I have to discuss the entropy solution to the following initial value problem
$ \begin{cases} u_t+f(u)_x = 0 \\ u(x,0) = u_0(x) = \begin{cases} u_{-},\quad x<0\\ u_+,\quad x>0 \end{cases} \end{cases} $
where the flux function is $f(u) = \sqrt{|u|}$. I just have a question about the case $|u_+|<|u_-|$ and $u_-u_+\geq 0$ where, in order to satisfy the Oleinik entropy condition, it should happen that the segment joining the 2 points $[u_-,f(u_-)]$ and $[u_+,f(u_+)]$ should be over the graph in order to say that the unique entropy condition is a shock wave moving with speed given by Rankine Hugoniot condition, which is not our case. So my idea is to propagate a rarefraction wave solution of the kind:
$ \begin{cases} u_-,\quad x<f'(u_-)t \\ (f')^{-1}(x/t),\quad f'(u_-)<x/t<f'(u_+) \\ u_+,\quad x>f'(u_+)t, \end{cases} $
but the problem is that I cannot invert everywhere the derivative of the flux function but just almost everywhere since even the derivative is not defined everywhere.
How should I solve this problem?
By the way here is the plot of the geometric representation of Oleinik condition in the case I'm interested in: $0\leq u_+ \leq u_-$
UPDATE I think I've managed to solve the case in which for instance $0<u_+\leq u_-$, where the derivative is well defined and being a monotone function is even invertible. My solution is the following:
$ u(x,t) = \begin{cases} u_-,\quad x/t<\frac{1}{2\sqrt{u_-}} \\ \frac{t^2}{4x^2},\quad \frac{1}{2\sqrt{u_-}}<x/t<\frac{1}{2\sqrt{u_+}} \\ u_+,\quad x/t>\frac{1}{2\sqrt{u_+}} \end{cases} $
obtained as usual by self similarity of solutions to conservation laws with flux function smooth enough, like the one obtained by restricting the domain of such a a function to the $u>0$.
But of course now the problem still remains since if I fix $u_+=0$ I don't know how to solve the Riemann problem with an arbitrary $u_->0$.
Maybe I should focus on the weak formulation so that the derivatives of the flux function are deloaded on some smooth test function but I really don't know how to proceed.