Math 6410 • ODE

\textbf{Conservation Laws.}
Consider the first order equation
\[
\begin{cases}
  \dfrac{\partial u}{\partial t}+(\tan u)\dfrac{\partial u}{\partial x}=0,
    & \text{for }x\in\R\text{ and }t\ge0,\\
  u(x,0)=\varphi(x), & \text{for }x\in\R.
\end{cases}
\qquad(\diamond)
\]
\begin{enumerate}
\item[(a.)] Find a solution \(u(x,t)\) to \((\diamond)\) where
\(\varphi(x)=\operatorname{Arctan}(x)\).
\item[(b.)] For
\[
  \varphi(x)=
  \begin{cases}
    0, & \text{if }x<0,\\
    \frac{\pi}{4}, & \text{if }x>0,
  \end{cases}
\]
find two weak solutions to \((\diamond)\). Prove that one of them is a weak solution.
\item[(c.)] State the entropy (Rankine-Hugoniot) condition. Which weak solution of part
(b.) satisfies this condition?
\end{enumerate}