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

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 $\epsilon$-sized initial perturbation of Sobolev regularity $H^{3^{+}}$ on a timescale $\mathcal{O}({\epsilon }^{-4\mathrm{/}3})$. 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.