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\begin{document}

\title{Partial Differential Equations}
\author{Homework 9}
\date{due December 3, 2008}
\maketitle


\begin{enumerate}

\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?

\emph{Notes:} You are not required to show that $u$ solves Burger's
equation in any sense.  This would require a much more involved
analysis.  You can use, without proof, the fact that if $x_n
\rightharpoonup x$ weakly on a Hilbert space, then
\[
  \lVert x \rVert \leq \liminf_{n \to \infty} \lVert x_n \rVert \,.
\]

\end{enumerate}

\item (From Evans, p. 164, Question 13.) Let $u \in C(\R \times
  [0,T])$ for some $T>0$ be a weak solution to the scalar conservation
  law
  \begin{gather*}
    u_t + F(u)_x = 0 \qquad \text{in } \R \times (0,T) \,, \\
    u = g \qquad \text{on } \R \times \{ t=0 \} \,.
  \end{gather*}
  Assume further that for any fixed $t \in [0,T]$, $u(\,\cdot\,,t)$ has
  compact support in $\R$, and that $F(0)=0$.  Show that
\[
  \int_\R u(x,t) \, dx = \int_\R g(x) \, dx 
\]
for every $t \in [0,T]$.

\item (From Evans, p. 164, Question 14.) Compute explicitly the unique
  entropy solution of 
  \begin{gather*}
    u_t + u \, u_x = 0 \qquad \text{in } \R \times (0,\infty) \,, \\
    u = g \qquad \text{on } \R \times \{ t=0 \} \,.
  \end{gather*}
for
\[
  g(x) = 
  \begin{cases}
    1 & \text{if } x < -1 \\
    0 & \text{if } -1 < x < 0 \\
    2 & \text{if } 0 < x < 1 \\
    0 & \text{if } 1 < x \,.
  \end{cases}
\]
Draw the characteristic curves in an $(x,t)$-plot, being sure to document
all qualitative changes in the solution for $t>0$.
\end{enumerate}



\end{document}


