One-dimensional compressible Navier-Stokes equations with large density oscillation

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September 2013, 6(3): 649-670. doi: 10.3934/krm.2013.6.649

One-dimensional compressible Navier-Stokes equations with large density oscillation

Tao Wang ^1, , Huijiang Zhao ^1, and Qingyang Zou ^1,

1.	School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China, China

Received December 2012 Revised February 2013 Published May 2013

Abstract
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This paper is concerned with nonlinear stability of viscous shock profiles for the one-dimensional isentropic compressible Navier-Stokes equations. For the case when the diffusion wave introduced in [6, 7] is excluded, such a problem has been studied in [5, 11] and local stability of weak viscous shock profiles is well-established, but for the corresponding result with large initial perturbation, fewer results have been obtained. Our main purpose is to deduce the corresponding nonlinear stability result with large initial perturbation by exploiting the elementary energy method. As a first step toward this goal, we show in this paper that for certain class of ``large" initial perturbation which can allow the initial density to have large oscillation, similar stability result still holds. Our analysis is based on the continuation argument and the technique developed by Kanel' in [4].

Keywords: large density oscillation., One-dimensional compressible Navier-Stokes equations, viscous shock profiles, nonlinear stability.

Mathematics Subject Classification: Primary: 35Q35; Secondary: 35B35, 76L05, 76N1.

Citation: Tao Wang, Huijiang Zhao, Qingyang Zou. One-dimensional compressible Navier-Stokes equations with large density oscillation. Kinetic and Related Models, 2013, 6 (3) : 649-670. doi: 10.3934/krm.2013.6.649

References:

[1]	R. Duan, H.-X. Liu and H.-J. Zhao, Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation, Trans. Amer. Math. Soc., 361 (2009), 453-493. doi: 10.1090/S0002-9947-08-04637-0.
[2]	J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal., 95 (1986), 325-344. doi: 10.1007/BF00276840.
[3]	F.-M. Huang and A. Matsumura, Stability of a composite wave of two viscous shock waves for the full compressible Navier-Stokes equation, Comm. Math. Phys., 289 (2009), 841-861. doi: 10.1007/s00220-009-0843-z.
[4]	Ja. Kanel', A model system of equations for the one-dimensional motion of a gas, (Russian) Differencial'nye Uravnenija, 4 (1968), 721-734; English translation in Diff. Eqns., 4 (1968), 374-380.
[5]	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.
[6]	T.-P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56 (1985), v+108 pp.
[7]	T.-P. Liu, Shock waves for compressible Navier-Stokes equations are stable, Comm. Pure Appl. Math., 39 (1986), 565-594. doi: 10.1002/cpa.3160390502.
[8]	T.-P. Liu and Y.-N. Zeng, On Green's function for hyperbolic-parabolic systems, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 1556-1572. doi: 10.1016/S0252-9602(10)60003-3.
[9]	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.
[10]	A. Matsumura and M. Mei, Convergence to travelling fronts of solutions of the $p$-system with viscosity in the presence of a boundary, Arch. Ration. Mech. Anal., 146 (1999), 1-22. doi: 10.1007/s002050050134.
[11]	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.
[12]	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.
[13]	A. Matsumura and K. Nishihara, Global stability of the rarefaction wave 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 asymptotics toward the rarefaction wave for solutions of viscous $p$-system with boundary effect, Quart. Appl. Math., 58 (2000), 69-83.
[15]	A. Matsumura and K. Nishihara, "Global Solutions for Nonlinear Differential Equations-Mathematical Analysis on Compressible Viscous Fluids," (Japanese), Nippon Hyoronsha, 2004.
[16]	K. Nishihara, T. Yang and H.-J. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597. doi: 10.1137/S003614100342735X.
[17]	J. Smoller, "Shock Waves and Reaction-Diffusion Equations," $2^{nd}$ edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258, Springer-Verlag, New York, 1994.
[18]	K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, With an appendix by Helge Kristian Jenssen and Gregory Lyng, in "Handbook of Mathematical Fluid Dynamics," Vol. III, North-Holland, Amsterdam, (2004), 311-533.

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References:

[1]	R. Duan, H.-X. Liu and H.-J. Zhao, Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation, Trans. Amer. Math. Soc., 361 (2009), 453-493. doi: 10.1090/S0002-9947-08-04637-0.
[2]	J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal., 95 (1986), 325-344. doi: 10.1007/BF00276840.
[3]	F.-M. Huang and A. Matsumura, Stability of a composite wave of two viscous shock waves for the full compressible Navier-Stokes equation, Comm. Math. Phys., 289 (2009), 841-861. doi: 10.1007/s00220-009-0843-z.
[4]	Ja. Kanel', A model system of equations for the one-dimensional motion of a gas, (Russian) Differencial'nye Uravnenija, 4 (1968), 721-734; English translation in Diff. Eqns., 4 (1968), 374-380.
[5]	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.
[6]	T.-P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56 (1985), v+108 pp.
[7]	T.-P. Liu, Shock waves for compressible Navier-Stokes equations are stable, Comm. Pure Appl. Math., 39 (1986), 565-594. doi: 10.1002/cpa.3160390502.
[8]	T.-P. Liu and Y.-N. Zeng, On Green's function for hyperbolic-parabolic systems, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 1556-1572. doi: 10.1016/S0252-9602(10)60003-3.
[9]	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.
[10]	A. Matsumura and M. Mei, Convergence to travelling fronts of solutions of the $p$-system with viscosity in the presence of a boundary, Arch. Ration. Mech. Anal., 146 (1999), 1-22. doi: 10.1007/s002050050134.
[11]	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.
[12]	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.
[13]	A. Matsumura and K. Nishihara, Global stability of the rarefaction wave 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 asymptotics toward the rarefaction wave for solutions of viscous $p$-system with boundary effect, Quart. Appl. Math., 58 (2000), 69-83.
[15]	A. Matsumura and K. Nishihara, "Global Solutions for Nonlinear Differential Equations-Mathematical Analysis on Compressible Viscous Fluids," (Japanese), Nippon Hyoronsha, 2004.
[16]	K. Nishihara, T. Yang and H.-J. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597. doi: 10.1137/S003614100342735X.
[17]	J. Smoller, "Shock Waves and Reaction-Diffusion Equations," $2^{nd}$ edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258, Springer-Verlag, New York, 1994.
[18]	K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, With an appendix by Helge Kristian Jenssen and Gregory Lyng, in "Handbook of Mathematical Fluid Dynamics," Vol. III, North-Holland, Amsterdam, (2004), 311-533.

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