Abstract. We consider a class of doubly nonlinear degenerate hyperbolicparabolic 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 Omnès [41]) 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 some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" L ∞ weak-⋆ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, . . . ). Our results cover the case of non-Lipschitz nonlinearities.

We characterize the vanishing viscosity limit for multi-dimensional conservation laws of the form) is assumed locally Lipschitz continuous in the unknown u and piecewise constant in the space variable x; the discontinuities of f(•, u) are contained in the union of a locally finite number of sufficiently smooth hypersurfaces of R N . We define "G V V -entropy solutions" (this formulation is a particular case of the one of [3]); the definition readily implies the uniqueness and the L 1 contraction principle for the G V V -entropy solutions. Our formulation is compatible with the standard vanishing viscosity approximation0, of the conservation law. We show that, provided u ε enjoys an ε-uniform L ∞ bound and the flux f(x, •) is non-degenerately nonlinear, vanishing viscosity approximations u ε converge as ε ↓ 0 to the unique G V V -entropy solution of the conservation law with discontinuous flux.

We study the approximation by finite volume methods of the model parabolicelliptic problem b(v)t = div (|Dv| p−2 Dv) on (0, T) × Ω ⊂ R × R d with an initial condition and the homogeneous Dirichlet boundary condition. Because of the nonlinearity in the elliptic term, a careful choice of the gradient approximation is needed. We prove the convergence of discrete solutions to the solution of the continuous problem as the discretization step h tends to 0, under the main hypotheses that the approximation of the operator div (|Dv| p−2 Dv) provided by the finite volume scheme is still monotone and coercive, and that the gradient approximation is exact on the affine functions of x ∈ Ω. An example of such a scheme is given for a class of two-dimensional meshes dual to triangular meshes, in particular for structured rectangular and hexagonal meshes. The proof uses the rewriting of the discrete problem under a "continuous" form. This permits us to directly apply the Alt-Luckhaus variational techniques which are known for the continuous case.

The notion of Kruzhkov entropy solution was extended by the first author in 2007 to conservation laws with a fractional laplacian diffusion term; this notion led to well-posedness for the Cauchy problem in the $L^\infty$-framework. In the present paper, we further motivate the introduction of entropy solutions, showing that in the case of fractional diffusion of order strictly less than one, uniqueness of a weak solution may fail.Comment: 23 page