The effect of linear stratification on the stability of a rest state in the 2D inviscid Boussinesq system

TL;DR

This paper studies how linear stratification affects the stability duration of a rest state in the 2D inviscid Boussinesq system, revealing that stratification induces dispersion which prolongs stability under Sobolev regularity assumptions.

Contribution

It demonstrates that stratification induces dispersion in the 2D inviscid Boussinesq system, enabling stability results with minimal regularity assumptions and extending findings to the dispersive SQG equation.

Findings

Stability of the rest state persists for timescales of O(ε^{-4/3}) with small initial perturbations.

Stratification induces dispersion, which is key to controlling nonlinear effects.

Results apply to both Boussinesq and dispersive SQG equations.

Abstract

We investigate and quantify the effect of stratification on the stability time of a stably stratified rest state for the 2D inviscid Boussinesq system on $\mathbb{R}^2$. As an important consequence, we obtain stability of the steady state starting from an $\varepsilon$-sized initial perturbation of Sobolev regularity $H^{3^+}$ on a timescale $\mathcal{O}(\varepsilon^{-4/3})$. In our setting, stratification induces dispersion and at the core of our approach are inhomogeneous Strichartz estimates used to control nonlinear contributions. This allows to keep only $L^2-$based regularity assumptions on the initial perturbation, whereas previous works impose additional localizations to achieve this timescale. We prove the analogous result for the related dispersive SQG equation.