I am a beginner in shallow water equation.
I am interested in the equation
$$h_t+(hu)_x=0$$ $$(hu)_t+(hu^2+\frac{1}{2}gh^2)_x=0$$
I have the following doubts
1)Weak solutions are not unique in general, what is the entropy condition to choose the solution?
2)What is the condition on the initial data for the existence of weak solution?
3)Since each of the equation is conservation law, can we expect the solution to satisfy all the properties of the solutions of the conservation law such as monotonicity etc at least at the Riemann problem level if not for the general initial data
Please suggest me some books which concentrate on such theoretic aspects of the solution.
Best Answer
As far as I know there is no uniqueness results (like kruzkhov uniqueness for scalar conservation laws) for the system of conservation laws.\ Depending on the physics of the problem we expect the solution to satisfy some additional conditions. However there is no proofs as such to show that these additional conditions give uniqueness for the $L^{\infty}$ initial data.