On parabolic partial differential equations with H\"older continuous diffusion coefficients

TL;DR

This paper establishes existence and regularity results for 1D parabolic PDEs with H"older continuous diffusion coefficients, relevant to liquid crystal models, explicitly linking solution regularity to coefficient regularity.

Contribution

It provides new regularity results for parabolic equations with H"older continuous coefficients, specifically applied to the Ericksen-Leslie model for nematic liquid crystals.

Findings

Existence of weak solutions under H"older continuity assumptions.

Explicit dependence of solution regularity on the diffusion coefficient's H"older exponent.

Application to liquid crystal modeling.

Abstract

We investigate existence and regularity of weak solutions of a 1-dimensional parabolic differential equation with a non-constant H\"older 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 H\"older continuity of the diffusion coefficient which comes from the physical background and the analysis of the Ericksen-Leslie model. Moreover, the dependence of the H\"older exponent of the solution is explicit on the H\"older exponent of the diffusion coefficient.