Math 6410 • ODE

\textbf{Conservation Laws.}
A model for flow of fluid in an infinite pipe says that the density \(\rho(x,t)\) satisfies
\[
  \frac{\partial\rho}{\partial t}+\rho\frac{\partial\rho}{\partial x}=-\rho^2,\qquad
  \rho(x,0)=
  \begin{cases}
    1, & \text{if }x\le0,\\
    \frac{1}{2}, & \text{if }x>0.
  \end{cases}
\]
\begin{enumerate}
\item[(a)] Find a solution \(\rho\).
\item[(b)] Define what it means for your solution \(\rho\) from part (a) to be a weak
solution of the system. (Do not carry out the calculations to prove that \(\rho\) is a weak
solution.)
\end{enumerate}