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

Author:
Zhou Ping Xin

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

MSC:
Primary 35L65; Secondary 76L05, 76N10

MathSciNet review:
970270

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 -energy method.

**[1]**Eduard Harabetian,*Rarefactions and large time behavior for parabolic equations and monotone schemes*, Comm. Math. Phys.**114**(1988), no. 4, 527–536. MR**929127****[2]**Jonathan Goodman,*Nonlinear asymptotic stability of viscous shock profiles for conservation laws*, Arch. Rational Mech. Anal.**95**(1986), no. 4, 325–344. MR**853782**, 10.1007/BF00276840**[3]**Jonathan Goodman,*Stability of viscous scalar shock fronts in several dimensions*, Trans. Amer. Math. Soc.**311**(1989), no. 2, 683–695. MR**978372**, 10.1090/S0002-9947-1989-0978372-9**[4]**A. M. Il'in and O. A. Oleinik,*Behavior of the solution of the Cauchy problem for certain quasilinear equations for unbounded increase of the time*, Amer. Math. Soc. Transl. (2)**42**(1964), 19-23.**[5]**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****[6]**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**0499781****[7]**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****[8]**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****[9]**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**, 10.1007/BF03167816**[10]**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**, 10.1007/BF03167088**[11]**Joel Smoller,*Shock waves and reaction-diffusion equations*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR**688146****[12]**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**, 10.1016/0022-0396(88)90117-9**[13]**Zhou Ping Xin,*Asymptotic stability of rarefaction waves for 2×2 viscous hyperbolic conservation laws—the two-modes case*, J. Differential Equations**78**(1989), no. 2, 191–219. MR**992146**, 10.1016/0022-0396(89)90063-6**[14]**O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva,*Linear and quasi-linear equations of parabolic type*, Amer. Math. Soc., Providence, R. I., 1968.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35L65,
76L05,
76N10

Retrieve articles in all journals with MSC: 35L65, 76L05, 76N10

Additional Information

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

Keywords:
Nonlinear stable,
viscous conservation law,
planar rarefaction wave,
-energy method

Article copyright:
© Copyright 1990
American Mathematical Society