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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
References:
[1]

R. E. Caflisch, Navier-Stokes and Boltzmann shock profiles for a model of gas dynamics, Comm. Pure Appl. Math., 32 (1979), 521-554. doi: 10.1002/cpa.3160320404.

[2]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Ration. Mech. Anal., 194 (2009), 221-251. doi: 10.1007/s00205-009-0220-2.

[3]

J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal., 95 (1986), 325-344. doi: 10.1007/BF00276840.

[4]

Y. Hu and R. Racke, Compressible Navier-Stokes equations with hyperbolic heat conduction, J. Hyperbolic Differ. Equ., 13 (2016), 233-247. doi: 10.1142/S0219891616500077.

[5]

A. M. Il'in and O. A. Oleĭnik, Asymptotic behavior of solutions of the Cauchy problem for some quasi-linear equations for large values of the time, Mat. Sb. (N.S.), 51 (1960), 191-216.

[6]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.

[7]

S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Comm. Pure Appl. Math., 47 (1994), 1547-1569. doi: 10.1002/cpa.3160471202.

[8]

A. Matsumura and M. Mei, Nonlinear stability of viscous shock profile for a non-convex system of viscoelasticity, Osaka J. Math., 34 (1997), 589-603.

[9]

A. Matsumura and K. Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2 (1985), 17-25. doi: 10.1007/BF03167036.

[10]

A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys., 165 (1994), 83-96. doi: 10.1007/BF02099739.

[11]

M. Mei, Stability of shock profiles for nonconvex scalar viscous conservation laws, Math. Models Methods Appl. Sci., 5 (1995), 279-296. doi: 10.1142/S0218202595000188.

[12]

S. Nishibata and M. Suzuki, Relaxation limit and initial layer to hydrodynamic models for semiconductors, J. Differential Equations, 249 (2010), 1385-1409. doi: 10.1016/j.jde.2010.06.008.

[13]

T. Nishida, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Comm. Math. Phys., 61 (1978), 119-148. doi: 10.1007/BF01609490.

[14]

R. Racke, Thermoelasticity with second sound---exponential stability in linear and non-linear 1-d, Math. Methods Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298.

[15]

S. Ukai and K. Asano, The Euler limit and initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J., 12 (1983), 311-332. doi: 10.14492/hokmj/1470081009.

show all references

References:
[1]

R. E. Caflisch, Navier-Stokes and Boltzmann shock profiles for a model of gas dynamics, Comm. Pure Appl. Math., 32 (1979), 521-554. doi: 10.1002/cpa.3160320404.

[2]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Ration. Mech. Anal., 194 (2009), 221-251. doi: 10.1007/s00205-009-0220-2.

[3]

J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal., 95 (1986), 325-344. doi: 10.1007/BF00276840.

[4]

Y. Hu and R. Racke, Compressible Navier-Stokes equations with hyperbolic heat conduction, J. Hyperbolic Differ. Equ., 13 (2016), 233-247. doi: 10.1142/S0219891616500077.

[5]

A. M. Il'in and O. A. Oleĭnik, Asymptotic behavior of solutions of the Cauchy problem for some quasi-linear equations for large values of the time, Mat. Sb. (N.S.), 51 (1960), 191-216.

[6]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.

[7]

S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Comm. Pure Appl. Math., 47 (1994), 1547-1569. doi: 10.1002/cpa.3160471202.

[8]

A. Matsumura and M. Mei, Nonlinear stability of viscous shock profile for a non-convex system of viscoelasticity, Osaka J. Math., 34 (1997), 589-603.

[9]

A. Matsumura and K. Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2 (1985), 17-25. doi: 10.1007/BF03167036.

[10]

A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys., 165 (1994), 83-96. doi: 10.1007/BF02099739.

[11]

M. Mei, Stability of shock profiles for nonconvex scalar viscous conservation laws, Math. Models Methods Appl. Sci., 5 (1995), 279-296. doi: 10.1142/S0218202595000188.

[12]

S. Nishibata and M. Suzuki, Relaxation limit and initial layer to hydrodynamic models for semiconductors, J. Differential Equations, 249 (2010), 1385-1409. doi: 10.1016/j.jde.2010.06.008.

[13]

T. Nishida, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Comm. Math. Phys., 61 (1978), 119-148. doi: 10.1007/BF01609490.

[14]

R. Racke, Thermoelasticity with second sound---exponential stability in linear and non-linear 1-d, Math. Methods Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298.

[15]

S. Ukai and K. Asano, The Euler limit and initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J., 12 (1983), 311-332. doi: 10.14492/hokmj/1470081009.

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