Math 6410 • ODE

\textbf{Conservation Laws.}
Consider the first-order equation
\[
  (G(u))_x+u_y=0,
\]
where \(G\in C^\infty\), with initial condition \(u(x,0)=h(x)\).
\begin{enumerate}
\item[(a)] Using the method of characteristics, derive the general solution of the
initial value problem. Assuming \(h\in C^\infty\), find a condition on \(G'(h(x))\) which
implies the solution \(u(x,y)\) will develop a singularity in finite time \(y>0\).
\item[(b)] Set \(G(u)=\frac{1}{2}u^2\). Given \(u_0\in\R\), find explicit weak solution(s)
for
\[
  h(x)=
  \begin{cases}
    u_0, & x<0,\\
    0, & x>0.
  \end{cases}
\]
Show that your solutions are weak solutions. Which properties do your solutions exhibit:
shocks, rarefaction, nonuniqueness?
\end{enumerate}