About the relative entropy method for hyperbolic systems of conservation laws

Abstract

We review the relative entropy method in the context of first-order hyperbolic systems of conservation laws, in one space-dimension. We prove that contact discontinuities in full gas dynamics are uniformly stable. Generalizing this calculus, we derive an infinite-dimensional family of Lyapunov functions for the system of full gas dynamics. 1 Systems of conservation laws and entropies We are interested in vector fields u(x, t) obeying first-order PDEs. The space variable x and time t run over the physical domain R d × (0, T). The field takes values in a convex open subset U of R n. A conservation law is a first-order PDE of the form ∂ t a + div x b = 0. The terminology refers to the fact that weak solutions obey the identity d dt Ω a(x, t) dx + ∂Ω b · ν ds(x) = 0, for every regular open subdomain Ω ⊂ R d. Hereabove ν is the outer normal and ds is the area element over the boundary. Actually, the PDE is often derived from the latter identity, which expresses a physical principle such as conservation of mass, momentum, species, charge, energy, ... See C. Dafermos's book [9] for a thorough description of this correspondance. *