\documentclass[11pt,reqno]{amsart} \usepackage{geometry} % See geometry.pdf to learn the layout options. There are lots. \geometry{letterpaper} % ... or a4paper or a5paper or ... %\geometry{landscape} % Activate for for rotated page geometry %\usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx} \usepackage{amssymb} \usepackage{epstopdf} \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} \usepackage{eucal} \usepackage{mathrsfs} \usepackage{enumerate} \usepackage{nicefrac} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\exd}{\mathrm{d}} \newcommand{\supp}{\mathrm{supp}} \newcommand{\singsupp}{\mathrm{sing\,supp}} \newcommand{\tstD}{\mathscr{D}} \newcommand{\tstE}{\mathscr{E}} \newcommand{\tstS}{\mathscr{S}} \title{Math 581 Assignment 6} \author{Due Friday March 30} \date{Winter 2012} \begin{document} \maketitle \begin{enumerate}[1.] \item \begin{enumerate}[a)] \item {\em Maxwell's equations} for 3 dimensional electromagnetism in vacuum are \begin{equation}\label{e:Maxwell-evol} \partial_t E = \nabla\times B, \qquad \partial_t B = - \nabla\times E, \end{equation} and \begin{equation}\label{e:Maxwell-const} \nabla\cdot E = 0, \qquad \nabla\cdot B = 0, \end{equation} where $E,B:\R^3\times\R\to\R^3$ are the electric and magnetic fields, respectively. Show that the system \eqref{e:Maxwell-evol} is symmetric hyperbolic. Then show that the constraints \eqref{e:Maxwell-const} are preserved by the evolution, i.e., that if one starts with initial data satisfying the constraints \eqref{e:Maxwell-const}, and if $E$ and $B$ evolve according to \eqref{e:Maxwell-evol}, then \eqref{e:Maxwell-const} will be satisfied for all time. \item Consider the second order system $$ \partial_t^2 u = \sum_{j,k=1}^{n}A_{j,k}\partial_j\partial_ku, \qquad u|_{t=0} = f, \qquad \partial_tu|_{t=0} = g, $$ which generalizes the wave equation. Here we suppose $u:\R^n\times\R\to\R^m$, and that $A_{j,k}$ are $m\times m$ matrices. Let us say that the system is {\em symmetric hyperbolic} if the matrices $A_{j,k}$ are symmetric and positive definite, which then we assume. Prove (either by reducing it to a first order system or directly) that the above Cauchy problem is well-posed in the following sense: For any initial data $(f,g)\in H^s\times H^{s-1}$ with some $s\in\R$, there exists a unique solution $u\in C^0(\R,H^s)$ with $\partial_tu\in C^0(\R,H^{s-1})$, satisfying $$ \|u(t)\|_{H^s} \leq C \|f\|_{H^s} + \alpha(t) \|g\|_{H^{s-1}}, \qquad \|\partial_tu(t)\|_{H^{s-1}} \leq C( \|f\|_{H^s} + \|g\|_{H^{s-1}}), $$ where $C$ is a constant, and the function $\alpha(t)$ grows slower than some polynomial for large $t$. For isotropic and homogeneous materials, the {\em elastodynamics equations} are given by $$ \partial_t^2 u = \mu \Delta u + \lambda\nabla(\nabla\cdot u), $$ where $u:\R^n\times\R\to\R^n$ is the displacement field, and $\mu$ and $\lambda$ are real parameters. In components, it reads $$\partial_t^2 u_k = \mu \Delta u_k + \lambda\partial_k(\partial_1 u_1 + \ldots \partial_n u_n),\qquad k=1,\ldots,n.$$ Determine the values of the parameters $\mu$ and $\lambda$ for which the system is symmetric hyperbolic. \end{enumerate} \item Consider the $n$-dimensional {\em Navier-Stokes equations} \begin{equation}\label{e:NS} \partial_t u = \Delta u - u\cdot\nabla u - \nabla p, \qquad \nabla\cdot u = 0, \end{equation} where $u:\R^n\times\overline\R_+\to\R^n$ is the velocity field, and $p:\R^n\times\overline\R_+\to\R$ is the pressure field. For clarity, in components the first of the Navier-Stokes equations would read $$\partial_t u_k + (u_1\partial_1+\ldots+u_n\partial_n) u_k = \Delta u_k - \partial_k p,\qquad k=1,\ldots,n.$$ Let us overload the notation $H^s(\R^n)$ so that it denotes also the space of vector fields with each component lying in $H^s(\R^n)$. Define the {\em Leray projector} $P:H^s(\R^n)\to H^s(\R^n)$ by $$ \widehat{Pu}(\xi) = \hat{u}(\xi) - (\hat{u}(\xi)\cdot\xi)\xi/|\xi|^2. $$ In components, it is $$\widehat{(Pu)}_k(\xi) = \hat{u}_k(\xi) - \frac{\xi_1\hat{u}_1(\xi)+\ldots+\xi_n\hat{u}_n(\xi)}{\xi_1^2+\ldots+\xi_n^2}\xi_k = (\delta_{km}-\frac{\xi_k\xi_m}{|\xi|^2})\hat{u}_m(\xi),$$ where the summation convention is understood in the latter expression. In Fourier space, the divergence free condition $\nabla\cdot u = 0$ is simply $\xi\cdot\hat{u}(\xi)=0$, so the Leray projector projects onto the divergence free space. A formal application of $P$ to \eqref{e:NS} gives \begin{equation}\label{e:NSL} \partial_t u = \Delta u - P(u\cdot\nabla u). \end{equation} \begin{enumerate}[a)] \item For which values of $s\in\R$ is $P:H^s\to H^s$ bounded? \item Show that in an appropriate sense, the two formulations \eqref{e:NS} and \eqref{e:NSL} are equivalent. \item Prove the local well-posedness of \eqref{e:NSL} in $H^s$ for $s>\frac{n}2+1$. \item In order to update the above result to a global well-posedness result, what kind of bound would you need? \item {\em Bonus problem}: Prove the global well-posedness of the 2 dimensional Navier-Stokes equations in $H^s$ for $s>2$. \end{enumerate} \item Let $\Omega\subset\R^n$ be an open set, let $k\geq0$ be an integer, and let $1\leq p\leq \infty$. Then the {\em Sobolev space} $W^{k,p}(\Omega)$ by definition consists of those $u\in\tstD'(\Omega)$ such that $\partial^\alpha u\in L^p(\Omega)$ for each $\alpha$ with $|\alpha|\leq k$. Equip it with the norm $$ \|u\|_{W^{k,p}(\Omega)} = N( \{ \|\partial^\alpha u\|_{L^{p}(\Omega)}: |\alpha|\leq k \} ), $$ where $N$ is a norm on the finite dimensional space $\{\lambda_\alpha\in\R:|\alpha|\leq k\}$. Obviously, the topology of $W^{k,p}(\Omega)$ does not depend on the choice of $N$, so one can pick $N$ at their convenience. \begin{enumerate}[a)] \item Show that $W^{k,p}(\Omega)$ is a Banach space for any $k\geq 0$ and $1\leq p\leq \infty$. \item Show that $\tstD(\R^n)$ is a dense subspace of $W^{k,p}(\R^n)$, for any $k\geq 0$ and $1\leq p<\infty$. \end{enumerate} \item Recall that the {\em Sobolev inequality} \begin{equation}\label{e:Sob-ineq} \|u\|_{L^q}\leq C \|u\|_{W^{1,p}}, \qquad u\in \tstD(\R^n), \end{equation} with some constant $C=C(p,q)$, is valid when $1 \leq p \leq q < \infty$, and $\frac1p \leq \frac1q+\frac1n$. \begin{enumerate}[a)] \item By way of a counterexample, show that the inequality \eqref{e:Sob-ineq} fails whenever $q
\frac1q+\frac1n$. \item Show that \eqref{e:Sob-ineq} fails for $p=n$ and $q=\infty$ when $n\geq2$. \item Derive sufficient conditions on the exponents $p,q,k,m$ under which the inequality $$ \|u\|_{W^{m,q}}\leq C \|u\|_{W^{k,p}}, \qquad u\in \tstD(\R^n), $$ is valid. \end{enumerate} \item \begin{enumerate}[a)] \item Let $\Omega\subset\R^n$ be a domain and let $\chi$ be a smooth function satisfying $\chi\in W^{\ell,\infty}(\Omega)$ for all $\ell$. Show that $u\mapsto \chi u : W^{k,p}(\Omega) \to W^{k,p}(\Omega)$ is bounded for $k\geq0$ and $1\leq p\leq\infty$. \item Consider the differential operator $$ L=\sum_{|\alpha|\leq m}a_{\alpha}\partial^\alpha, $$ on some domain $\Omega\subset\R^n$, where the coefficients satisfy $a_\alpha\in C^{\infty}(\Omega)\cap W^{\ell,\infty}(\Omega)$ for all $\ell$. Show that $L : W^{k+m,p}(\Omega) \to W^{k,p}(\Omega)$ is bounded for $k\geq0$ and $1\leq p\leq\infty$. \item Let $\Omega,\Omega'\subset\R^n$ be two bounded domains, with a diffeomorphism $\phi:\Omega\to\Omega'$ that can be extended to a diffeomorphism from a neighbourhood of $\Omega$ to a neighbourhood of $\Omega'$. Prove that the pullback $\phi^*:W^{k,p}(\Omega')\to W^{k,p}(\Omega)$ is a bounded linear operator for $k\geq0$ and $1\leq p\leq\infty$. \end{enumerate} \end{enumerate} \end{document}