# American Institute of Mathematical Sciences

August  2019, 12(4): 923-944. doi: 10.3934/krm.2019035

## Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws

 1 Graduate School of Mathematics, Kyushu University, Fukuoka 819-0395, Japan 2 Department of Applied Mathematics, Kumamoto University, Kumamoto 860-8555, Japan 3 Faculty of Science and Engineering, Waseda University, Tokyo 169-8555, Japan

* Corresponding author: Kenta Nakamura

Received  November 2018 Published  May 2019

This paper is concerned with the rarefaction waves for a model system of hyperbolic balance laws in the whole space and in the half space. We prove the asymptotic stability of rarefaction waves under smallness assumptions on the initial perturbation and on the amplitude of the waves. The proof is based on the $L^2$ energy method.

Citation: Kenta Nakamura, Tohru Nakamura, Shuichi Kawashima. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinetic and Related Models, 2019, 12 (4) : 923-944. doi: 10.3934/krm.2019035
##### References:
 [1] E. Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Comm. Math. Phys., 114 (1988), 527-536.  doi: 10.1007/BF01229452. [2] 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. [3] A. M. Il'in and O. A. Oleinik, Asymptotic behavior of the solutions of Cauchy problem for certain quasilinear equations for large time, (in Russian), Mat. Sb., 51 (1960), 191-216. [4] 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. [5] S. Kawashima, A. Matsumura and K. Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat conductive gas, Proc. Japan Acad., 62 (1986), 249-252.  doi: 10.3792/pjaa.62.249. [6] S. Kawashima and Y. Nikkuni, Stability of rarefaction waves of for the discrete Boltzmann equations, Adv. Math. Sci. Appl., 12 (2002), 327-353. [7] S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas, Kyushu J, Math., 58 (2004), 211-250.  doi: 10.2206/kyushujm.58.211. [8] S. Kawashima, T. Yanagisawa and Y. Shizuta, Mixed problems for quasi-linear symmetric hyperbolic systems, Proc. Japan Acad., 63 (1987), 243-246.  doi: 10.3792/pjaa.63.243. [9] S. Kawashima and P. Zhu, Asymptotic stability of rarefaction wave for the Navier-Stokes equations for a compressible fluid wave in the half space, Arch. Rat. Mech. Anal., 194 (2009), 105-132.  doi: 10.1007/s00205-008-0191-8. [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] A. Matsumura, Asymptotic toward rarefaction wave of solutions of the Broadwell model of a discrete velocity gas, Japan J. Appl. Math., 4 (1987), 489-502.  doi: 10.1007/BF03167816. [12] A. Matsumura and K. Nishihara, Asymptotic toward the rarefaction waves of solutions of a one-dimensional model system for compressible viscous gas, Japan. J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088. [13] A. Matsumura and K. Nishihara, Global stability of the rarefaction waves of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys., 144 (1992), 325-335.  doi: 10.1007/BF02101095. [14] A. Matsumura and K. Nishihara, Global Solutions to Nonlinear Differential Equation - Mathematical Analysis of Compressible Viscous Flow (in Japanese), Amazon (POD), 2015. [15] 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. [16] 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. [17] K. Nishihara, T. Yang and H. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597.  doi: 10.1137/S003614100342735X. [18] S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75.  doi: 10.1007/BF01210792.

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##### References:
 [1] E. Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Comm. Math. Phys., 114 (1988), 527-536.  doi: 10.1007/BF01229452. [2] 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. [3] A. M. Il'in and O. A. Oleinik, Asymptotic behavior of the solutions of Cauchy problem for certain quasilinear equations for large time, (in Russian), Mat. Sb., 51 (1960), 191-216. [4] 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. [5] S. Kawashima, A. Matsumura and K. Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat conductive gas, Proc. Japan Acad., 62 (1986), 249-252.  doi: 10.3792/pjaa.62.249. [6] S. Kawashima and Y. Nikkuni, Stability of rarefaction waves of for the discrete Boltzmann equations, Adv. Math. Sci. Appl., 12 (2002), 327-353. [7] S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas, Kyushu J, Math., 58 (2004), 211-250.  doi: 10.2206/kyushujm.58.211. [8] S. Kawashima, T. Yanagisawa and Y. Shizuta, Mixed problems for quasi-linear symmetric hyperbolic systems, Proc. Japan Acad., 63 (1987), 243-246.  doi: 10.3792/pjaa.63.243. [9] S. Kawashima and P. Zhu, Asymptotic stability of rarefaction wave for the Navier-Stokes equations for a compressible fluid wave in the half space, Arch. Rat. Mech. Anal., 194 (2009), 105-132.  doi: 10.1007/s00205-008-0191-8. [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] A. Matsumura, Asymptotic toward rarefaction wave of solutions of the Broadwell model of a discrete velocity gas, Japan J. Appl. Math., 4 (1987), 489-502.  doi: 10.1007/BF03167816. [12] A. Matsumura and K. Nishihara, Asymptotic toward the rarefaction waves of solutions of a one-dimensional model system for compressible viscous gas, Japan. J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088. [13] A. Matsumura and K. Nishihara, Global stability of the rarefaction waves of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys., 144 (1992), 325-335.  doi: 10.1007/BF02101095. [14] A. Matsumura and K. Nishihara, Global Solutions to Nonlinear Differential Equation - Mathematical Analysis of Compressible Viscous Flow (in Japanese), Amazon (POD), 2015. [15] 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. [16] 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. [17] K. Nishihara, T. Yang and H. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597.  doi: 10.1137/S003614100342735X. [18] S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75.  doi: 10.1007/BF01210792.
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