Long-time stability of a stably stratified rest state in the inviscid 2D Boussinesq equation

2024/08/27 by Jurja, Catalina, Widmayer, Klaus · 2 citations
#35B35 #35Q35 #35Q86 #76B15 #76B55 #76E20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.15154

Abstract

We establish the nonlinear stability on a timescale O(ε^-2) of a linearly, stably stratified rest state in the inviscid Boussinesq system on ℝ². Here ε>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 [EW15]. 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 [GPW23].

Cited by

• The effect of linear stratification on the stability of a rest state in the 2D inviscid Boussinesq system
• Instabilities of internal gravity waves in the two-dimensional Boussinesq system

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