In this paper we investigate the asymptotic stability of a composite wave consisting of two viscous shock waves for the full compressible Navier-Stokes equation. By introducing a new linear diffusion wave special to this case, we successfully prove that if the strengths of the viscous shock waves are suitably small with same order and also the initial perturbations which are not necessarily of zero integral are suitably small, the unique global solution in time to the full compressible Navier-Stokes equation exists and asymptotically tends toward the corresponding composite wave whose shifts (in space) of two viscous shock waves are uniquely determined by the initial perturbations. We then apply the idea to study a half space problem for the full compressible Navier-Stokes equation and obtain a similar result.
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Courant R., Friedrichs K.O.: Supersonic Flows and Shock Waves. Wiley-Interscience, New York (1948)
Goodman J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal. 95(4), 325–344 (1986)
Huang F.M., Matsumura A., Shi X.: On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary. Osaka J. Math. 41(1), 193–210 (2004)
Huang F.M., Matsumura A.: Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations. Arch. Rat. Mech. Anal. 179, 55–77 (2005)
Huang F.M., Xin Z.P., Yang T.: Contact discontinuity with general perturbation for gas motion. Adv. Math. 219(4), 1246–1297 (2008)
Huang F.M., Zhao H.J.: On the global stability of contact discontinuity for compressible Navier-Stokes equations. Rend. Sem. Mat. Univ. Padova 109, 283–305 (2003)
Kawashima S., Matsumura A.: Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Commun. Math. Phys. 101(1), 97–127 (1985)
Kawashima, S., Matsumura, A., Nishihara, K.: Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas. Proc. Japan Acad. 62, Ser. A, 249–252 (1986)
Liu T.-P.: Linear and nonlinear large time behavior of general systems of hyperbolic conservation laws. Comm. Pure Appl. Math. 30, 767–796 (1977)
Liu T.-P.: Nonlinear stability of shock waves for viscous conservation laws. Mem. Amer. Math. Soc. 56(329), 1–108 (1985)
Liu T.-P.: Shock waves for compressible Navier-Stokes equations are stable. Comm. Pure Appl. Math. 39, 565–594 (1986)
Liu T.-P.: Pointwise convergence to shock waves for viscous conservation laws. Comm. Pure Appl. Math. 50(11), 1113–1182 (1997)
Liu T.-P., Xin Z.P.: Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations. Commun. Math. Phys. 118(3), 451–465 (1988)
Liu T.-P., Xin Z.P.: Pointwise decay to contact discontinuities for systems of viscous conservation laws. Asian J. Math. 1, 34–84 (1997)
Matsumura A., Mei M.: Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch. Rat. Mech. Anal. 146, 1–22 (1999)
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. 2(1), 17–25 (1985)
Matsumura A., Nishihara K.: Asymptotics toward the rarefaction waves of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 3(1), 3–13 (1985)
Matsumura A., Nishihara K.: Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas. Commun. Math. Phys. 144, 325–335 (1992)
Matsumura, A., Nishihara, K.: Global Solutions for Nonlinear Differential Equations–Mathematical Analysis on Compressible Viscous Fluids (In Japanese). Nippon Hyoronsha, 2004
Nishihara K., Yang T., Zhao H.-J.: Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations. SIAM J. Math. Anal. 35(6), 1561–1597 (2004)
Smoller J.: Shock Waves and Reaction-Diffusion Equations, Second Edition. Springer-Verlag, New York (1994)
Szepessy A., Xin Z.P.: Nonlinear stability of viscous shock waves. Arch. Rat. Mech. Anal. 122, 53–103 (1993)
Xin, Z.P.: On nonlinear stability of contact discontinuities. In: Hyperbolic Problems: Theory, Numerics, Applications (Stony Brook, NY, 1994), River Edge, NJ: World Sci. Publishing, 1996, pp. 249–257
Zumbrun, K.: Stability of large-amplitude shock waves of compressible Navier-Stokes equations. In: Handbook of Mathematical Fluid Dynamics, Vol. III, Amsterdam: North-Holland, 2004, pp. 311–533
Research is supported in part by NSFC Grant No. 10471138, NSFC-NSAF Grant No. 10676037 and 973 project of China, Grant No. 2006CB805902, in part by Japan Society for the Promotion of Science, the Invitation Fellowship for Research in Japan (Short-Term).
Research is supported in part by Grant-in-Aid for Scientific Research (B) 19340037, Japan.
Communicated by P. Constantin
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Huang, F., Matsumura, A. Stability of a Composite Wave of Two Viscous Shock Waves for the Full Compressible Navier-Stokes Equation. Commun. Math. Phys. 289, 841–861 (2009). https://doi.org/10.1007/s00220-009-0843-z
- Shock Wave
- Asymptotic Stability
- Travel Wave Solution
- Rarefaction Wave
- Riemann Problem