No abstract

The usual weak formulation of parabolic problems, in the case where the data are in L 1 , does not ensure the uniqueness of the solution, thus we give here an "entropy" formulation, which allows us to achieve existence and uniqueness.

Abstract. We consider the nonlinear heat equation (with Leray-Lions operators) on an open bounded subset of R N with Dirichlet homogeneous boundary conditions. The initial condition is in L 1 and the right hand side is a smooth measure. We extend a previous notion of entropy solutions and prove that they coincide with the renormalized solutions.2000 Mathematics Subject Classification: 35K55, 35D99.

International audienceWe are interested here in the numerical approximation of a family of probability measures, solution of the Chapman-Kolmogorov equation associated to some non-diffusion Markov process with Uncountable state space. Such an equation contains a transport term and another term, which implies redistribution Of the probability mass on the whole space. All implicit finite Volume scheme is proposed, which is intermediate between an upstream weighting scheme and a modified Lax-Friedrichs one. Due to the seemingly unusual probability framework, a new weak bounded variation inequality had to be developed, in order to prove the convergence of the discretised transport term. Such an inequality may be used in other contexts, such as for the study of finite Volume approximations of scalar linear or nonlinear hyperbolic equations with initial data in $L^1$. Also, due to the redistribution term, the tightness of the family of approximate probability measures had to be proven. Numerical examples are provided, showing the efficiency of the implicit finite volume scheme and its potentiality to be helpful in an industrial reliability context