×

Stability of rarefaction waves in viscous media. (English) Zbl 0861.35037

This is a study of the large time asymptotic behaviour of a weak rarefaction wave for the parabolic system \(u_t+f(u)_x=u_{xx}\) with a strictly hyperbolic first member. Solutions of perturbed rarefaction data converge to an approximate Burgers rarefaction wave for suitably small and localized initial perturbations.

MSC:

35K45 Initial value problems for second-order parabolic systems
35L67 Shocks and singularities for hyperbolic equations
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Friedman, A., Partial Differential Equation of Parabolic Type, Prentice-Hall, 1964. · Zbl 0144.34903
[2] Goodman, J., Szepessy, A. & Zumbrun, K., A remark on the stability of viscous shocks, preprint (1992) TRITA-NA-9211, Royal Inst. of Technology, S-100 44 Stockholm, to appear in SIAM J. Math. Anal
[3] Goodman, J., Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986) 325-344. · Zbl 0631.35058 · doi:10.1007/BF00276840
[4] Harebetian, E., Rarefactions and large-time behavior for parabolic equations and monotone schemes, Comm. Math. Phys. 114 (1986) 527-536. · Zbl 0645.65052 · doi:10.1007/BF01229452
[5] Hoff, D. & Smoller, J., Solutions in the large for certain nonlinear parabolic systems, Analyse non Lin. 2 (1985) 213-235. · Zbl 0578.35044
[6] Hopf, E., The partial differential equation u t +uu x =? xx , Comm. Pure Appl. Math. 3 (1950) 201-230. · Zbl 0039.10403 · doi:10.1002/cpa.3160030302
[7] Il’in, A. M. & Oleinik, O. A., Behavior of the solution of the Cauchy problem for certain quasi linear equations for unbounded increase of time, Amer. Math. Soc. Translations, Ser. 2, 42 (1964) 19-23.
[8] Kawashima, S., Matsumura, A. & Nishihara, K., Asymptotic behavior of the solutions for the equations of a viscous head-conductive gas. Proc. Japan Acad. 62 (1986) 249-252. · Zbl 0624.76097 · doi:10.3792/pjaa.62.249
[9] Levi, E. E., Sulle equazioni lineari totalmente ellittiche alle derivate parziali, Rend. Circ. Mat. Palermo, 24 (1907) 275-317. · JFM 38.0402.01 · doi:10.1007/BF03015067
[10] Liu, T.-P., Linear and nonlinear large-time behavior of solutions to general systems of hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977) 767-797. · Zbl 0358.35014 · doi:10.1002/cpa.3160300605
[11] Liu, T.-P., Interaction of nonlinear hyperbolic waves, in Nonlinear Analysis, Eds. F.-C. Liu & T.-P. Liu, World Scientific, 1991, 171-184.
[12] Liu, T.-P., Nonlinear Stability of Shock Waves for Viscous Conservation Laws, Memoirs of Amer. Math. Soc., 328 (1986).
[13] Liu, T.-P. Pointwise convergence to shock waves for the system of viscous conservation laws, to appear.
[14] Liu, T.-P. & Zeng, Y., Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, preprint (1994).
[15] Liu, T.-P. & Zumbrun, K., Nonlinear stability of an undercompressive shock of complex Burgers equation, to appear, Comm. Math. Phys. · Zbl 0821.35123
[16] Liu, T.-P. & Zumbrun, K., On stability of general undercompressive viscous shock waves, preprint (1994). · Zbl 0840.76028
[17] Matsumura, A. & Nishihara K., Asymptotics towards the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan. J. Appl. Math. 3 (1986) 1-13. · Zbl 0612.76086 · doi:10.1007/BF03167088
[18] Matsumura, A. & Nishihara, K. Global stability of the rarefaction wave of a one-dimensional model system for compressible gas, Comm. Math. Phys. 144 (1992), 335-335. · Zbl 0745.76069 · doi:10.1007/BF02101095
[19] Smoller, J., Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983. · Zbl 0508.35002
[20] Szepessy, A. & Xin, Z., Nonlinear stability of viscous shock waves, Arch. Rational Mech. Anal. 122 (1993) 53-103. · Zbl 0803.35097 · doi:10.1007/BF01816555
[21] Xin, Z., Asymptotic stability of rarefaction waves for 2{\(\times\)}2 viscous hyperbolic conservation laws, J. Diff. Eqs. 73 (1988) 45-77. · Zbl 0669.35072 · doi:10.1016/0022-0396(88)90117-9
[22] Zingano, P., Thesis, New York University, 1990.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.