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