Asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions

Xin, Zhou Ping

doi:10.1090/S0002-9947-1990-0970270-8

Asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions
HTML articles powered by AMS MathViewer

by Zhou Ping Xin

Trans. Amer. Math. Soc. 319 (1990), 805-820

DOI: https://doi.org/10.1090/S0002-9947-1990-0970270-8

PDF | Request permission

Abstract:

This paper concerns the large time behavior toward planar rarefaction waves of the solutions for scalar viscous conservation laws in several dimensions. It is shown that a planar rarefaction wave is nonlinearly stable in the sense that it is an asymptotic attractor for the viscous conservation law. This is proved by using a stability result of rarefaction wave for scalar viscous conservation laws in one dimension and an elementary ${L^2}$-energy method.

References

Eduard Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Comm. Math. Phys. 114 (1988), no. 4, 527–536. MR 929127
Jonathan Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986), no. 4, 325–344. MR 853782, DOI 10.1007/BF00276840
Jonathan Goodman, Stability of viscous scalar shock fronts in several dimensions, Trans. Amer. Math. Soc. 311 (1989), no. 2, 683–695. MR 978372, DOI 10.1090/S0002-9947-1989-0978372-9

Behavior of the solution of the Cauchy problem for certain quasilinear equations for unbounded increase of the time

42

Shuichi Kawashima and Akitaka Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985), no. 1, 97–127. MR 814544
Tai-Ping Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977), no. 6, 767–796. MR 499781, DOI 10.1002/cpa.3160300605
Tai-Ping Liu and Zhou Ping Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys. 118 (1988), no. 3, 451–465. MR 958806
A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. MR 748308, DOI 10.1007/978-1-4612-1116-7
Akitaka Matsumura, Asymptotics toward rarefaction wave of solutions of the Broadwell model of a discrete velocity gas, Japan J. Appl. Math. 4 (1987), no. 3, 489–502. MR 925621, DOI 10.1007/BF03167816
Akitaka Matsumura and Kenji 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), no. 1, 1–13. MR 899210, DOI 10.1007/BF03167088
Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
Zhou Ping Xin, Asymptotic stability of rarefaction waves for $2*2$ viscous hyperbolic conservation laws, J. Differential Equations 73 (1988), no. 1, 45–77. MR 938214, DOI 10.1016/0022-0396(88)90117-9
Zhou Ping Xin, Asymptotic stability of rarefaction waves for $2\times 2$ viscous hyperbolic conservation laws—the two-modes case, J. Differential Equations 78 (1989), no. 2, 191–219. MR 992146, DOI 10.1016/0022-0396(89)90063-6

Linear and quasi-linear equations of parabolic type