Symmetrizability of shallow water equations Consider the shallow water equation
 \begin{equation}h_t+(hu)_x=0\\
(hu)_t+\left(hu^2+\frac{g}{2}h^2 \right)_x=0
\end{equation}
I want to know the entropy of this system?
I understood that if their exists a change of variable which symmetrizes the system, then system admits strictly convex entropy..
But I am unable to proceed... Please help

Follow the steps in Sec 3.2 of (1). Let's subtract $u$ times the first equation to the second one. After division by $h$, we get the following conservation law for $u$:
$$
u_t + (\tfrac12 u^2 + gh)_x = 0 \, .
$$
Now, multiply the conservation law for $h$ by $\frac12 u ^2 + gh$, multiply the conservation law for $u$ by $hu$, and add the results. We have the additional conservation law
$$
\eta_t + G_x \leq 0
$$
in the week sense, where $\eta = \tfrac12 h u^2 + \tfrac12 g h^2$ is a convex entropy and $G =  (\tfrac12 h u^2 + gh^2)\, u$ is the corresponding entropy flux (cf. Eqs. (1.25) and (1.27) of (1)). You may be able to conclude, see (2).
(1) F Bouchut: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws — and Well-Balanced Schemes for Sources, Birkhäuser, 2000. doi:10.1007/b93802
(2) KO Friedrichs, PD Lax: "Systems of conservation equations with a convex extension", Proc Natl Acad Sci U S A 68.8 (1971), 1686-8. doi:10.1073/pnas.68.8.1686
