American Institute of Mathematical Sciences

Advanced
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

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 & 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.  doi: 10.1090/S0002-9947-08-04637-0.  Google Scholar

[2]

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

[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.  doi: 10.1007/s00220-009-0843-z.  Google Scholar

[4]

Ja. Kanel', A model system of equations for the one-dimensional motion of a gas,, (Russian) Differencial'nye Uravnenija, 4 (1968), 721.   Google Scholar

[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.  doi: 10.1007/BF01212358.  Google Scholar

[6]

T.-P. Liu, Nonlinear stability of shock waves for viscous conservation laws,, Mem. Amer. Math. Soc., 56 (1985).   Google Scholar

[7]

T.-P. Liu, Shock waves for compressible Navier-Stokes equations are stable,, Comm. Pure Appl. Math., 39 (1986), 565.  doi: 10.1002/cpa.3160390502.  Google Scholar

[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.  doi: 10.1016/S0252-9602(10)60003-3.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s002050050134.  Google Scholar

[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.  doi: 10.1007/BF03167036.  Google Scholar

[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.  doi: 10.1007/BF03167088.  Google Scholar

[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.  doi: 10.1007/BF02101095.  Google Scholar

[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.   Google Scholar

[15]

A. Matsumura and K. Nishihara, "Global Solutions for Nonlinear Differential Equations-Mathematical Analysis on Compressible Viscous Fluids,", (Japanese), (2004).   Google Scholar

[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.  doi: 10.1137/S003614100342735X.  Google Scholar

[17]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", $2^{nd}$ edition, 258 (1994).   Google Scholar

[18]

K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations,, With an appendix by Helge Kristian Jenssen and Gregory Lyng, (2004), 311.   Google Scholar

show all references

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.  doi: 10.1090/S0002-9947-08-04637-0.  Google Scholar

[2]

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

[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.  doi: 10.1007/s00220-009-0843-z.  Google Scholar

[4]

Ja. Kanel', A model system of equations for the one-dimensional motion of a gas,, (Russian) Differencial'nye Uravnenija, 4 (1968), 721.   Google Scholar

[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.  doi: 10.1007/BF01212358.  Google Scholar

[6]

T.-P. Liu, Nonlinear stability of shock waves for viscous conservation laws,, Mem. Amer. Math. Soc., 56 (1985).   Google Scholar

[7]

T.-P. Liu, Shock waves for compressible Navier-Stokes equations are stable,, Comm. Pure Appl. Math., 39 (1986), 565.  doi: 10.1002/cpa.3160390502.  Google Scholar

[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.  doi: 10.1016/S0252-9602(10)60003-3.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s002050050134.  Google Scholar

[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.  doi: 10.1007/BF03167036.  Google Scholar

[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.  doi: 10.1007/BF03167088.  Google Scholar

[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.  doi: 10.1007/BF02101095.  Google Scholar

[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.   Google Scholar

[15]

A. Matsumura and K. Nishihara, "Global Solutions for Nonlinear Differential Equations-Mathematical Analysis on Compressible Viscous Fluids,", (Japanese), (2004).   Google Scholar

[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.  doi: 10.1137/S003614100342735X.  Google Scholar

[17]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", $2^{nd}$ edition, 258 (1994).   Google Scholar

[18]

K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations,, With an appendix by Helge Kristian Jenssen and Gregory Lyng, (2004), 311.   Google Scholar

[1]

Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic & Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409
[2]

Xulong Qin, Zheng-An Yao, Hongxing Zhao. One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries. Communications on Pure & Applied Analysis, 2008, 7 (2) : 373-381. doi: 10.3934/cpaa.2008.7.373
[3]

Ben Duan, Zhen Luo. Dynamics of vacuum states for one-dimensional full compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2543-2564. doi: 10.3934/cpaa.2013.12.2543
[4]

Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004
[5]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675
[6]

Yuming Qin, Lan Huang, Zhiyong Ma. Global existence and exponential stability in $H^4$ for the nonlinear compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1991-2012. doi: 10.3934/cpaa.2009.8.1991
[7]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[8]

Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595
[9]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041
[10]

Xinhua Zhao, Zilai Li. Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1421-1448. doi: 10.3934/cpaa.2020052
[11]

Xin Liu, Yongjin Lu, Xin-Guang Yang. Stability and dynamics for a nonlinear one-dimensional full compressible non-Newtonian fluids. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020071
[12]

Qi S. Zhang. An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5521-5523. doi: 10.3934/dcds.2013.33.5521
[13]

Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021
[14]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
[15]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[16]

Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609
[17]

Dong Li, Xinwei Yu. On some Liouville type theorems for the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4719-4733. doi: 10.3934/dcds.2014.34.4719
[18]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907
[19]

Fang Li, Bo You. Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 55-80. doi: 10.3934/dcdsb.2019172
[20]

Fang Li, Bo You, Yao Xu. Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4267-4284. doi: 10.3934/dcdsb.2018137

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences