Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity

TL;DR

This paper investigates the long-term behavior of solutions to the 2D Boussinesq equations with zero diffusivity across various domains, providing improved bounds on velocity and vorticity norms over time.

Contribution

It establishes new exponential bounds on the norms of solutions, enhancing previous results for the 2D Boussinesq equations with zero diffusivity.

Findings

Bounded the $H^2$ and $H^1$ norms of velocity and density by a single exponential

Extended long-time behavior analysis to different domain types

Improved upon earlier bounds for solution norms

Abstract

We address long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity in the cases of the torus, ${\mathbb R}^2$, and on a bounded domain with Lions or Dirichlet boundary conditions. In all the cases, we obtain bounds on the long time behavior for the norms of the velocity and the vorticity. In particular, we obtain that the norm $\Vert (u,\rho)\Vert_{H^2\times H^{1}}$ is bounded by a single exponential, improving earlier bounds.