Finite Volume Schemes for Nonlinear Parabolic Problems: Another Regularization Method

Abstract. On one hand, the existence of a solution to degenerate parabolic equations, without a nonlinear convection term, can be proven using the results of Alt and Luckhaus, Minty and Kolmogorov. On the other hand, the proof of uniqueness of an entropy weak solution to a nonlinear scalar hyperbolic equation, first provided by Krushkov, has been extended in two directions: Carrillo has handled the case of degenerate parabolic equations including a nonlinear convection term, whereas Di Perna has proven the uniqueness of weaker solutions, namely Young measure entropy solutions. All of these results are reviewed in the course of a convergence result for two regularizations of a degenerate parabolic problem including a nonlinear convective term. The first regularization is classicaly obtained by adding a minimal diffusion, the second one is given by a finite volume scheme on unstructured meshes. The convergence result is therefore only based on L∞(Ω × (0, T )) and L2(0, T ;H1(Ω)) estimates, associated with the uniqueness result for a weaker sense for a solution.