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

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