Existence of solutions to the Poisson--Nernst--Planck system with singular permanent charges in $\mathbb{R}^2$

TL;DR

This paper establishes the existence of solutions for the Poisson-Nernst-Planck system with singular permanent charges in two dimensions, using weighted spaces and energy estimates to handle non-integrable singularities.

Contribution

It introduces a novel approach transforming the system into a weighted parabolic form to prove local and global existence of solutions with singular charges.

Findings

Local existence of solutions via fixed-point argument.

Global existence under small initial data.

Handling non-integrable singularities with weighted spaces.

Abstract

In this paper, we study the well-posedness of Poisson-Nernst-Planck system with no-flux boundary condition and singular permanent charges in two dimension. The main difficulty comes from the lack of integrability of singular permanent charges. In order to overcome the difficulty, the main idea is to transform the system into another weighted parabolic system. By choosing suitable weighted spaces, local existence of solutions can be obtained based on a fixed-point argument. Moreover, we also proved global existence by energy estimates under the smallness assumption on initial data.