The effect of linear stratification on the stability of a rest state in the 2D inviscid Boussinesq system
Catalina Jurja, Haram Ko
TL;DR
The paper investigates the long-time stability of a stratified rest state for the 2D inviscid Boussinesq system, revealing how stratification-induced dispersion governs the lifespan of solutions. By reformulating the system in dispersive unknowns $Z_\pm$ with dispersion $\Lambda_\kappa(\xi)=\kappa\xi_1/|\xi|$, the authors prove a lifespan $T \ge C_n\kappa^{1/3}\varepsilon^{-4/3}$ for initial data of size $\varepsilon$ in $H^n$ with $n>3$, using a bootstrap argument that combines energy methods with inhomogeneous Strichartz estimates. A Besov-frequency localization framework is developed to handle the unbounded Riesz transform in $L^\infty$ and to control nonlinearities, yielding an analogous result for the dispersive SQG equation. The work connects small-data stability with strong-dispersion limits, extends long-time solvability results in this regularity class, and provides a robust method that avoids extra localization assumptions while capturing the dispersive mechanism driving stability.
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.