ASYMPTOTIC BEHAVIOR OF SOLUTIONS TOWARD THE SUPERPOSITION OF CONTACT DISCONTINUITY AND SHOCK WAVE FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

doi:10.1016/S0252-9602(12)60025-3

Abstract

Abstract:

A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated. The asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave is established under some smallness conditions. To do this, we first construct a new viscous contact wave such that the momentum equation is satisfied exactly and then determine the shift of the viscous shock wave. By using them
together with an inequality concerning the heat kernel in the half space, we obtain the desired a priori estimates. The proof is based on the elementary energy method by the anti-derivative argument.

Key words: compressible Navier-Stokes equations, free boundary, superposition of shock wave and contact discontinuity, stability

CLC Number:

35Q30

Hakho Hong, Feimin Huang. ASYMPTOTIC BEHAVIOR OF SOLUTIONS TOWARD THE SUPERPOSITION OF CONTACT DISCONTINUITY AND SHOCK WAVE FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY[J].Acta mathematica scientia,Series B, 2012, 32(1): 389-412.

Trendmd

References

[1] Atkinson F V, Peletier L A. Similariry solutions of the nonlinear di?usion equation. Arch Rat Mech Anal, 1974, 54: 373–392

[2] Duan R, Liu H G, 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

[3] Duyn C T, Peletier L A. A class of similariry solution of the nonlinear di?usion equation. Nonlinear Anal, T M A, 1977, 1: 223–233

[4] Huang F, Li J, Matsumura A. Asymptotic stability of combination of viscous contact wave with rarefaction
waves for one-dimentional compressible Navier-Stokes system. Arch Rat Mech Anal, 2010, 197: 89–116

[5] Huang F, Matsumura A. Stability of a composite wave of two viscous shock waves for full compressible
Navier-Stokes equation. Commun Math Phys, 2009, 298: 841–861

[6] Huang F, Matsumura A, Shi X. Viscous shock wave and boundary layer solution to an inflow problem for
compressible viscous gas. Comm Math Phys, 2003, 239: 261–285

[7] Huang F, Matsumura A, Shi X. On the stability of contact discontinuity for compressible Navier-Stokes
equations with free boundary. Osaka J Math, 2004, 41: 193–210

[8] Huang F, Matsumura A, Xin Z P. Stability of contact discontinuity for the 1-D compressible Navier-Stokes
equations. Arch Rat Mech Anal, 2006, 179(1): 55–77

[9] Huang F, Shi X, Wang Y. Stability of viscous shock wave for compressible Navier-Stokes equations with
free boundary. Kinet Relat Models, 2010, 3(3): 409–425

[10] Huang F, Wang Y, Zhai X Y. Stability of viscous contact wave for compressible Navier-Stokes equations of general gas with free boundary. Acta Math Sci, 2010, 30B(6): 1906–1916

[11] Huang F, Xin Z P, Yang T. Contact discontinuity with general perturbation for gas motion. Adv Math, 2008, 219: 1246–1297

[12] Huang F, Zhao H. On the global stability of viscous discontinuity for the compressible Navier-Stokes
equations. Rend Sem Mat Univ Padova, 2003, 109: 283–305

[13] Kawashima S,Matsumura A.Asymptotic stabilityof traveling wave solutions ofsystems ofone-dimensional
gas motion. Comm Math Phys, 1985, 101: 97–127

[14] Kawashima S, Matsumura A, Nishihara K. Asymptotic behavior of solutions for the equations of a viscous
heat-conductive gas. Proc Japan Acad, Ser A, 1986, 62: 249–252

[15] Liu T P. Shock waves for compressible Navier-Stokes equations are stable. Comm Pure Appl Math, 1986,
39: 565–594

[16] Liu T P, Xin Z P. Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations. Comm
Math Phys, 1988, 118: 451–465

[17] Matsumura A, Mei M. Convergence to traveling fronts of solutions of the p-system with viscosity in the
presence of a boundary. Arch Rat Mech Anal, 1999, 146: 1–22

[18] Matsumura A, Nishihara K. On the stability of traveling waves of a one-dimensional model system for
compressible viscous gas. Japan J Appl Math, 1985, 2: 17–25

[19] Matsumura A, Nishihara K. Asymptotic toward the rarefaction waves of a one dimensional model system
for compressible viscous gas. Japan J Appl Math, 1986, 3: 1–13

[20] Matsumura A, Nishihara K. Global stability of the rarefaction waves of a one dimensional model system
for compressible viscous gas. Comm Math Phys, 1992, 144: 325–335

[21] Nishihara K, Yang T, Zhao H J. Nonlinear stability of strong rarefaction waves for compressible Navier-
Stokes equations. SIAM J Math Anal, 2004, 35: 1561–1597

[22] Pan T, Liu H, Nishihara K. Asymptotic behavior of a one-dimensional compressible viscous gas with free
boundary. SIAM J Math Anal, 2002, 34: 172–291