Convexity and entropy for conservation laws I am reading Hyperbolic Systems of Conservation Laws by Godlewski and Raviart (1).


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*While defining the entropy why do we consider only convex flux?
Can it be extended to general entropy flux? Please suggest me the reference.

*I understood that entropy conditions are the conditions which assures the uniqueness of the solution. Why do we call them as entropy condition? Is it related to the entropy  which we study in Thermodynamics?



(1) E. Godlewski, P.-A. Raviart, Hyperbolic Systems of Conservation Laws. Ellipses, 1991.

Let us consider the scalar one-dimensional case $u_t + f(u)_x = 0$ for sake of simplicity, as described in Section 3.8 of (1).
An entropy function $\eta$ is a convex function of $u$ which satisfies the conservation law $\eta(u)_t + \psi(u)_x = 0$, where $\psi$ is the entropy flux. In the definition, neither the flux $f$ nor the entropy flux $\psi$ is convex, but only $\eta$ is convex. This is a conventional choice with important consequences, "though one may consider a more general
framework" (2). If $\eta$ is chosen convex, then the vanishing viscosity weak solution -- i.e., the physically relevant weak solution -- makes the entropy decrease in the weak sense. This leads to the definition of the entropy solution, which is a weak solution $u$ such that $\eta(u)_t + \psi(u)_x \leq 0$ weakly for all convex entropy function $\eta$.
As such, the definition of an entropy function $\eta$ is not related to the thermodynamical notion of entropy. Firstly, it decreases over time, while the entropy increases in thermodynamics. However, there may be a link with the thermodynamical entropy for some particular systems. In the case of Eulerian gas dynamics, a natural mathematical entropy function is defined with respect to the thermodynamical entropy (cf. Section II.2 of (2)).

(1) R.J. LeVeque, Numerical Methods for Conservation Laws. Birkhäuser, 1992. doi:10.1007/978-3-0348-8629-1
(2) E. Godlewski, P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, 1996. doi:10.1007/978-1-4612-0713-9
