On the Three-dimensional Nernst-Planck-Boussinesq System

TL;DR

This paper investigates a complex three-dimensional Nernst-Planck-Boussinesq system modeling ionic electrodiffusion with temperature effects, proving global existence of solutions and their exponential decay to steady states.

Contribution

It introduces a new 3D model incorporating temperature variations and buoyancy, and establishes global weak solutions and their long-term behavior.

Findings

Proved global existence of weak solutions for large initial data.

Established exponential decay of solutions to steady states.

Analyzed the nonlinear structure influenced by temperature in the electromigration term.

Abstract

In this paper, we analyze a three-dimensional Nernst-Planck-Boussinesq (NPB) system that describes ionic electrodiffusion in an incompressible viscous fluid. This new model incorporates variational temperature and is forced by buoyancy force stemming from temperature and salinity fluctuations, enhancing its generality and realism. The electromigration term in the NPB system displays a complex nonlinear structure influenced by the reciprocal of the temperature that distinguishes its mathematical aspects from other electrodiffusion models studied in the literature. We address the global existence of weak solutions to the NPB system on the three-dimensional torus for large initial data. In addition, we study the long-time dynamics of these weak solutions and the associated relative entropies and establish their exponential decay in time to steady states.