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<p>Our study focuses on the convergence behavior of invariant measures associated with Navier–Stokes equations forced by cylindrical <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10259_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-stable noises, which could be either non-degenerate or degenerate. Despite the absence of uniqueness in the invariant measures, we successfully demonstrate the convergence of invariant measures from both non-degenerate and degenerate noise cases to their corresponding Navier–Stokes equations driven by Brownian motions as <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10259_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> tends to 2, under the Wasserstein metric. Especially, for the non-degenerate case, the convergence of invariant measures is established in the sense of the Wasserstein-1 metric.</p>

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The \(\alpha \)-Dependence of the Invariant Measure for the Stochastic Navier–Stokes Equation Driven by \(\alpha \)-Stable Lévy Processes

  • 期刊论文

  • 发表日期：

    2025年4月30日

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  • Ting Li,
  • Xianming Liu

摘要

Our study focuses on the convergence behavior of invariant measures associated with Navier–Stokes equations forced by cylindrical \(\alpha \) α -stable noises, which could be either non-degenerate or degenerate. Despite the absence of uniqueness in the invariant measures, we successfully demonstrate the convergence of invariant measures from both non-degenerate and degenerate noise cases to their corresponding Navier–Stokes equations driven by Brownian motions as \(\alpha \) α tends to 2, under the Wasserstein metric. Especially, for the non-degenerate case, the convergence of invariant measures is established in the sense of the Wasserstein-1 metric.