Well-posedness of a Debye type system endowed with a full wave equation

TL;DR

This paper establishes the well-posedness of a coupled transport-diffusion and wave system, using advanced functional analysis techniques, with results varying based on the spatial dimension and initial data size.

Contribution

It introduces a novel approach to prove well-posedness for a Debye-type system coupled with a wave equation, extending results to large initial data in one dimension.

Findings

Well-posedness proven for small initial data in multiple dimensions.

Global well-posedness in one dimension for arbitrarily large initial data.

Application of Chemin-Lerner spaces and Gagliardo-Nirenberg inequalities.

Abstract

We prove well-posedness for a transport-diffusion problem coupled with a wave equation for the potential. We assume that the initial data are small. A bilinear form in the spirit of Kato's proof for the Navier-Stokes equations is used, coupled with suitable estimates in Chemin-Lerner spaces. In the one dimensional case, we get well-posedness for arbitrarily large initial data by using Gagliardo-Nirenberg inequalities.