On the Convergence of the Parabolic Approximation of a Conservation Law in Several Space Dimensions

. We give a proof of the convergence of the solution of the parabolic approximation u " t + div f(x; t; u " ) = "\Deltau " towards the entropic solution of the scalar conservation law u t + div f(x; t; u) = 0 in several space dimensions. For any initial condition u 0 2 L 1 (R N ) and for a large class of flux f , we prove the strong convergence in any L p loc space, using the notion of entropy process solution, which is a generalization of the measure-valued solutions of DiPerna. 1 Introduction We give a proof of the strong convergence in L p loc ; p ! 1 of the solution of the parabolic approximation: u " t + div f(x; t; u " ) = "\Deltau " ; x 2 R N ; t ? 0 (1) u " (x; 0) = u " 0 (x); x 2 R N (2) towards the entropic solution to the scalar conservation law u t + div f(x; t; u) = 0; x 2 R N ; t ? 0 (3) u(x; 0) = u 0 (x); x 2 R N (4) where u 0 2 L 1 (R N ), u " 0 denotes some approximation of u 0 such that ae ku " 0 k L 1 M u " 0 ! u 0 stro..