Let $$ \frac{\partial \boldsymbol{u}}{\partial t} + \sum_{j=1}^d \frac{\partial}{\partial x_j} \boldsymbol{f}_j(\boldsymbol{u}) = \boldsymbol{0}, $$ be a system of conservation laws and let us assume that we are able to find an additional conservation law $$ \frac{\partial U(\boldsymbol{u})}{\partial t} + \sum_{j=1}^d \frac{\partial}{\partial x_j} F_j(\boldsymbol{u}) = 0, $$ where $U$ is a strictly convex function. (This is common for many physical system.) Can we now say that the function $U$ is a strictly convex mathematical entropy and conclude that the original system of conservation laws is symmetrizable (and thus locally well-posed)?
In Godlewski, Raviart: Numerical Approximation of Hyperbolic Systems of Conservation Laws (1996) the authors say that a convex function $U$ is a mathematical entropy if there exist functions $F_j$ (entropy fluxes) such that $$ U'(\boldsymbol{u})\boldsymbol{f}_j'(\boldsymbol{u}) = F_j'(\boldsymbol{u}). $$ If this condition holds then, of course, the additional conservation law is satisfied. However, this does not work the other way around. And yet, in Examples 3.1 and 3.2 the authors only check whether the additional conservation law, and not the condition above, holds. Could someone clear this up for me? Is it always sufficient to find an additional conservation law to conclude symmetrizability?
The two conditions are equivalent.
Using the chain rule we can write, $$ \frac{\partial u}{\partial t} + \sum_{j=1}^d \frac{\partial}{\partial x_j} f_j(u) = \frac{\partial u}{\partial t} + \sum_{j=1}^d f_j'(u) \frac{\partial u}{\partial x_j}. $$ Then multiply by $U'(u)$ to get $$ U'(u) \frac{\partial u}{\partial t} + \sum_{j=1}^d U'(u)f_j'(u) \frac{\partial u}{\partial x_j}. $$ and notice that the entropy equality will hold only when $U'(u)f'_j(u) = F'_j(u)$. This is discussed in the book "Shock Waves and Reaction-Diffusion Equations" by Joel Smoller (2nd Edition) (See Chapter 20 Part B).