Entropy Formulation for Parabolic Degenerate Equations with General Dirichlet Boundary Conditions and Application to the Convergence of FV Methods

This paper is devoted to the analysis and the approximation of parabolic hyperbolic degenerate problems defined on bounded domains with nonhomogeneous boundary conditions. It consists of two parts. The first part is devoted to the definition of an original notion of entropy solutions to the continuous problem, which can be adapted to define a notion of measure-valued solutions, or entropy process solutions. The uniqueness of such solutions is established. In the second part, the convergence of the finite volume method is proved. This result relies on (weak) estimates and on the theorem of uniqueness of the first part. It also entails the existence of a solution to the continuous problem.