Degenerate regularization of forward-backward parabolic equations: the vanishing viscosity limit

Abstract

We address the vanishing viscosity limit of the regularized problem studied in Smarrazzo and Tesei [Arch Rat Mech Anal 2012 (in press)]. We show that the limiting points in a suitable topology of the family of solutions of the regularized problem can be regarded as suitably defined weak measure-valued solutions of the original problem. In general, these solutions are the sum of a regular term, which is absolutely continuous with respect to the Lebesgue measure, and a singular term, which is a Radon measure singular with respect to the other. By using a family of entropy inequalities, we prove that the singular term is nondecreasing in time. We also characterize the disintegration of the Young measure associated with the regular term, proving that it is a superposition of two Dirac masses with support on the branches of the graph of the nonlinearity $${\varphi}$$ .