Parabolic Equations with Singular Coefficients and Boundary Data: Analysis and Numerical Simulations
Arshyn Altybay, Alibek Yeskermessuly
TL;DR
We study a linear parabolic equation in divergence form with singular coefficients and non-smooth boundary data, where products of distributions render classical approaches ill-posed. The authors develop a very weak solution framework based on regularisation and moderate nets, proving existence, uniqueness (via negligibility), and consistency with classical solutions in the smooth regime. They establish a Galerkin-based construction and uniform energy estimates for the homogeneous Dirichlet case, then extend the theory to non-homogeneous Dirichlet data using regularised problems and a lifting to handle boundary conditions. Numerical experiments in one spatial dimension demonstrate robustness to delta-type singularities and distributional boundary traces, validating the analytical results and illustrating the framework’s practical viability for highly singular inputs.
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.