Long Time Behavior of Solutions of an Electroconvection Model in $\R^2$

TL;DR

This paper analyzes the long-term decay behavior of solutions to a 2D electroconvection model, showing solutions decay at the same rate as the linear system and establishing bounds in various function spaces.

Contribution

It proves sharp $L^2$ decay rates for the nonlinear electroconvection system by comparing it to the linear case and establishing new bounds in $H^2$ and quadratic moments.

Findings

Solutions decay in $L^2$ at the same rate as the linear system.

The difference between nonlinear and linear solutions decays faster.

Bounds are established for decay in $H^2$ and quadratic moments.

Abstract

We consider a two dimensional electroconvection model which consists of a nonlinear and nonlocal system coupling the evolutions of a charge distribution and a fluid. We show that the solutions decay in time in $L^2(\Rr^2)$ at the same sharp rate as the linear uncoupled system. This is achieved by proving that the difference between the nonlinear and linear evolution decays at a faster rate than the linear evolution. In order to prove the sharp $L^2$ decay we establish bounds for decay in $H^2(\Rr^2)$ and a logarithmic growth in time of a quadratic moment of the charge density.