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

Abstract

We establish the nonlinear stability on a timescale \(O(\varepsilon ^{-2})\) of a linearly, stably stratified rest state in the inviscid Boussinesq system on \(\mathbb {R}^2\). Here, \(\varepsilon >0\) 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).