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.

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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

We study the Cauchy problem in R N for the parabolic equationwhich can degenerate into a hyperbolic equation for some intervals of values of u. In the context of conservation laws (the case ϕ ≡ 0), it is known that an entropy solution can be non-unique when F has singularities. We show the uniqueness of an entropy solution to the general parabolic problem for all L ∞ initial datum, under the isotropic condition on the flux F known for conservation laws. The only assumption on the diffusion term is that ϕ is a non-decreasing continuous function. Mathematics Subject Classification (2000). Primary 35K65; Secondary 35A05.

This work is intended to provide a convenient tool for the mathematical analysis of a particular kind of finite volume approximation which can be used, for instance, in the context of nonlinear and/or anisotropic diffusion operators in 3D. Following the so-called DDFV (Disctere Duality Finite Volume) approach developed by F. Hermeline and by K. Domelevo and P. Omnès in 2D, we consider a "double" covering T of a three-dimensional domain by a rather general primal mesh and by a well-chosen "dual" mesh. The associated discrete divergence operator div T is obtained by the standard finite volume approach. A simple and consistent discrete gradient operator ∇ T is defined by local affine interpolation that takes into account the geometry of the double mesh. Under mild geometrical constraints on the choice of the dual volumes, we show that −div T , ∇ T are linked by the "discrete duality property", which is an analogue of the integration-by-parts formula. The primal mesh need not be conformal, and its interfaces can be general polygons. We give several numerical examples for anisotropic linear diffusion problems; good convergence properties are observed. The sequel [3] of this paper will summarize some key discrete functional analysis tools for DDFV schemes and give applications to proving convergence of DDFV schemes for several nonlinear degenerate parabolic PDEs.

19 pagesWe consider the general degenerate hyperbolic-parabolic equation: \begin{equation}\label{E}\tag{E} u_t+\div f(u)-\Delta\phi(u)=0 \mbox{ in } Q = (0,T)\times\Omega,\;\;\;\; T>0,\;\;\;\Omega\subset\mathbb R^N ; \end{equation} with initial condition and the zero flux boundary condition. Here $\phi$ is a continuous non decreasing function. Following [B\"{u}rger, Frid and Karlsen, J. Math. Anal. Appl, 2007], we assume that $f$ is compactly supported (this is the case in several applications) and we define an appropriate notion of entropy solution. Using vanishing viscosity approximation, we prove existence of entropy solution for any space dimension $N\geq 1$ under a partial genuine nonlinearity assumption on $f$. Uniqueness is shown for the case $N=1$, using the idea of [Andreianov and Bouhsiss, J. Evol. Equ., 2004], nonlinear semigroup theory and a specific regularity result for one dimension

Abstract. We survey recent developments and give some new results concerning uniqueness of weak and renormalized solutions for degenerate parabolic problems of the form ut − div (a 0 (∇w) + F (w)) = f , u ∈ β(w) for a maximal monotone graph β, a Leray-Lions type nonlinearity a 0 , a continuous convection flux F , and an initial condition u| t=0 = u 0 . The main difficulty lies in taking boundary conditions into account. Here we consider Dirichlet or Neumann boundary conditions or the case of the problem in the whole space.We avoid the degeneracy that could make the problem hyperbolic in some regions; yet our starting point is the notion of entropy solution, notion that underlies the theory of general hyperbolic-parabolic-elliptic problems. Thus, we focus on techniques that are compatible with hyperbolic degeneracy, but here they serve to treat only the "parabolic-elliptic aspects". We revisit the derivation of entropy inequalities inside the domain and up to the boundary; technique of "going to the boundary" in the Kato inequality for comparison of two solutions; uniqueness for renormalized solutions obtained via reduction to weak solutions. On several occasions, the results are achieved thanks to the notion of integral solution coming from the nonlinear semigroup theory.1. Introduction 1.1. A survey of literature. Study of degenerate parabolic problems has undergone a considerable progress in the last ten years, thanks to the fundamental paper of J. Carrillo [26] in which the Kruzhkov device of doubling of variables was extended to hyperbolic-parabolic-elliptic problems of the form j(v)−div(f (v)+∇ϕ(v)) = 0, and a technique for treating the homogeneous Dirichlet boundary conditions was put forward. In [26], the appropriate notion of entropy solution was established, and this definition (or, sometimes, parts of the uniqueness techniques of [26]) led to many developments; among them, let us mention [2,3,5,4,6,8,9,10,11,12,13,18,19,24,25,27,29,30,34,37,38,39,41,42,45,46,47,48,52,53,57,58,59]. Also numerical aspects of the problem were investigated; see, e.g., [7,32,33,35,40,49].The notion of entropy solution (or, as in the present paper, entropy solutions techniques used on weak solutions) was retained by most of the authors; yet, let us mention the version of Bendahmane and Karlsen [18,19] We thank Safimba Soma for fruitful discussions on the subject of this note, and the anonymous referee for a careful reading and very pertinent remarks.