期刊
JOURNAL OF HYPERBOLIC DIFFERENTIAL EQUATIONS
卷 7, 期 1, 页码 1-67出版社
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0219891610002062
关键词
Degenerate hyperbolic-parabolic equation; conservation law; Leray-Lions type operator; non-Lipschitz flux; entropy solution; existence; uniqueness; finite volume scheme; discrete duality; convergence
资金
- FONDECYT [1070682]
- Research Council of Norway
We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnes [43]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and hyperbolic L-infinity weak-star compactness arguments (i.e. propagation of compactness along the lines of Tartar, DiPerna,...). Our results cover the case of non-Lipschitz nonlinearities.
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