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\title{Introduction to Partial Differential Equations}
\author{Homework 8}
\date{due May 7, 2013}
\maketitle

\begin{enumerate}

\item Evans, p.\ 164, Question 10.

\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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