Parabolic Equations with Singular Coefficients and Boundary Data: Analysis and Numerical Simulations

TL;DR

This paper develops a framework for analyzing and numerically approximating linear parabolic equations with highly singular coefficients and boundary data, extending classical solution concepts to handle distributions.

Contribution

It introduces a very weak solution framework using regularization, establishing existence, uniqueness, and consistency with classical solutions for singular data.

Findings

Numerical simulations confirm robustness with delta-type potentials.

Framework handles highly singular boundary conditions.

Existence and uniqueness proven under minimal regularity.

Abstract

We investigate linear parabolic equations in divergence form with singular coefficients and non-smooth boundary data. When the diffusion, drift, or potential terms, as well as the initial or boundary conditions, are distributions rather than functions, classical and weak solution concepts become inadequate due to the ill-posedness of products involving distributions. To overcome this, we introduce a framework of very weak solutions based on regularization techniques and the theory of moderate nets. Existence of very weak solutions is established under minimal regularity assumptions. We further prove consistency with classical solutions when the data are smooth and demonstrate uniqueness via negligibility arguments. Finally, we present numerical computations that illustrate the robustness of the very weak solution framework in handling highly singular inputs, including delta-type potentials and distributional boundary traces.