Linear and nonlinear stability for the $3$D stratified Boussinesq equations with the horizontal viscosity and diffusivity

TL;DR

This paper studies the stability and decay properties of solutions to the 3D stratified Boussinesq equations with horizontal viscosity and diffusivity, revealing enhanced dissipation and anisotropic decay behaviors.

Contribution

It introduces a novel analysis of the 3D stratified Boussinesq equations using anisotropic Littlewood--Paley theory, highlighting the impact of stratification-induced dispersion on decay rates.

Findings

Decay rates are faster due to stratification dispersion.

The third velocity component exhibits enhanced dissipation.

Solutions show anisotropic and dispersive decay behavior.

Abstract

In this manuscript, we consider the $3$D Boussinesq equations for stably stratified fluids with the horizontal viscosity and thermal diffusivity and investigate the large time behavior of the solutions. Making use of the anisotropic Littlewood--Paley theory, we obtain their precise $L^1$-$L^p$ decay estimates, which provide us information on both the anisotropic and dispersive structure of the system. More precisely, we reveal that the dispersion from the skew symmetric terms of stratification makes the decay rates of some portions of the solutions faster and furthermore the third component of the velocity field exhibit the enhanced dissipative effect, which provides the additional fast decay rate.