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\begin{document}

\title{Introduction to Partial Differential Equations}
\author{Homework 7}
\date{due May 12, 2011}
\maketitle

\begin{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 $t>0$.


\item (From Evans, p.\ 163, Question 5.)
\begin{enumerate}
\item For $1 < r < \infty$, let $H \colon \R^n \to \R$ be defined as
\[
  H(p) = \frac1p \, \lvert p \rvert^r \,.
\]
Show that its Legendre transform satisfies
\[
  H^*(q) = \frac1s \, \lvert q \rvert^s \,,
  \qquad \text{where } \frac1r + \frac1s = 1 \,.
\]

\item Let $H(p) = p^T A p + b^T p$ where $A \in M(n \times n)$ is
symmetric and positive definite, and $b \in \R^n$.  Compute $H^*(q)$. 
\end{enumerate}

\item Consider the initial-value problem
\begin{gather*}
  u_t + \lvert u_x \rvert^2 = 0 \qquad \text{in } \R \times (0,\infty)
  \,, \\
  u = 0 \qquad \text{on } \R \times \{ t = 0 \} \,.
\end{gather*}
One non-trivial candidate solution which is Lipshitz and satisfies the
equation almost everywhere, is given by Evans, p.\ 129:
\begin{gather*}
  u (x,t) =
  \begin{cases}
    0 & \text{if } \lvert x \rvert \geq t \,, \\
    x-t & \text{if } 0 \leq x \leq t \,, \\
    -x-t & \text{if } -t \leq x \leq 0 \,.
  \end{cases}
\end{gather*}

\begin{enumerate}
\item Show that $u$ is not semiconcave.
\item Give another nontrivial solution which is Lipshitz and satisfies
the equation almost everywhere.
\item What is the solution given by the Hopf--Lax formula?
\end{enumerate}

\item Fill in the gaps in the proof of Evans, p.\ 131, Lemma 4.

\item Evans, p.\ 164, Question 10.
\end{enumerate}
\end{document}
