Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. (English) Zbl 0763.35058

Hsiao, Ling; Liu, Tai-Ping

Commun. Math. Phys. 143, No. 3, 599-605 (1992).

Summary: We consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time- asymptotically. The hyperbolic model may be viewed as isentropic Euler equations with friction term added to the momentum equation to model gas flow through a porous media. In this case our result justifies Darcy’s law time-asymptotically. Our model may also be viewed as an elastic model with damping.

MSC:

35L65	Hyperbolic conservation laws
35Q35	PDEs in connection with fluid mechanics
76S05	Flows in porous media; filtration; seepage
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

isentropic Euler equations; Darcy’s law; elastic model with damping

References:

[1]	Duyn, C.T., Van Peletier, L.A.: Nonlinear analysis. T.M.A.1, 223–233 (1977)
[2]	Liu, T.-P.: Nonlinear hyperbolic-parabolic partial differential equations. Nonlinear Analysis, Proceedings, 1989 Conference. Liu, F.C., Liu, T.P. (eds.), pp. 161–170. Academia Sinica, Taipei, R.O.C.: World Scientific · Zbl 0819.35101
[3]	Matzumura, A.: Nonlinear hyperbolic equations and related topics in fluid dynamics. Nishida, T. (ed.). Pub. Math. D’Orsay, 53–57 (1978)
[4]	Nishida, T.: Nonlinear hyperbolic equations and related topics in fluid dynamics. Nishida, T. (ed.) Pub. Math. D’Orsay, 46–53 (1978) · Zbl 0392.76065