Consider the Riemann Problem
I am only going to deal with the case that there is a shock curve ( ), since I can figure out the rarefaction wave once I understand the shock. The solution for this PDE using the Rankine Hugoniot Condition is
Now, I want to determine if this solution satisfies the definition of the weak solution. So I assume that I would want to take a test function with compact support and multiply it by the PDE and then integrate over , i.e.
For some reason this just doesn't seem right.
Another thing that's bugging me is: if then your solution looks a lot like Heaviside's function (which at a first glance would have me believe it will not be a weak solution in the sense above, or at least not in )