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

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