Long time behavior of solutions of an electroconvection model in $${\mathbb {R}}^2$$

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 \(L^2({{\mathbb {R}}}^2)\) 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 \(H^2({{\mathbb {R}}}^2)\) and a logarithmic growth in time of a quadratic moment of the charge density.