{ "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", "### https://github.com/colliand/ascona2016\n", "\n", "***" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Overview of Lecture 1\n", "\n", "* Initial Value Problem for NLS\n", "* Conserved Quantities\n", "* Well-posedness Theory" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Initial Value Problem for NLS" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "# Initial Value Problem for NLS\n", "\n", "$$\n", "\\begin{equation*}\n", "\\tag{$NLS^{\\pm}_p (\\Omega)$}\n", " \\left\\{\n", " \\begin{matrix}\n", " (i \\partial_t + \\Delta) u = \\pm |u|^{p-1} u \\\\\n", " u(0,x) = u_0 (x), ~ x \\in \\Omega.\n", " \\end{matrix}\n", " \\right.\n", "\\end{equation*}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Studies explore interplay between:\n", "\n", "* Dispersion\n", "* Nonlinearity" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Dilation invariance quantifies the balance between these effects." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Dilation Invariance\n", "\n", "One solution $u$ generates parametrized family $\\{u^\\lambda\\}_{\\lambda > 0}$ of solutions:\n", "\n", "$$u:[0,T) \\times \\mathbb{R}^d_x \\rightarrow \\mathbb{C} ~{\\mbox{solves}}~ NLS_p^{\\pm}(\\mathbb{R}^d)$$ \n", "\n", "$${\\iff}$$\n", "\n", "$$u^\\lambda: [0,\\lambda^2 T )\\times \\mathbb{R}^d_x \\rightarrow \\mathbb{C} ~{\\mbox{solves}}~ NLS_p^{\\pm}(\\mathbb{R}^d)$$ \n", "where\n", "$$\n", "u^\\lambda (\\tau, y) = \\lambda^{-2/(p-1)} u( \\lambda^{-2} \\tau, \\lambda^{-1} y).\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Norms which are invariant under $u \\longmapsto u_\\lambda$ are **critical**. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Critical Regularity\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "In the $L^2$-based Sobolev scale,\n", "$$\n", "\\| D^s u^\\lambda (t) \\|_{L^2} = \\lambda^{-\\frac{2}{p-1} - s + \\frac{d}{2}} \\| D^s u (t)\\|_{L^2}.\n", "$$\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "The **critical Sobolev index** for $NLS_p^{\\pm}(\\mathbb{R}^d)$ is\n", "$$\n", "s_c := \\frac{d}{2} - \\frac{2}{p-1}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "\n", "| critical Sobolev index | Regime | \n", "|:---------------:|:-------------------------------:| \n", "| $ s_c < 0$ | mass subcritical |\n", "| $0 < s_c < 1$ | mass super/energy subcritical | \n", "| $s_c = 1$ | energy critical | \n", "| $1 < s_c < \\frac{d}{2}$| energy supercritical |\n", "\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "\n", "### The spatial domain $\\Omega$?\n", "\n", "\n", "* Infinite measure space like $\\mathbb{R}^d$ \n", "* Finite measure space like $\\mathbb{T}^d$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### The Choice of Sign?\n", " \n", "* $+$ **defocusing**\n", "* $-$ **focusing**\n", "\n", "Large data dynamics are completely different." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "> [numerical simulations of $NLS_5^+ (\\mathbb{R}^5)$](https://wwejubwfy.s3.amazonaws.com/chirped-data-NLS5R5-simulation.pdf)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Conserved Quantities\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "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": [ "\n", "## Remarks\n", "\n", "* Conserved quantities constrain the dynamics.\n", "* NLS defines a flow on a sphere in $L^2$.\n", "* Energy vividly reveals **focusing vs. defocusing** difference.\n", "* Mass is $L^2$; Momentum scales like $H^{1/2}$; Energy involves $H^1$.\n", "* Local conservation laws express **how** a quantity is\n", " conserved: \n", " $\\partial_t |u|^2= \\nabla \\cdot 2 \\Im (\\overline{u}\n", " \\nabla u)$. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "# Variations on Conserved Quantities?" ] }, { "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": [ "* Energy/Compactness methods for building solutions.\n", "* Globalizing control to extend local-in-time solutions." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ " \n", "## Almost Conserved##\n", "$$\\big| \\partial_t Q[u] \\big| ~\\mbox{is small}.$$\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* Bourgain's High/Low Frequency Decomposition\n", "* $I$-Method\n", "* Multilinear Correction Terms\n", "* Applications" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Monotone ## \n", "$$\\partial_t Q[u] > 0.$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* Virital identity $\\implies$ blow-up.\n", "* Morawetz-type inequalities $\\implies$ decay." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Well-posedness Theory" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "# Local Well-posedness Theory\n", "\n", "* Fixed point argument based on Contraction Mapping\n", "* Show the \"Picard iterates\" converge\n", "* Key innovation for the analysis: identify the right space!" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "# Free Schrödinger Evolution" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "$$\n", "\\begin{equation*}\n", "\\tag{$LS(\\mathbb{R}^d)$}\n", " \\left\\{\n", " \\begin{matrix}\n", " (i \\partial_t + \\Delta) u = 0 \\\\\n", " u(0,x) = u_0 (x).\n", " \\end{matrix}\n", " \\right.\n", "\\end{equation*}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "## Explicit Solution Formula\n", "\n", "$$ u_0 \\longmapsto u(t,x) = e^{it \\Delta} u_0$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Explicit Solution Formula" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "\n", "* Fourier Multiplier Representation:\n", "$$\n", "e^{it \\Delta} u_0 (x) = c_\\pi\\int_{\\mathbb{R}^d} e^{i x \\cdot \\xi} e^{-i t |\\xi|^2}\n", "\\widehat{u_0} (\\xi) d\\xi.\n", "$$\n", "* Convolution Representation:\n", "$$\n", "e^{it \\Delta} u_0 (x) = k_\\pi \\frac{1}{(it)^{d/2}} \\int_{\\mathbb{R}^d} e^{i\n", " \\frac{|x-y|^2}{4t}} u_0 (y) dy.\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Estimates for Schrödinger Propagator $e^{it \\Delta} u_0$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* Fourier Multiplier Representation $\\implies$ Unitary in $H^s$:\n", "$$\n", "\\| D_x^s e^{it \\Delta} u_0 \\|_{L^2_x} = \\| D_x^s u_0 \\|_{L^2_x}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* Convolution Representation $\\implies$ Dispersive estimate:\n", "$$\n", "\\| e^{it \\Delta} u_0 \\|_{L^\\infty_x} \\leq \\frac{C}{t^{d/2}} \\| u_0 \\|_{L^1_x}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* Spacetime estimates? **Strichartz estimates** hold, for example,\n", "$$\n", "\\| e^{it \\Delta} u_0 \\|_{L^4 ( \\mathbb{R}_t \\times \\mathbb{R}^2_x)} \\leq C \n", "\\| u_0 \\|_{L^2 (\\mathbb{R}^2_x)}.\n", " $$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "# Schrödinger Evolution with Forcing" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "$$\n", "\\begin{equation*}\n", " \\left\\{\n", " \\begin{matrix}\n", " (i \\partial_t + \\Delta) u = F \\\\\n", " u(0,x) = u_0 (x).\n", " \\end{matrix}\n", " \\right.\n", "\\end{equation*}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "### Strichartz Estimates\n", "\n", "$$ \\| u \\|_{L^q_t L^r_x (\\mathbb{R}_t \\times \\mathbb{R}^d_x )} \\leq C \\| u_0 \\|_{L^2_x} \n", "+ \\| F \\|_{L^{Q'}_t L^{R'}_x(\\mathbb{R}_t \\times \\mathbb{R}^d_x )}.\n", "$$\n", "***\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Admissibility for *independent* pairs $(q,r), (Q,R)$\n", "$$\n", "\\frac{2}{q} + \\frac{d}{r} = \\frac{d}{2}, ~ q > 2.\n", "$$\n", "The $'$ denotes Hölder dual exponent." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "# Local-in-time theory for $NLS^{\\pm}_3 (\\mathbb{R}^2)$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* $\\forall ~u_0 \\in L^2 (\\mathbb{R}^2)~\\exists ~T_{lwp} ( u_0 ) $ determined by \n", "$$\n", "\\| e^{it \\Delta} u_0 \\|_{L^4_{tx} ([0,T_{lwp} ] \\times \\mathbb{R}^2)} <\n", "\\frac{1}{100} ~{\\mbox{such that}}~\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\exists$ unique $u \\in C([0, T_{lwp} ]; L^2 ) \\cap L^4_{tx} ([0,T_{lwp}] \\times \\mathbb{R}^2)$ solving $NLS_3^{+} (\\mathbb{R}^2)$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "* $\\forall ~ u_0 \\in H^s (\\mathbb{R}^2), s>0$, $T_{lwp} \\thicksim \\| u_0 \\|_{H^s}^{-\\frac{2}{s}}$ and regularity persists: $u \\in C([0,T_{lwp}]; H^s (\\mathbb{R}^2))$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* Define the **maximal forward existence time** $T^* (u_0)$ by \n", "$$\n", "\\| u \\|_{L^4_{tx} ([0,T^* -\\delta] \\times \\mathbb{R}^2) }< \\infty\n", "$$\n", "for all $\\delta > 0$ but diverges to $\\infty$ as $\\delta \\searrow 0$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* $\\exists ~$ **small data scattering threshold** $\\mu_0 > 0$\n", "$$\n", "\\| u_0 \\|_{L^2} < \\mu_0 \\implies \\|u \\|_{L^4_{tx} (\\mathbb{R} \\times \\mathbb{R}^2)} < 2 \\mu_0.\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "# $H^1$ Global-in-time theory for $NLS^{+}_3 (\\mathbb{R}^2)$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "* $\\forall ~ u_0 \\in H^1 (\\mathbb{R}^2), s>0$, $T_{lwp} \\thicksim \\| u_0 \\|_{H^1}^{-2}$ and regularity persists: $u \\in C([0,T_{lwp}]; H^1 (\\mathbb{R}^2))$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* Conserved Quantities $\\implies ~T_{lwp} > C(u_0)$\n", "$$\n", "{\\mbox{Energy}} = H[u(t)] = \\frac{1}{2} \\int_{\\mathbb{R}^2} |\\nabla u(t) |^2 dx + \\frac{2}{p+1} |u(t)|^{p+1}\n", "$$\n", "$$\n", "{\\mbox{Mass}} = \\| u \\|_{L^2_x}^2 = \\int_{\\mathbb{R}^d} |u(t,x)|^2 dx\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* Globalize by iteration using $ ~T_{lwp} > C(u_0)$.\n" ] } ], "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": "3.4.4" } }, "nbformat": 4, "nbformat_minor": 0 }