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

where

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.