Asymptotic properties of the Boussinesq Equations with Dirichlet Boundary Conditions

TL;DR

This paper investigates the long-term behavior of solutions to the Boussinesq equations with zero thermal diffusivity under no-slip boundary conditions, revealing dissipation, convergence, and boundedness properties.

Contribution

It provides new insights into the asymptotic behavior of the Boussinesq equations with Dirichlet boundary conditions, including dissipation and convergence results.

Findings

Dissipation of the $L^2$ norm of velocity and its gradient.

Convergence of the $L^2$ norm of $Au$.

Polynomial bounds on the gradient of vorticity in the interior.

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.