Math 6420 • PDE

Find the entropy solution of the following scalar conservation
law, arising in traffic flow modeling,

\[
\begin{cases}
\partial_t u+\partial_x(u(1-u))=0 & \text{for } x\in\mathbb{R},\ t>0,\\
u(x,0)=g(x)
\end{cases}
\]

with initial data

\[
g(x)=
\begin{cases}
0, & x<-1,\\
1, & -1<x<0,\\
0, & 0<x<1,\\
\frac12, & 1<x.
\end{cases}
\]

After the rarefaction hits the shock you should write the explicit ODE IVP
for the shock location \(\gamma(t)\) and a piecewise definition of the solution using
this shock location \(\gamma(t)\), but you don't actually need to solve the ODE IVP
to get the explicit formula for \(\gamma(t)\).