<p>We establish the nonlinear stability on a timescale <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$O(\varepsilon ^{-2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>ε</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of a linearly, stably stratified rest state in the inviscid Boussinesq system on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>. Here, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> denotes the size of an initially sufficiently small, Sobolev regular and localized perturbation. A similar statement also holds for the related dispersive SQG equation.</p><p>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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$t^{-1/2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>t</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, 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).</p>

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

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2026年4月6日

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Catalina Jurja,
Klaus Widmayer

摘要

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).