Existence of Solutions for the Debye-H\"{u}ckel System with Low Regularity Initial Data

TL;DR

This paper proves the existence and uniqueness of solutions for the Debye-Hückel system with low regularity initial data, extending previous results by lowering the regularity requirements and establishing conditions for global solutions.

Contribution

The paper introduces new existence results for the Debye-Hückel system with initial data in lower regularity Besov spaces, improving upon prior regularity assumptions.

Findings

Existence of local solutions for initial data in specific Besov spaces.

Global solutions exist if initial data is sufficiently small.

Improved regularity index compared to previous studies.

Abstract

In this paper we study existence of solutions for the Cauchy problem of the Debye-H\"{u}ckel system with low regularity initial data. By using the Chemin-Lerner time-space estimate for the heat equation, we prove that there exists a unique local solution if the initial data belongs to the Besov space $\dot{B}^{s}_{p,q}(\mathbb{R}^{n})$ for $-3/2<s\leq-2+\frac{n}{2}$, $p=\frac{n}{s+2}$ and $1\leq q\leq \infty$, and furthermore, if the initial data is sufficiently small then the solution is global. This result improves the regularity index of the initial data space in previous results on this model.