The question I'm working on is:
$$u_t + uu_x = 0, t > 0$$
with
$$u(x,0)=f(x)=\begin{cases} 0 & x< 0 \\ 1 & 0\leq x\leq1 \\ 0&x>1 \end{cases}$$ I'm trying to find the weak/shock solution. So far I've solved the characteristic equations and I got:
$$T = \tau, u = f(s), x=\tau f(s)+s$$
After getting these I then did: $$x=\begin{cases} s, & s< 0 \\ \tau+s, & 0\leq s\leq1 \\ s, & s>1 \end{cases}$$
$$ => T =\tau,\space \space \space s=\begin{cases} x & x< 0 \\ x-t, & 0\leq x-t\leq 1 \\x & x>1 \end{cases}$$
Giving me a solution of: $$u=\begin{cases} 0 & x<0 \\ 1 & 0\leq x-t\leq1 \\0 & x>1 \end{cases}$$
I'm confused on where to go from here to get the shock curve. I know the Rankine condition is
$$[u]f'(t) = [q],\space where\space u=u, q=(u^2/2) $$
But in this case $u_L$ & $u_R$ are both $=0$ so I don't know how to proceed from here as that leaves me with a $0$ on both sides. Any help on what to do next or where I may have gone wrong would be appreciated
