Long-Time Stability of a Stably Stratified Rest State in the Inviscid 2D Boussinesq Equation
Abstract
Abstract We establish the nonlinear stability on a timescale $$O(\varepsilon ^{-2})$$ O ( ε - 2 ) of a linearly, stably stratified rest state in the inviscid Boussinesq system on $$\mathbb {R}^2$$ R 2 . Here, $$\varepsilon >0$$ 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 $$t^{-1/2}$$ 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).
Fields of Study & Subjects
Advanced Mathematical Physics Problems
Quantum chaos and dynamical systems
advanced mathematical theories
Publication details
- Journal / source
- Archive for Rational Mechanics and Analysis
- Publisher
- Springer Science and Business Media LLC
- Volume
- 250
- Issue
- 3
- Pages
- 26–26
- Publication date
- 6 April 2026
- Language
- EN