{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Nonlinear Schrödinger as a Dynamical System\n", "\n", "> #### [J. Colliander](http://colliand.com) ([UBC](http://www.math.ubc.ca))\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" ] }, { "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", "\n", "\n", "### https://github.com/colliand/ascona2016\n", "\n", "\n", "***\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Overview of Lecture 3\n", "\n", "* Generalized Virial Identity\n", "* A Priori Spacetime Estimates\n", " * Lin-Strauss Morawetz Estimate\n", " * Interaction Morawetz Estimates" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "# Variations on Conserved Quantities?\n", "\n", "## Monotone ## \n", "$$\\partial_t Q[u] > 0.$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "\n", "# Generalized Virial Identities" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", "\n", "\n", "## Lagrangian NLS\n", " \n", " \n", " \n", " \n", "$$\n", "\\begin{equation*}\n", "\\left\\{\n", "\\begin{matrix}\n", "(i \\partial_t + \\Delta) u = \\pm F'(|u|^2) u \\\\\n", "u(0,x) = u_0 (x) \\\\\n", "\\end{matrix}\n", "\\right.\n", "\\end{equation*}\n", "$$\n", "\n", "## Generalized NLS Equation (GNLS)\n", "\n", "\n", "\n", "$$\n", "\\begin{equation*}\n", "\\left\\{\n", "\\begin{matrix}\n", "(i \\partial_t + \\Delta ) \\phi = \\mathcal{N} \\\\\n", "\\phi(0,x) = \\phi_0 (x) \\\\\n", "\\end{matrix}\n", "\\right.\n", "\\end{equation*}\n", "$$\n", "\n", "\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "**Remarks:**\n", "\n", "* Assume $F' \\geq 0$.\n", "The $+$ case is **defocusing** $-$ is **focusing**.\n", "* Generalized NLS with Lagrangian derivation. \n", "* $U(1)$ solution symmetry: $u \\rightarrow e^{i\\theta} u$. \n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "## Time Invariant Quantities\n", "\n", "The following quantities do not change with time:\n", "$$\n", "{\\mbox{Mass}} = \\int_{{\\mathbb{R}}^d} |u(t,x)|^2 dx. \n", "$$\n", "$$\n", "{\\mbox{Momentum}} = 2 \\Im \\int_{{\\mathbb{R}}^2} {\\overline{u}(t)} \\nabla u (t)\n", "dx. \n", "$$\n", "$$\n", "{\\mbox{Energy}} = H[u(t)] = \\frac{1}{2} \\int_{R^2} |\\nabla u(t) |^2 dx {\\mp} F(|u(t)|^2) dx .\n", "$$\n", "$\\implies$ a priori **conservation controls** (defocusing case):\n", "$$\n", "\\| u \\|_{L^\\infty_t L^2_x} \\leq \\| u_0 \\|_{L^2} \n", "$$\n", "$$\n", "\\| \\nabla u \\|_{L^\\infty_t L^2_x } \\leq E[u_0]. \n", "$$\n", "These are very useful bounds but do not give any decay in time.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", " ## Local Conservation Laws\n", "\n", "We consider an even more general NLS equation. \n", "\n", "\n", "* Suppose $\\phi:[0,T] \\times {\\mathbb{R}}^d \\rightarrow \\mathbb{C}$ solves **generalized\n", " NLS (GNLS)**\n", "\n", "$$\n", "(i \\partial_t + \\Delta ) \\phi = \\mathcal{N}\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "for $\\mathcal{N} = \\mathcal{N} (t, x, \\phi ):[0,T] \\times {\\mathbb{R}}^d \\times \\mathbb{C} \\rightarrow \\mathbb{C}$. Assume $\\phi$ is nice. \n", "\n", "* Not necessarily Lagrangian; No $U(1)$ symmetry.\n", "\n", "* Express mass and momentum (non)conservation for $GNLS$.\n", "\n", "\n", "\n", " \n", "Write $\\partial_{x_j} \\phi = \\partial_j \\phi = \\phi_j$.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ " ## Local mass/momentum (non)conservation\n", " \n", "\n", " \n", " * mass density: \n", "$T_{00} = |\\phi|^2 $\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "* momentum density/mass current: \n", "$T_{0j} = T_{j0} = 2 \\Im (\\overline{\\phi} \\phi_j)$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "* (linear part of the) momentum current: \n", "$L_{jk} = L_{kj} = - \\partial_j \\partial_k |\\phi|^2 + 4 \\Re (\\overline{\\phi_j} \\phi_k)$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "* mass bracket: \n", "$\\{ f, g\\}_m = \\Im (f \\overline{g})$\n", "\n", "* momentum bracket: \n", "$\\{ f, g\\}^j_{p} = \\Re (f \\partial_j \\overline{g} - g \\partial_j \\overline{f} )$\n", " " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "\n", "**Local mass (non)conservation identity:** \n", "$$\n", "\\partial_t T_{00} + \\partial_j T_{0j} = 2 \\{ \\mathcal{N} , \\phi \\}_m\n", "$$\n", "**Local momentum (non)conservation identity:** \n", "$$\\partial_t T_{0j} + \\partial_k L_{kj} = 2 \\{ \\mathcal{N} , \\phi\n", "\\}_p^j\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ " ## Local mass/momentum (non)conservation\n", "\n", "Consider $\\mathcal{N} = F' (|\\phi|^2) \\phi$ for polynomial $F:{\\mathbb{R}}^+ \\rightarrow \\mathbb{R}$.\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "* We calculate the mass bracket\n", "$$\n", "\\{ F' (|\\phi|^2) \\phi , \\phi \\}_m = \\Im ( F' (|\\phi|^2) \\phi\n", "\\overline{\\phi}) =0.\n", "$$\n", "\n", "Thus mass is conserved for these nonlinearities. \n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "* We calculate the momentum bracket\n", "$$\n", "\\{ F' (|\\phi|^2) \\phi , \\phi \\}_p^j = - \\partial_j G (|\\phi|^2)\n", "$$\n", "where $G(z) = z F'(z) - F(z) \\thicksim F(z)$. \n", "\n", "\n", "Thus the momentum bracket contributes a divergence and momentum is conserved for these nonlinearities." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", " \n", " ## Generalized Virial Identity\n", "\n", " \n", "Let $a: {\\mathbb{R}}^d \\rightarrow \\mathbb{R}$ be a **virial weight function**. \n", "Form the **virial potential**\n", "$$V_a (t) = \\int_{{\\mathbb{R}}^d} a(x) |\\phi(t,x)|^2 dx.$$\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Form the **Morawetz action**\n", "$$\n", "M_a (t) = \\int_{{\\mathbb{R}}^d} \\nabla a \\cdot 2 \\Im (\\overline{\\phi} \\nabla \\phi) dx.\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "Conservation identities lead to the **generalized virial identities**\n", "$$\n", "\\partial_t V_a = M_a + \\int_{{\\mathbb{R}}^d} a (x) \\{ \\mathcal{N}, \\phi\\}_m (t,x) dx,\n", "$$\n", "$$\n", "\\partial_t M_a = \\int_{{\\mathbb{R}}^d} (-\\Delta \\Delta a) |\\phi|^2 + 4 a_{jk}\n", "\\Re (\\overline{\\phi_j} \\phi_k) + 2 a_j \\{ \\mathcal{N}, \\phi\\}_p^j dx.\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "## Remarks on Virial Identities\n", "\n", "* The virial potential is a weighted average of the mass density against the virial weight $a$.\n", "\n", "* The Morawetz action is a contraction of the momentum density against $\\nabla a$. Vector fields not arising as gradients could also be considered.\n", "\n", "* Useful estimates emerge from monotonicity and boundedness of terms in the virial identities.\n", " \n", "* Monotone quantities provide dynamical insights.\n", "\n", "* **Idea of Morawetz Estimates:** *Cleverly choose* the weight\n", "function $a$ so that $\\partial_t M_a \\geq 0$ but $M_a \\leq C (\\phi_0)$ to\n", "obtain spacetime control on $\\phi$. This strategy imposes various\n", "constraints on $a$ which suggest choosing $a (x) = |x|$.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "## Variance Identity\n", " \n", "\n", "* Consider $GNLS$ with $\\mathcal{N} = \\pm |u|^{4/d} u$. This is the **$L^2$ critical focusing** equation $NLS_{1+{\\frac{4}{d}}}^{\\pm} ({\\mathbb{R}}^d)$. \n", "* Choose $a(x) = |x|^2$. Calculations reveal that\n", "$$\n", "\\partial_t^2 \\int_{{\\mathbb{R}}^d} |x|^2 |u(t,x)|^2 dx = 16 H[u(t)].\n", "$$\n", "* In the focusing case, we can consider initial data $u_0$ with $H[u_0] <0$ and finite variance. Such data must blow up in finite time." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# A Priori Spacetime Estimates" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", " \n", " ## [Lin-Strauss] Morawetz identity\n", "\n", "\n", "Consider $(i \\partial_t + \\Delta ) \\phi = F' (|\\phi|^2) \\phi$ with $F' \\geq 0$ and $x \\in {\\mathbb{R}}^3$. Choose $a(x) = |x|$. Observe that $a$ is weakly convex, $\\nabla a = \\frac{x}{|x|}$ is bounded, and $-\\Delta \\Delta a = 4 \\pi \\delta_0$. From monotonicity $\\partial_t M_a \\geq 0$ and the bound $|M_a| \\leq \\sqrt{H[u_0]}$ emerges the **Lin-Strauss Morawetz identity**\n", "$$\n", "M_a (T) - M_a (0) = \\int\\limits_0^T \\int\\limits_{{\\mathbb{R}}^3} 4 \\pi \\delta_0 (x) |\\phi (t,x)|^2 + (\\geq 0) + 4\n", "\\frac{G(|\\phi|^2)}{|x|} dx dt.\n", "$$\n", "\n", "This implies the spacetime control estimate (centered at $x=0$)\n", "$$\n", "(H[u_0])^{1/2} \\|u_0\\|_{L^2} \\gtrsim \\int\\limits_0^T \\int\\limits_{{\\mathbb{R}}^3}\n", "\\frac{G(|\\phi|^2)}{|x|} dx dt. \n", "$$ \n", " \n", "[Morawetz] \"Reward and Anchor.\" \n", "\n", "[Ginibre-Velo] $H^1$-Scattering." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "## [Bourgain] and [Grillakis] truncation\n", "\n", "* Let $\\chi_{B_R}$ denote a smooth cutoff adapted to $B_R = \\{ |x| < R \\}$.\n", "* Choose cutoff virial weight\n", "$\n", "a(x) = \\chi_{B_R} (x) |x|.\n", "$\n", "and calculate\n", "$$\n", "M_a \\Big|_0^T \\geq \\int\\limits_0^T \\int\\limits_{{\\mathbb{R}}^3} 4 \\pi \\delta_0 (x) |\\phi (t,x)|^2 + 4 \\int\\limits_0^T \\int\\limits_{|x| < R/2} \n", "\\frac{G(|\\phi|^2)}{|x|} dx dt\n", "$$\n", "* \n", "$\n", "| M_a \\Big|_0^T | \\leq R^{-1} T H[u_0] + R H[u_0] \\implies$ choose $R \\thicksim T^{1/2} \\implies$\n", "$$\n", "\\int\\limits_0^T \\int\\limits_{|x| < T^{1/2}} \\frac{G(|\\phi|^2)}{|x|} dx \\lesssim T^{1/2} \\|\\nabla \\phi \\|_{L^\\infty_{[0,T]} L^2_x}^2.\n", "$$\n", "\n", "\n", "[Bourgain], [Grillakis]: Energy critical bubbles sparse along time axis." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Interaction Morawetz Estimates" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", " \n", "\n", "## Averaging over [Lin-Strauss] center?\n", "\n", "* Translation invariance? Weight $|x|^{-1}$ difficult in proofs.\n", "* Recenter [L-S] at fixed $y \\in {\\mathbb{R}}^d$. Set $a(x) = |x-y|$.\n", "* Recentered Morawetz action can be expressed\n", "$$\n", "M_y [u] (t) = \\int_{{\\mathbb{R}}^d} \\frac{(x-y)}{|x-y|} 2 \\Im ( u \\nabla \\overline{u}) (t,x) dx.\n", "$$\n", "* Monotonicity $\\partial_t M_y [u] \\geq 0$: mass repelled from $y \\in {\\mathbb{R}}^d$.\n", "* Can we **average with respect to center $y$** and obtain new translation invariant spacetime control?\n", "* Yes, if we average against the natural density $|u(t,y)|^2$.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "\n", "## Interaction Morawetz via Averaging\n", "\n", "* Define the **Morawetz interaction potential**\n", "$$\n", "M[u](t) = \\int_{{\\mathbb{R}}^d_y} |u(t,y)|^2 M_y [u] (t) dy.\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "It is bounded: \n", "$$\\Big|M[u](t)\\Big| \\lesssim \\| u(t) \\|_{L^2_x}^3 \\| \\nabla u(t) \\|_{L^2_x}.$$ " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "We calculate\n", "$$\n", "\\partial_t M[u] = \\int_{{\\mathbb{R}}^d_y} |u(t,y)|^2 \\{\\partial_t M_y [u] \\} + \\{\\partial_t |u(y)|^2\\} M_y [u] dy.\n", "$$\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* Local conservation and [Lin-Strauss] Morawetz $\\implies$ monotonicity: \n", "$\\exists ~I, II, III, IV$ such that $I, III \\geq 0$ and $II + IV \\geq 0$ and \n", "$ \\partial_t M[u] = I + II + III + IV. $ Integrating in time gives\n", " $$\\int_0^T \\int_{{\\mathbb{R}}^3} |u(t,x)|^4 dx dt \\lesssim \\| u(t) \\|_{L^\\infty_T L^2_x}^3 \\| \\nabla u(t) \\|_{L^\\infty_T L^2_x}~ .\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", " \n", "## 2-particle interaction Morawetz \n", " (Hassell 04)\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", " \n", "* Suppose $\\phi_1, \\phi_2$ are two solutions of $(i \\partial_t + \\Delta ) \\phi = F' (|\\phi|^2) \\phi$ with $F' \\geq 0$ and $x \\in {\\mathbb{R}}^3$. The **2-particle wave function**\n", "$$\n", "\\Psi (t, x_1, x_2) = \\phi_1 (t, x_1) \\phi_2 (t, x_2)\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "satisfies an NLS-type equation on ${\\mathbb{R}}^{1+6}$\n", "$$\n", "(i \\partial_t + \\Delta_1 + \\Delta_2) \\Psi = [F' (|\\phi_1 |^2) + F'\n", "(|\\phi_2 |^2)] \\Psi.\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "\n", "* Note that $ [F' (|\\phi_1 |^2) + F' (|\\phi_2 |^2)] \\geq 0$ so defocusing.\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "* Reparametrize ${\\mathbb{R}}^6$ using center-of-mass coordinates $(\\overline{x}, y)$ with $\\overline{x} = \\frac{1}{2} (x_1 + x_2) \\in {\\mathbb{R}}^3$. Note that\n", "$y=0$ corresponds to the diagonal $x_1 = x_2 = \\overline{x}$. Apply the\n", "generalized virial identity with the **choice** $a(x_1, x_2) = |y|$. Dismissing terms with favorable\n", "signs, one obtains...\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ " \n", " ## 2-particle interaction Morawetz\n", " \n", "$$\n", " \\| \\nabla u \\|_{L^\\infty_{[0,T]} L^2_x} \\|u_0\\|_{L^2}^3 \\geq \\int_0^T \\int_{{\\mathbb{R}}^6}\n", " (-\\Delta_6 \\Delta_6 |y|) |\\Psi ( x_1, x_2)|^2 d x_1 dx_2 dt \n", "$$\n", "$$\n", "\\geq c\\int_0^T \\int_{{\\mathbb{R}}^6} \\delta_{\\{y=0\\}} (x_1, x_2) |\\phi_1(x_1) \\phi_2( x_2)|^2 dx_1 dx_2 dt \n", "$$\n", "$$\n", " \\geq c\\int_0^T \\int_{{\\mathbb{R}}^3} |\\phi_1(t, \\overline{x}) \\phi_2(t, \\overline{x})|^2 d \\overline{x}\n", "dt.\n", "$$\n", " \n", "Specializing to $\\phi_1 = \\phi_2$ gives the **2-particle Morawetz estimate**\n", "$$\n", "\\int_0^T \\int_{{\\mathbb{R}}^3} |\\phi(t,x)|^4 dx dt \\leq C \\| \\nabla \\phi\n", "\\|_{L^\\infty_{[0,T]} L^2_x} \\| \\phi_0 \\|_{L^2_x}^3\n", "$$\n", "valid uniformly for all defocusing NLS equations on ${\\mathbb{R}}^3$.\n", " " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ " \n", "## \"The\" 2-particle Morawetz Estimate\n", "Efforts to extend the $L^4(\\mathbb{R}_t \\times {\\mathbb{R}}^3_x)$ interaction Morawetz to the ${\\mathbb{R}}^2_x$ setting led to...\n", "\n", "**Theorem:**\n", "Finite energy solutions of any defocusing $NLS^+ ({\\mathbb{R}}^d)$ satisfy\n", "$$\n", "\t\\| D^{\\frac{3-d}{2}} |u|^2 \\|^2_{L^2_{t,x}} \\lesssim \n", "\t\\| u_0 \\|_{L^2_x}^3 \\| \\nabla u \\|_{L^\\infty_t L^2_x}.\n", "$$\n", "\n", "\n", "\n", "* [C-Grillakis-Tzirakis], [Planchon-Vega]: independently\n", "* Simple proof of $H^1$-scattering in mass supercritical case.[Nakanishi]\n", "* Simplified proof extends to $H^s$ for certain $s<1$.\n", "* Applied by Dodson to resolve the $L^2$ scattering conjecture." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## 4-particle Morawetz Estimate\n", "\n", "(Hassel-Tao) [C-Holmer-Visan-Zhang]\n", "\n", "* ${\\mathbb{R}}^4 = \\{ {\\bf{x}}=(x_1, x_2, x_3, x_4): x_i \\in \\mathbb{R}; i=\\{ 1,2,3,4 \\}$: \n", "$\\overline{x}=$ **center of mass** $ = \\frac{1}{4}(x_1 + x_2 + x_3 + x_4)$. \n", " Define $y = (x_1 - \\overline{x}, x_2 - \\overline{x}, x_3 - \\overline{x}, x_4 - \\overline{x})$. Here $y \\in {\\mathbb{R}}^3$. $ {\\mathbb{R}}^4 \\ni {\\bf{x}}= (x_1, x_2, x_3, x_4) \\iff (\\overline{x}, y) \\in \\mathbb{R} \\times {\\mathbb{R}}^3$\n", " " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "* The 4-particle wave function \n", "$$ \\Psi (t, {\\bf{x}}) = \\prod_{i=1}^4 \\phi_1 (t, x_i) $$\n", "satisfies a defocusing NLS equation on ${\\mathbb{R}}^{1+4}$.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "* Choice of virial weight $a({\\bf{x}}) = |y|$ spawns\n", "$$\n", "\\int_0^T \\int_{\\mathbb{R}} |\\phi|^8 d{\\overline{x}} dt \\lesssim \\| \\phi \\|^7_{L^\\infty_T L^2_x} \\|\\nabla \\phi \\|_{L^\\infty_T L^2_x}.\n", "$$\n", "* **Q:** How does this estimate generalize to other dimensions?\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Applications" ] } ], "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 }