Long time behavior of solutions of an electroconvection model in R²

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 L²(R²) 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 L² decay we establish bounds for decay in H²(R²) and a logarithmic growth in time of a quadratic moment of the charge density.