# American Institute of Mathematical Sciences

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August  2018, 11(4): 795-819. doi: 10.3934/krm.2018032

## Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law

 1 Department of Applied Mathematics, Kumamoto University, Kumamoto 860-8555, Japan 2 Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan

Received  August 2017 Revised  December 2017 Published  April 2018

Fund Project: The first author's work was supported in part by Grant-in-Aid for Scientific Research (C) 16K05237 of Japan Society for the Promotion of Science. The second author's work was supported in part by Grant-in-Aid for Scientific Research (S) 25220702 of Japan Society for the Promotion of Science.

In the current paper, we consider large time behavior of solutions to scalar conservation laws with an artificial heat flux term. In the case where the heat flux is governed by Fourier's law, the equation is scalar viscous conservation laws. In this case, existence and asymptotic stability of one-dimensional viscous shock waves have been studied in several papers. The main concern in the current paper is a $2 × 2$ system of hyperbolic equations with relaxation which is derived by prescribing Cattaneo's law for the heat flux. We consider the one-dimensional Cauchy problem for the system of Cattaneo-type and show existence and asymptotic stability of viscous shock waves. We also obtain the convergence rate by utilizing the weighted energy method. By letting the relaxation time zero in the system of Cattaneo-type, the system is formally deduced to scalar viscous conservation laws of Fourier-type. This is a singular limit problem which occurs an initial layer. We also consider the singular limit problem associated with viscous shock waves.

Citation: Tohru Nakamura, Shuichi Kawashima. Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law. Kinetic & Related Models, 2018, 11 (4) : 795-819. doi: 10.3934/krm.2018032
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