# American Institute of Mathematical Sciences

July & August  2021, 20(7&8): 2441-2474. doi: 10.3934/cpaa.2021049

## Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law

 1 School of Mathematics and Statistics and Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, China 2 College of Mathematics, Sichuan University, Chengdu 610065, China

* Corresponding author

Dedicated to Professor Shuxing Chen on the occasion of his 80th birthday

Received  December 2020 Revised  February 2021 Published  July & August 2021 Early access  March 2021

Fund Project: Lin He was supported by the National Natural Science Foundation of China under contract 12001388 and the Fundamental Research Funds for the Central Universities under contract YJ201962 and the Sichuan Youth Science and Technology Foundation under contract 2021JDTD0024. Huijiang Zhao was supported by two grants from the National Natural Science Foundation of China under contracts 11731008 and 11671309, respectively

This paper is concerned with the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem and the initial-boundary value problem in the half space with impermeable wall boundary condition for a scalar conservation laws with an artificial heat flux satisfying Cattaneo's law. In our results, although the $L^2\cap L^\infty-$norm of the initial perturbation is assumed to be small, the $H^1-$norm of the first order derivative of the initial perturbation with respect to the spatial variable can indeed be large. Moreover the far fields of the artificial heat flux can be different. Our analysis is based on the $L^2$ energy method.

Citation: Yinsong Bai, Lin He, Huijiang Zhao. Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2441-2474. doi: 10.3934/cpaa.2021049
##### References:
 [1] 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. [2] H. Freistuhler and D. Serre, $L^1$ stability of shock waves in scalar viscous conservation laws, Commun. Pure Appl. Math., 51 (1998), 291-301.  doi: 10.1002/(SICI)1097-0312(199803)51:3<291::AID-CPA4>3.3.CO;2-S. [3] L. Hsiao and T. P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Commun. Math. Phys., 143 (1992), 599-605. [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] E. Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Commun. Math. Phys., 114 (1988), 527-536. [6] Y. Hattori and K. Nishihara, A note on the stability of the rarefaction wave of the Burgers equation, Japan J. Indust. Appl. Math., 8 (1991), 85-96.  doi: 10.1007/BF03167186. [7] A. M. Il'in and O. A. Oleinik, Asymptotic behavior of solutions of the Cauchy problem for some quasilinear equations for large values of the time, Mat. Sb. (N.S.), 51 (1960), 191-216. [8] S. Kawashima and K. Kurata, Hardy type inequality and application to the stability of degenerate stationary waves, J. Funct. Anal., 257 (2009), 1-19.  doi: 10.1016/j.jfa.2009.04.003. [9] S. Kawashima, T. Yanagisawa and Y. Shizuta, Mixed problems for quasilinear symmetric hyperbolic systems, Proc. Japan Acad., 63 (1987), 243-246. [10] T. P. Liu, A. Matsumura and K. Nishihara, Behavior of solutions for the Burgers equations with boundary corresponding to rarefaction waves, SIAM. J. Math. Anal., 29 (1998), 293-308.  doi: 10.1137/S0036141096306005. [11] T. P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect, J. Differ. Equ., 133 (1997), 296-320.  doi: 10.1006/jdeq.1996.3217. [12] T. P. Liu and S. H. Yu, Propagation of a stationary shock layer in the presence of a boundary, Arch. Ration. Mech. Anal., 139 (1997), 57-82.  doi: 10.1007/s002050050047. [13] A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, Methods Appl. Anal., 8 (2001), 645-666.  doi: 10.4310/MAA.2001.v8.n4.a14. [14] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088. [15] A. Matsumura and K. Nishihara, Global stability of the rarefaction waves of a one-dimensional model system for compressible viscous gas, Commun. Math. Phys., 144 (1992), 325-335. [16] 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. [17] K. Nakamura, T. Nakamura and S. Kawashima, Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws, Kinet. Relat. Models, 12 (2019), 923-944.  doi: 10.3934/krm.2019035. [18] T. Nakamura, Asymptotic decay toward the rarefaction waves of solutions for viscous conservation laws in a one dimensional half space, SIAM J. Math. Anal., 34 (2003), 1308-1317.  doi: 10.1137/S003614100240693X. [19] T. Nakamura and S. Kawashima, Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law, Kinet. Relat. Models, 11 (2018), 795–819. doi: 10.3934/krm. 2018032. [20] K. Nishihara, Boundary effect on a stationary viscous shock wave for scalar viscous conservation laws, J. Math. Anal. Appl., 255 (2001), 535-550.  doi: 10.1006/jmaa.2000.7255. [21] K. Nishihara and H. J. Zhao, Convergence rates to viscous shock profile for general scalar viscous conservation laws with large initial disturbance, J. Math. Soc. Japan, 54 (2002), 447-466.  doi: 10.2969/jmsj/05420447. [22] R. Racke, Thermoelasticity with second sound–exponential stability in linear and nonlinear 1-d, Math. Methods Appl. Sci., 25 (2002), 409-441.  doi: 10.1002/mma.298. [23] S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Commun. Math. Phys., 104 (1986), 49-75. [24] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^nd$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [25] W. A. Strauss, Decay and asymptotics for $\Box u=F(u)$, J. Funct. Anal., 2 (1968), 409-457.  doi: 10.1016/0022-1236(68)90004-9. [26] L. C. Tsai, Viscous shock propagation with boundary effect, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 1-25. [27] Y. Ueda, T. Nakamura and and S. Kawashima, Stability of degenerate stationary waves for viscous gases, Arch. Ration. Mech. Anal., 198 (2010), 735-762.  doi: 10.1007/s00205-010-0369-8.

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##### References:
 [1] 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. [2] H. Freistuhler and D. Serre, $L^1$ stability of shock waves in scalar viscous conservation laws, Commun. Pure Appl. Math., 51 (1998), 291-301.  doi: 10.1002/(SICI)1097-0312(199803)51:3<291::AID-CPA4>3.3.CO;2-S. [3] L. Hsiao and T. P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Commun. Math. Phys., 143 (1992), 599-605. [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] E. Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Commun. Math. Phys., 114 (1988), 527-536. [6] Y. Hattori and K. Nishihara, A note on the stability of the rarefaction wave of the Burgers equation, Japan J. Indust. Appl. Math., 8 (1991), 85-96.  doi: 10.1007/BF03167186. [7] A. M. Il'in and O. A. Oleinik, Asymptotic behavior of solutions of the Cauchy problem for some quasilinear equations for large values of the time, Mat. Sb. (N.S.), 51 (1960), 191-216. [8] S. Kawashima and K. Kurata, Hardy type inequality and application to the stability of degenerate stationary waves, J. Funct. Anal., 257 (2009), 1-19.  doi: 10.1016/j.jfa.2009.04.003. [9] S. Kawashima, T. Yanagisawa and Y. Shizuta, Mixed problems for quasilinear symmetric hyperbolic systems, Proc. Japan Acad., 63 (1987), 243-246. [10] T. P. Liu, A. Matsumura and K. Nishihara, Behavior of solutions for the Burgers equations with boundary corresponding to rarefaction waves, SIAM. J. Math. Anal., 29 (1998), 293-308.  doi: 10.1137/S0036141096306005. [11] T. P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect, J. Differ. Equ., 133 (1997), 296-320.  doi: 10.1006/jdeq.1996.3217. [12] T. P. Liu and S. H. Yu, Propagation of a stationary shock layer in the presence of a boundary, Arch. Ration. Mech. Anal., 139 (1997), 57-82.  doi: 10.1007/s002050050047. [13] A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, Methods Appl. Anal., 8 (2001), 645-666.  doi: 10.4310/MAA.2001.v8.n4.a14. [14] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088. [15] A. Matsumura and K. Nishihara, Global stability of the rarefaction waves of a one-dimensional model system for compressible viscous gas, Commun. Math. Phys., 144 (1992), 325-335. [16] 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. [17] K. Nakamura, T. Nakamura and S. Kawashima, Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws, Kinet. Relat. Models, 12 (2019), 923-944.  doi: 10.3934/krm.2019035. [18] T. Nakamura, Asymptotic decay toward the rarefaction waves of solutions for viscous conservation laws in a one dimensional half space, SIAM J. Math. Anal., 34 (2003), 1308-1317.  doi: 10.1137/S003614100240693X. [19] T. Nakamura and S. Kawashima, Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law, Kinet. Relat. Models, 11 (2018), 795–819. doi: 10.3934/krm. 2018032. [20] K. Nishihara, Boundary effect on a stationary viscous shock wave for scalar viscous conservation laws, J. Math. Anal. Appl., 255 (2001), 535-550.  doi: 10.1006/jmaa.2000.7255. [21] K. Nishihara and H. J. Zhao, Convergence rates to viscous shock profile for general scalar viscous conservation laws with large initial disturbance, J. Math. Soc. Japan, 54 (2002), 447-466.  doi: 10.2969/jmsj/05420447. [22] R. Racke, Thermoelasticity with second sound–exponential stability in linear and nonlinear 1-d, Math. Methods Appl. Sci., 25 (2002), 409-441.  doi: 10.1002/mma.298. [23] S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Commun. Math. Phys., 104 (1986), 49-75. [24] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^nd$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [25] W. A. Strauss, Decay and asymptotics for $\Box u=F(u)$, J. Funct. Anal., 2 (1968), 409-457.  doi: 10.1016/0022-1236(68)90004-9. [26] L. C. Tsai, Viscous shock propagation with boundary effect, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 1-25. [27] Y. Ueda, T. Nakamura and and S. Kawashima, Stability of degenerate stationary waves for viscous gases, Arch. Ration. Mech. Anal., 198 (2010), 735-762.  doi: 10.1007/s00205-010-0369-8.
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