A Kinetic Approach in Nonlinear Parabolic Problems with $L^1$-Data

Abstract

We consider the Cauchy-Dirichlet problem for a nonlinear parabolic equation with $L^1$ data. We show how the concept of kinetic formulation for conservation laws introduced by P.-L. Lions, B. Perthame and E. Tadmor [A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. 7 (1994), 169–191]<\i> can be be used to give a new proof of the existence of renormalized solutions. To illustrate this approach, we also extend the method to the case where the equation involves an additional gradient term.