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. *
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Communication dans un congrès
Carlos M. da Fonseca, Dinh Van Huynh, Steve Kirkland and Vu Kim Tuan. Conference on Mathematics and its Applications , Nov 2014, Safat, Kuwait. American Mathematical Society, Contemporary Mathematics, 658, pp.237 - 248, 2016, A panorama of mathematics: pure and applied. 〈10.1090/conm/658/13123〉
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Denis Serre, Alexis Vasseur. About the relative entropy method for hyperbolic systems of conservation laws. Carlos M. da Fonseca, Dinh Van Huynh, Steve Kirkland and Vu Kim Tuan. Conference on Mathematics and its Applications , Nov 2014, Safat, Kuwait. American Mathematical Society, Contemporary Mathematics, 658, pp.237 - 248, 2016, A panorama of mathematics: pure and applied. 〈10.1090/conm/658/13123〉. 〈ensl-01402415〉

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