Asymptotic stability and sharp decay rates to the linearly stratified Boussinesq equations in horizontally periodic strip domain

TL;DR

This paper establishes the global existence, convergence, and sharp decay rates of solutions to the multi-dimensional linearly stratified Boussinesq equations in a horizontally periodic strip, using spectral analysis and energy estimates.

Contribution

It provides the first sharp decay rates for temperature fluctuation and vertical velocity in all intermediate norms for these equations.

Findings

Proved global-in-time classical solutions in high Sobolev spaces.

Established convergence of temperature to an asymptotic profile.

Derived sharp decay rates for velocity and temperature fluctuations.

Abstract

We consider an initial boundary value problem of the multi-dimensional Boussinesq equations in the absence of thermal diffusion with velocity damping or velocity diffusion under the stress free boundary condition in horizontally periodic strip domain. We prove the global-in-time existence of classical solutions in high order Sobolev spaces satisfying high order compatibility conditions around the linearly stratified equilibrium, the convergence of the temperature to the asymptotic profile, and sharp decay rates of the velocity field and temperature fluctuation in all intermediate norms based on spectral analysis combined with energy estimates. To the best of our knowledge, our results provide first sharp decay rates for the temperature fluctuation and the vertical velocity to the linearly stratified Boussinesq equations in all intermediate norms.