NONLINEAR STABILITY OF VISCOUS SHOCK WAVES FOR ONE-DIMENSIONAL NONISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH A CLASS OF LARGE INITIAL PERTURBATION

doi:10.1016/S0252-9602(18)30797-5

Abstract

Abstract:

We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier-Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous shock waves are shown to be time asymptotically stable under large initial perturbation with no restriction on the range of the adiabatic exponent provided that the strengths of the viscous shock waves are assumed to be sufficiently small.The proofs are based on the nonlinear energy estimates and the crucial step is to obtain the positive lower and upper bounds of the density and the temperature which are uniformly in time and space.

Key words: One-dimensional nonisentropic compressible Navier-Stokes equations, viscous shock waves, nonlinear stability, large initial perturbation

Shaojun TANG, Lan ZHANG. NONLINEAR STABILITY OF VISCOUS SHOCK WAVES FOR ONE-DIMENSIONAL NONISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH A CLASS OF LARGE INITIAL PERTURBATION[J].Acta mathematica scientia,Series B, 2018, 38(3): 973-1000.

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References

[1] Duan R, Liu H X, Zhao H J. Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation. Trans Amer Math Soc, 2009, 361(1):453-493
[2] Chapman S, Cowling T. The Mathematical Theory of Non-uniform Gases. 3rd ed. London:Cambrige University Press, 1970
[3] Fan L L, Liu H X, Wang T, et al. Inflow problem for the one-dimensional compressible Navier-Stokes equations under large initial perturbation. J Differential Equations, 2014, 257(10):3521-3553
[4] Goodman J. Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch Rational Mech Anal, 1986, 95:325-344
[5] Hong H. Global stability of viscous contact wave for 1-D compressible Navier-Stokes equations. J Differential Equations, 2012, 252(5):3482-3505
[6] Hong H H, Wang T. Stability of stationary to the in flow problem for full compressible Navier-Stokes equations with a large initial perturbation. SIAM J Math Anal, 2017, 49(3):2138-2166
[7] Hong H H, Wang T. Large-time behavior of solutions to the in flow problem offull compressible NavierStokes equations with large perturbation. Nonlinearity, 2017, 30:3010-3039
[8] Huang B K, Wang L S, Xiao Q H. Global nonlinear stability of rarefaction waves for compressible NavierStokes equations with temperature and density dependent transport coefficients. Kinet Relat Models, 2016, 9(3):469-514
[9] Huang B K, Liao Y K. Global stability of combination of viscous contact wave with rarefaction wave for compressible Navier-Stokes equations with temperature-dependent viscosity. Preprint, 2016
[10] Hong H, Huang F M. Asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave for compressible Navier-Stokes equations with free boundary. Acta Mathematica Scientia, 2012, 32B(1):389-412
[11] Huang F M, Li J, Matsumura A. Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system. Arch Ration Mech Anal, 2010, 197(1):89-116
[12] Huang F M, Li J, Shi X D. Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space. Commun Math Sci, 2010, 8(3):639-654
[13] Huang F M, Matsumura A. Stability of a composite wave of two viscous shock waves for the full compressible Navier-Stokes equation. Comm Math Phys, 2009, 289(3):841-861
[14] Huang F M, Matsumura A, Xin Z P. Stability of contact discontinuities for the 1-D compressible NavierStokes equations. Arch Ration Mech Anal, 2006, 179(1):55-77
[15] Huang F M, Shi X D, Wang Y. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinet Relat Models, 2010, 3(3):409-425
[16] Huang F M, Wang T. Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system. Indiana Univ Math J, 2016, 65:1833-1875
[17] Huang F M, Xin Z P, Yang T. Contact discontinuity with general perturbations for gas motions. Adv Math, 2008, 219(4):1246-1297
[18] Huang F M, Zhao H J. On the global stability of contact discontinuity for compressible Navier-Stokes equations. Rend Sem Mat Univ Padova, 2003, 109:283-305
[19] Jiang S. Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains. Comm Math Phys, 1999, 200:181-193
[20] Jiang S. Remarks on the asymptotic behaviour of solutions to the compressible Navier-Stokes equations in the half-line. Proc Roy Soc Edinburgh Sect A, 2002, 132:627-638
[21] Jiang S, Zhang P. Global weak solutions to the Navier-Stokes equations for a 1D viscous polytropic ideal gas. Quart Appl Math, 2003, 61:435-449
[22] Kanel' Ya. A model system of equations for the one-dimensional motion of a gas (Russian). Differencial'nye Uravnenija, 1968, 4:721-734
[23] Kawashima S, Matsumura A. Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Comm Math Phys, 1985, 101(1):97-127
[24] Kawashima S, Matsumura A, Nishihara K. Asymptotic behaviour of solutions for the equations of a viscous heat-conductive gas. Proc Japan Acad, 1986, 62A:249-252
[25] Kazhikhove A V. Cauchy problem for viscous gas equations. Siberian Math J, 1982, 23:44-49
[26] Kazhikhov A V, Shelukhin V V. Unique golbal solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. J Appl Math Mech, 1977, 41(2):273-282; translated from Prikl Mat Meh, 1977, 41(2):282-291(Russian)
[27] Li J, Liang Z L. Some uniform estimates and large-iime behavior of solutions to one-dimensional compressible Navier-Stokes system in unbounded domains with large data. Arch Ration Mech Anal, 2016, 220(3):1195-1208
[28] Liu T P. Shock waves for compressible Navier-Stokes equations are stable. Comm Pure Appl Math, 1986, 39(5):565-594
[29] Liu T P, Xin Z P. Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations. Comm Math Phys, 1988, 118:451-465
[30] Liu T P, Xin Z P. Pointwise decay to contact discontinuities for systems of viscous conservation laws. Asian J Math, 1997, 1:34-84
[31] Liu H X, Yang T, Zhao H J, et al. One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data. SIAM Journal on Mathematical Analysis, 2014, 46(3):2185-2228
[32] Matsumura A, Mei M. Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch Ration Mech Anal, 1999, 146(1):1-22
[33] Matsumura A, Nishihara K. On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J Appl Math, 1985, 2:17-25
[34] Matsumura A, Nishihara K. Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas. Japan J Appl Math, 1986, 3(1):1-13
[35] Matsumura A, Nishihara K. Global asymptotics toward the rarefaction wave for solutions of viscous psystem with boundary effect. Quart Appl Math, 2000, 58(1):69-83
[36] Matsumura A, Nishihara K. Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas. Comm Math Phys, 1992, 144(2):325-335
[37] Nishihara K, Yang T, Zhao H J. Nonlinear stability of strong rarefaction waves for compressible NavierStokes equations. SIAM J Math Anal, 2004, 35(6):1561-1597
[38] Smoller J. Shock Waves and Reaction-Diffusion Equations. 2nd edition. Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences]. New York:Springer-Verlag, 1994
[39] Tan A, Yang T, Zhao H J, et al. Global solutions to the one-dimensional compressible Navier-Stokes-Poisson equations with large data. SIAM Journal on Mathematical Analysis, 2013, 45(2):547-571
[40] Wan L, Wang T, Zou Q Y. Stability of stationary solutions to the outflow problem for full compressible Navier-Stokes equations with large initial perturbation. Nonlinearity, 2016, 29(4):1329-1354
[41] Wang T, Zhao H J, Zou Q Y. One-dimensional compressible Navier-Stokes equations with large density oscillation. Kinetic and Related Models, 2013, 6(3):649-670
[42] Zumbrun K. Stability of large-amplitude shock waves of compressible Navier-Stokes equations. With an appendix by Helge Kristian Jenssen and Gregory Lyng. Handbook of Mathematical Fluid Dynamics. Vol. Ⅲ. Amsterdam:North-Holland, 2004
[43] He L, Tand S J, Wang T. Stability viscous shock waves for the one dimensional compressitle Novier-Stokes equations with density-dependent viscosity. Acta Mathematica Scientia, 2016, 36B(1):34-48