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<p>We consider the Navier–Stokes Cauchy problem with an initial datum in a weighted Lebesgue space. The weight is a radial function increasing at infinity. Our study partially follows the ideas of the paper by Galdi and Maremonti (J Math Fluid Mech 25:7, 2023). The authors of the quoted paper consider a special study of stability of steady fluid motions. The results hold in 3D and for small data. Here, relatively to the perturbations of the rest state, we generalize the result. We study the <i>n</i>D Navier–Stokes Cauchy problem, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_923_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We prove the existence (local) of a unique regular solution. Moreover, the solution enjoys a spatial asymptotic decay whose order of decay is connected to the weight.</p>

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The Navier–Stokes Cauchy Problem in a Class of Weighted Function Spaces

  • 期刊论文

  • 发表日期：

    2025年2月20日

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  • Paolo Maremonti,
  • Vittorio Pane

摘要

We consider the Navier–Stokes Cauchy problem with an initial datum in a weighted Lebesgue space. The weight is a radial function increasing at infinity. Our study partially follows the ideas of the paper by Galdi and Maremonti (J Math Fluid Mech 25:7, 2023). The authors of the quoted paper consider a special study of stability of steady fluid motions. The results hold in 3D and for small data. Here, relatively to the perturbations of the rest state, we generalize the result. We study the nD Navier–Stokes Cauchy problem, \(n\ge 3\) n ≥ 3 . We prove the existence (local) of a unique regular solution. Moreover, the solution enjoys a spatial asymptotic decay whose order of decay is connected to the weight.