\Lambda_4 (\\frac{1}{2} ; u)$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", "## Time Dependence of Multilinear Forms\n", "\n", "Suppose $u$ nicely solves $NLS_3^+ (\\mathbb{R}^2)$; $M$ is time\n", "independent, symmetric. \n", "\n", "### How would you calculate\n", "\n", "$$\n", "\\partial_t \\Lambda_k( M; u(t) )?\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Time Differentiation Formula\n", "\n", "\n", "$$\n", "\\partial_t \\Lambda_k( M; u(t) ) =\n", "\\Lambda_k( i M \\alpha_k; u(t) ) - \\Lambda_{k+2}( i k X(M); u(t) ) \n", "$$\n", "$$\n", "=\n", "\\Lambda_k( i M \\alpha_k; u(t) ) - \\Lambda_{k+2}( [i k X(M)]_{sym}; u(t)\n", ").\n", "$$\n", "Here\n", "$$ \\alpha_k(\\xi_1,\\ldots,\\xi_k) := -|\\xi_1|^2 + |\\xi_2|^2 - \\ldots - |\\xi_{k-1}|^2 + |\\xi_k|^2$$\n", "(so $\\alpha_2 = 0$ on $\\Sigma_2$) and \n", "$$ X(M)(\\xi_1,\\ldots,\\xi_{k+2}) := M( \\xi_{123}, \\xi_4, \\ldots, \\xi_{k+2}).$$\n", "We use the notation $\\xi_{ab} := \\xi_a + \\xi_b$, $\\xi_{abc} := \\xi_a +\n", "\\xi_b + \\xi_c$, etc. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", "## AC Quantities via Multilinear Corrections\n", "\n", "\n", "* Abbreviate $m(\\xi_j)$ as $m_j$. Define $\\sigma_2$ s.t. $\\| I \\nabla u \\|_{L^2}^2 = \\Lambda_2 (\\sigma_2; u):$ \n", "$$\\sigma_2(\\xi_1,\\xi_2) := -\n", " \\frac{1}{2} \\xi_1 m_1 \\cdot \\xi_2 m_2 = \\frac{1}{2} |\\xi_1|^2 m_1^2$$\n", " \n", "* With $\\tilde \\sigma_4$ (symmetric, time independent) {{to be determined}}, set\n", "$$\n", "{\\widetilde{E}} := \\Lambda_2( \\sigma_2 ; u ) + \\Lambda_4( \\tilde\n", "\\sigma_4 ; u ).\n", "$$\n", "* Using the time differentiation formula, we calculate\n", "$$\n", "\\partial_t \\widetilde E = \\Lambda_4 ( {{\\left\\{i \\tilde\n", " \\sigma_4 \\alpha_4 - i 2[ X (\\sigma_2)]_{sym} \\right\\}}} ; u) -\n", "\\Lambda_6 ( [i 4 X(\\tilde \\sigma_4)]_{sym}; u).\n", "$$\n", "\n", "\n", "\n", "We'd like to define $\\tilde \\sigma_4$ to cancel away the $\\Lambda_4$\n", "contribution. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", " \n", " ## Natural Choice of $\\sigma_4$ Fails\n", " \n", "Here is the natural choice:\n", " $$ \\tilde \\sigma_4 =~ \\frac{[2 i X(\\sigma_2)]_{sym}}{i \\alpha_4}.$$\n", "On $\\Sigma_4$, we can reexpress $\\alpha_4 = -|\\xi_1|^2 + |\\xi_2|^2\n", "-|\\xi_3|^2 + |\\xi_4|^2$ as\n", "$$\n", "\\alpha_4 = -2 \\xi_{12} \\cdot \\xi_{14} = -2 |\\xi_{12}| |\\xi_{14}| \\cos\n", "\\angle(\\xi_{12},\\xi_{14}), \n", "$$\n", "and\n", "$$\n", "[2 i X(\\sigma_2)]_{sym} = \\frac{1}{4} ( - m_1^2 |\\xi_1|^2 + m_2^2\n", "|\\xi_2|^2 - m_3^2 |\\xi_3|^2 + m_4^2 |\\xi_4|^2 ).\n", "$$\n", "When all the $m_j = 1$ (so $\\max_{j} |\\xi_j | < N$), $\\tilde \\sigma_4$ is\n", "well-defined. However, $\\alpha_4$ can also vanish when $\\xi_{12}$ and\n", "$\\xi_{14}$ are orthogonal. \n", "\n", "### Small Divisor Problem\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ " \n", "## Speculation on Integrable Systems?\n", "\n", "For $NLS_3^+ (\\mathbb{R})$,\n", "the resonant obstruction disappears. Thus, \n", "$$ \\widetilde E^1 = \\Lambda_2 (\\sigma_2) + \\Lambda_4 (\\tilde \\sigma_4);$$\n", "$$ \\partial_t \\widetilde E^1 = - \\Lambda_6 ( [i 4 X(\\tilde \\sigma_4)]_{sym}).$$\n", "We can then define, with $\\tilde \\sigma_6$ to be determined,\n", "$$\\widetilde E^2 = \\widetilde E^1 + \\Lambda_6 (\\tilde \\sigma_6 );$$\n", "$$\n", "\\partial_t \\widetilde E^2 = \\Lambda_6 ( {{\\{ i \\tilde \\sigma_6\n", " \\alpha_6 - [i 4 X(\\tilde \\sigma_4)]_{sym}\\} }}) + \n", "\\Lambda_{8}( [i 6 X(\\tilde \\sigma_6)]_{sym}).\n", "$$\n", "Let's define\n", "$$\n", "\\tilde \\sigma_6 = \\frac{[i 4 X (\\tilde \\sigma_4)]_{sym}}{i \\alpha_6}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "## Speculation on Integrable Systems?\n", "\n", "Thus, we formally obtain a continued-fraction-like algorithm.\n", "$$\n", "\\tilde \\sigma_6 = \\frac{\\left[i 4 X \\left ( \\frac{[2 i\n", " X(\\sigma_2)]_{sym}}{i \\alpha_4}\\right)\\right]_{sym}}{i \\alpha_6}, \n", "$$\n", "$$\n", "\\tilde \\sigma_8 = \\frac{\\left[i 6 X \\left( \\frac{\\left[i 4 X \\left ( \\frac{[2 i\n", " X(\\sigma_2)]_{sym}}{i \\alpha_4}\\right)\\right]_{sym}}{i \\alpha_6}\n", "\\right)\\right]_{sym}}{i \\alpha_8}, \\ldots.\n", "$$\n", "Each step gains two derivatives but costs two more factors.\n", "\n", "**Speculation:** The multipliers $\\tilde \\sigma_6, \\tilde \\sigma_8,\n", "\\ldots$ are well defined and lead to better AC properties. Same for\n", "other integrable systems." ] } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.4.3" } }, "nbformat": 4, "nbformat_minor": 0 }