ASYMPTOTIC PROPERTIES OF THE BOUSSINESQ EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS

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 L2 norm of the velocity and its gradient, convergence of the L2 norm of Au, and an o(1)-type exponential growth for IIA3/2uIIL2. We also obtain that in the interior of the domain the gradient of the vorticity is bounded by a polynomial function of time.

Automated summary

We investigate the asymptotic properties of the Boussinesq equations with vanishing thermal diffusivity in a bounded domain with no-slip boundary conditions. The dissipations of the L2 norm of velocity and its gradient, the convergence of the L2 norm of Au, and an o(1)-type exponential growth for IIA3/2uIIL2 are shown. Additionally, we obtain that the gradient of vorticity in the interior of the domain is bounded by a polynomial function of time.