Asymptotic properties of the Boussinesq Equations with Dirichlet Boundary Conditions

Abstract

We address the asymptotic properties for the Boussinesq equations with vanishing thermal diffusivity in a bounded domain with no-slip boundary conditions. We show the dissipation of the $L^2$ norm of the velocity and its gradient, convergence of the $L^2$ norm of $Au$, and an $o(1)$-type exponential growth for $\Vert A^{3/2}u{\Vert }_{L^2}$. We also obtain that in the interior of the domain the gradient of the vorticity is bounded by a polynomial function of time.