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<p>We study a two-dimensional Navier–Stokes system with anisotropic viscosity, linear damping term, and an additive noise on the whole space <InlineEquation ID="IEq1"> <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>. For this model, we prove uniqueness of invariant measures when the damping coefficient is sufficiently large compared to the noise intensity. The argument is based on an asymptotic coupling method and relies on anisotropic energy estimates together with exponential-type estimates for the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-energy. Since no Poincaré inequality is available on <InlineEquation ID="IEq3"> <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>, the damping term is essential even for the existence of invariant measures. Our result applies to general additive noise without any non-degeneracy condition and remains valid even in the deterministic case <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\sigma \equiv 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>≡</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Uniqueness of invariant measures for stochastic damped anisotropic Navier–Stokes equations

  • 期刊论文

  • 发表日期：

    2026年3月23日

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  • Siyu Liang

摘要

We study a two-dimensional Navier–Stokes system with anisotropic viscosity, linear damping term, and an additive noise on the whole space \(\mathbb {R}^2\) R 2 . For this model, we prove uniqueness of invariant measures when the damping coefficient is sufficiently large compared to the noise intensity. The argument is based on an asymptotic coupling method and relies on anisotropic energy estimates together with exponential-type estimates for the \(H^1\) H 1 -energy. Since no Poincaré inequality is available on \(\mathbb {R}^2\) R 2 , the damping term is essential even for the existence of invariant measures. Our result applies to general additive noise without any non-degeneracy condition and remains valid even in the deterministic case \(\sigma \equiv 0\) σ ≡ 0 .