A local regularity result for Neumann parabolic problems with nonsmooth data

Abstract

In this work we analyze the relations between two different concepts of solution of the Neumann problem for a second order parabolic equation: the usual notions of weak solution and those of transposition solution, which allow well-posedness of problems with measure data. We give a regularity result for the transposition solution and we prove that, under smoothness assumptions for the principal part of the operator, the local regularity of the transposition solution is the same as that of the usual weak solution. As an interesting particular case, we present a rigorous proof of local continuity of the solution for a convection–diffusion problem with pointwise source term.