Long-Time Stability of a Stably Stratified Rest State in the Inviscid 2D Boussinesq Equation

We establish the nonlinear stability on a timescale $O\left({\epsilon }^{-2}\right)$ of a linearly, stably stratified rest state in the inviscid Boussinesq system on $R^2$ . Here, $\epsilon >0$ denotes the size of an initially sufficiently small, Sobolev regular and localized perturbation. A similar statement also holds for the related dispersive SQG equation. At the core of this result is a dispersive effect due to anisotropic internal gravity waves. At the linearized level, this gives rise to amplitude decay at a rate of $t^{-1/2}$ , as observed in Elgindi and Widmayer (SIAM J. Math. Anal. 47(6):4672-4684, 2015). We establish a refined version of this, and propagate nonlinear control via a detailed analysis of nonlinear interactions using the method of partial symmetries developed in Guo et al. (Invent. Math. 231(1):169-262, 2023).