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\title{Introduction to Partial Differential Equations}
\author{Homework 7}
\date{due April 22, 2015}
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\begin{enumerate}
\item  Evans, p.~163 problem 5
\item Let $L$ be convex with superlinear growth and suppose that $g$
is uniformly Lipshitz as defined in class (and in Evans, p.~124) via
\[
  \operatorname{Lip}(g) = \sup_{\substack{x,y \in \mathbb R^n\\x\neq y}}
  \frac{\lvert g(x)-g(y) \rvert}{\lvert x-y \rvert} < \infty \,.
\]
Show that if, for fixed $x \in \mathbb R^n$ and $t>0$,
\[
  u(x,t) = \inf_{y \in \mathbb R^n} 
   \biggl\{ t \, L \biggl( \frac{x-y}t \biggr) + g(y) \biggr\} \,,
\]
then the infimum in this expression actually is really a minimum.


\item (From Evans, p.\ 164, Question 13.) Let $u \in C(\R \times
[0,T])$ for some $T>0$ be an integral 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}
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