\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts}

\newcommand\PVint{\mathop{\setbox0\hbox{$\displaystyle\intop$}%
    \hskip0.2\wd0%
    \vcenter{\hrule width0.6\wd0height0.5pt depth0.5pt}%
    \hskip-0.8\wd0%
    }\mskip-\thinmuskip\intop\nolimits}
    
\newcommand*  {\T} {{\mathbb T}}
\newcommand*  {\N} {{\mathbb N}}
\newcommand*  {\R} {{\mathbb R}}

\newcommand*{\norm}[3][{\vphantom 1}]{\lVert #2 \rVert_{#3}^{#1}}
\newcommand*{\bnorm}[3][]{\bigl\lVert #2 \bigr\rVert_{#3}^{#1}}
\newcommand*{\tnorm}[3][]%
  {\smash[b]{\lVert #2 \rVert_{\smash[b]{#3}}^{#1}}}
\newcommand*{\abs}[1]{\lvert #1 \rvert}

\newcommand*{\Space}[3]{{#1}_{{\vphantom 1}#2}^{{\vphantom +}#3}}

\renewcommand*{\l}[2][]{\Space{L}{#1}{#2}}
\newcommand*{\h}[2][]{\Space{H}{#1}{#2}}
\newcommand*{\Lt}[1]{\Space{L}{}{#1}(\T)}
\newcommand*{\Ht}[1]{\Space{H}{}{#1}(\T)}
\newcommand*{\wh}[2][]{\Space{{\operatorname{w-}}H}{#1}{#2}}
\newcommand*{\wl}[2][]{\Space{{\operatorname{w-}}L}{#1}{#2}}

\newcommand*{\ini}{{\operatorname{in}}}

\begin{document}

\title{Partial Differential Equations}
\author{Homework 8}
\date{due November 26, 2008}
\maketitle

In the following, $\T$ denotes the 1-torus, i.e.\ $\T = \R \,
\operatorname{mod} 2\pi$.

\begin{enumerate}

\item Let $U \subset \mathbb R^n$ be open and let $u_n$ and $v_n$ be
two sequences in $\l2(U)$.
\begin{enumerate}
\item Suppose $u_n \to u$ strongly and $v_n \rightharpoonup v$
weakly, and that, moreover, $u_n v_n \in \l2(U)$ for every $n \in
\N$ and also $uv \in \l2(U)$.

Show that $u_n v_n \rightharpoonup uv$ weakly.
\item Give an example that when $u_n
  \rightharpoonup u$ only weakly in (a), then $u_n v_n$ may not
  converge to $uv$ weakly.
\end{enumerate}

\item
\begin{enumerate}
\item Show that, for every $u \in \Ht2$,
\[
  \norm[2]{u}{\h1} \leq \norm{u}{\l{2}} \, \norm{u}{\h{2}}\,. 
\]

\item Consider the Fisher--Kolmogorov equation on $\T$,
\begin{gather*}
  u_t = u_{xx} + (1-u) \, u^m \,, \\
  u(0) = u^\ini \,,
\end{gather*}
where $m$ is an even positive integer.  Use the result from (a), as
well as Question~4 of the previous homework set, to prove that that
\[
  \limsup_{t \to \infty} \, \norm{u(t)}{\h1} \leq C   
\]
where an explicit estimate for $C$ can be given which, in particular,
shows that $C$ does not depend on the initial data $u^\ini$.  You may
assume that $u$ is sufficiently differentiable so that all your formal
manipulations are justified.

\end{enumerate}

\item Prove the following version of the \emph{Poincar\'e inequality}:
For every $u \in \Ht1$ which has zero mean, i.e.\ where
\[
  \PVint_{\T} u \, dx = 0 \,,
\]
we have
\[
  \int_{\T} |u|^2 \, dx
  \leq C \, \int_{\T} |u_x|^2 \, d x \,.
\]
Find the best estimate for $C$.
\end{enumerate}
\end{document}

\item Consider the inviscid Burger's equation on $\T$, i.e.
\[
  u_t + u \, u_x = 0 \,.
\]
\begin{enumerate}
\item Define an approximate solution $u_n$ by applying the projector
$\mathbb{P}_n$ which projects onto modes up to wave number $n$ to
Burger's equation.  Show that
\[
  \norm{u_n(t)}{\l2} = \norm{u_n(0)}{\l2} \,.
\]
\item Conclude that $\{u_n\}$ has a subsequence that converges to some
$u$ weakly in $\l2(\T)$, and that
\[
  \norm{u(t)}{\l2} \leq \norm{u(0)}{\l2} \,.
\]
Why do you have an inequality rather than equality?

(Note: you are not required to show that $u$ solves Burger's equation
in any sense.  This would require a much more involved analysis.)

\end{enumerate}

\end{enumerate}



\end{document}


