Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-1990-0970270-8
MathSciNet review: 970270
Full-text PDF

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 $ {L^2}$-energy method.


References [Enhancements On Off] (What's this?)

  • [1] E. Harabetian, Rarefaction and large time behavior for parabolic equations and monotone schemes, Comm. Math. Phys. 114 (1988), 527-536. MR 929127 (89d:35084)
  • [2] J. Goodman, Nonlinear asymptotic stability of viscous shock profile for conservation laws, Arch. Rational Mech. Anal. 95 (1986), 325-344. MR 853782 (88b:35127)
  • [3] -, Stability of viscous scalar shock fronts in several dimensions, Trans. Amer. Math. Soc. 311 (1989), 683-695. MR 978372 (89j:35059)
  • [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] S. Kawashima and A. Mastumura, Asymptotic stability of traveling wave solutions of system for one-dimensional gas motion, Comm. Math. Phys. 101 (1985), 97-127. MR 814544 (87h:35035)
  • [6] T. P. Liu, Linear and nonlinear large time behavior of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977), 767-796. MR 0499781 (58:17556)
  • [7] T. P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys. 118 (1988), 451-465. MR 958806 (89i:35112)
  • [8] A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Springer-Verlag, New York, 1984. MR 748308 (85e:35077)
  • [9] A. Mastumura, Asymptotics toward rarefaction wave of solutions of the Broadwell model of a discrete velocity gas, Japan J. Appl. Math. 4 (1987), 489-502. MR 925621 (89h:35044)
  • [10] A. Mastumura and K. 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), 1-13. MR 899210 (88e:35173)
  • [11] J. Smoller, Shock waves and reaction diffusion equations, Springer-Verlag. New York and Berlin, 1983. MR 688146 (84d:35002)
  • [12] Z. P. Xin, Asymptotic stability of rarefaction waves for $ 2 \times 2$ viscous hyperbolic conservation laws, J. Differential Equations 73 (1988), 45-77. MR 938214 (89h:35203)
  • [13] -, Asymptotic stability of rarefaction waves for $ 2 \times 2$ viscous hyperbolic conservation laws--the two modes case, J. Differential Equations 78 (1989), 191-219. MR 992146 (90h:35155)
  • [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.

Similar Articles

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, $ {L^2}$-energy method
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society