  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.
