My question is similar to the one asked in this post. I am trying to find the weak form solution for the following 2D shallow water equation (SWE) with source terms: $$ \begin{cases} \partial_t h + \partial_x(hu) + \partial_y(hv) = 0\\ \partial_t(hu) + \partial_x(hu^2 + \frac{gh^2}{2}) + \partial_y(huv)= gh(S_{0,x} - S_{f,x})\\ \partial_t(hv) + \partial_x(huv) + \partial_y(hv^2 + \frac{gh^2}{2})= gh(S_{0,y} - S_{f,y}) \end{cases} $$
I have written the system in weak form, but I am struggling to derive the entropy equation to ensure the uniqueness of the weak solutions. I attempted to follow the steps outlined in François Bouchut's textbook, but was unsuccessful. Any guidance or assistance would be greatly appreciated