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

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