Communications in Mathematical Sciences

Volume 12 (2014)

Number 6

Relative entropy in hyperbolic relaxation for balance laws

Pages: 1017 – 1043



Alexey Miroshnikov (Department of Mathematics, University of Massachusetts, Amherst, Mass., U.S.A.)

Konstantina Trivisa (Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)


We present a general framework for the approximation of systems of hyperbolic balance laws. The novelty of the analysis lies in the construction of suitable relaxation systems and the derivation of a delicate estimate on the relative entropy. We provide a direct proof of convergence in the smooth regime for a wide class of physical systems. We present results for systems arising in materials science, where the presence of source terms presents a number of additional challenges and requires delicate treatment. Our analysis is in the spirit of the framework introduced by Tzavaras [A. Tzavaras, Commun. Math. Sci., 3(2), 119–132, 2005] for systems of hyperbolic conservation laws.


hyperbolic relaxation, balance laws, relative entropy, weak dissipation

2010 Mathematics Subject Classification

35B35, 76N10

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