Math 6410 • ODE

Consider Burger's equation
\[
\begin{aligned}
  u_t+\left(\frac{u^2}{2}\right)_x &= 0, && \text{in }\R\times(0,\infty),\\
  u &= g, && \text{on }\R\times\{0\}.
\end{aligned}
\]
\begin{enumerate}
\item[(a)] Construct an integral solution satisfying the Rankine-Hugoniot condition which
exhibits ``nonphysical shocks.''
\item[(b)] Write down the explicit entropy condition for this problem and explain why it
rules out the solution in part (a).
\item[(c)] Construct another integral solution which satisfies the entropy condition, and
explain why this condition does not allow shocks.
\end{enumerate}