# American Institute of Mathematical Sciences

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  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 & Applied Analysis, 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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [5] E. Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Commun. Math. Phys., 114 (1988), 527-536.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [9] S. Kawashima, T. Yanagisawa and Y. Shizuta, Mixed problems for quasilinear symmetric hyperbolic systems, Proc. Japan Acad., 63 (1987), 243-246.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [26] L. C. Tsai, Viscous shock propagation with boundary effect, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 1-25.   Google Scholar [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.  Google Scholar

show all references

##### 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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [5] E. Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Commun. Math. Phys., 114 (1988), 527-536.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [9] S. Kawashima, T. Yanagisawa and Y. Shizuta, Mixed problems for quasilinear symmetric hyperbolic systems, Proc. Japan Acad., 63 (1987), 243-246.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [26] L. C. Tsai, Viscous shock propagation with boundary effect, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 1-25.   Google Scholar [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.  Google Scholar
 [1] M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849 [2] Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021064 [3] Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3759-3779. doi: 10.3934/dcds.2021015 [4] Huancheng Yao, Haiyan Yin, Changjiang Zhu. Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1297-1317. doi: 10.3934/cpaa.2021021 [5] Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $H^1$. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019 [6] Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 [7] Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 [8] Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044 [9] Mikhail Dokuchaev, Guanglu Zhou, Song Wang. A modification of Galerkin's method for option pricing. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021077 [10] Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018 [11] Hong-Yi Miao, Li Wang. Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021012 [12] Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329 [13] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 [14] Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002 [15] Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035 [16] Chiun-Chuan Chen, Hung-Yu Chien, Chih-Chiang Huang. A variational approach to three-phase traveling waves for a gradient system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021055 [17] Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 [18] Yumi Yahagi. Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021099 [19] Tong Li, Nitesh Mathur. Riemann problem for a non-strictly hyperbolic system in chemotaxis. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021128 [20] Andrew Comech, Elena Kopylova. Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021063

2019 Impact Factor: 1.105

Article outline