On parabolic partial differential equations with Hölder continuous diffusion coefficients

Abstract

We investigate existence and regularity of weak solutions of a 1-dimensional parabolic differential equation with a non-constant Hölder diffusion coefficient and a rough forcing term. Such an equation appears in studying the 1-dimensional Ericksen-Leslie model for nematic liquid crystals where our result applies. The result presented here uses the Hölder continuity of the diffusion coefficient which comes from the physical background and the analysis of the Ericksen-Leslie model. Moreover, the dependence of the Hölder exponent of the solution is explicit on the Hölder exponent of the diffusion coefficient.