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Abstract <p> We discuss the occurrence of spatial asymptotic expansions of solutions to the Navier–Stokes equation on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3225_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\( {\mathbb{R}} ^d\)</EquationSource> </InlineEquation>. In particular, we prove that the Navier–Stokes equation is locally well-posed in a class of weighted Sobolev and asymptotic spaces. The solutions depend analytically on the initial data and time and (generically) develop nontrivial asymptotic terms as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3225_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">\(|x|\to\infty\)</EquationSource> </InlineEquation>. In addition, the solutions have a spatial smoothing property that depends on the order of the asymptotic expansion. </p> <p> <b> DOI</b> 10.1134/S1061920824601812 </p>

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Spatial Decay/Asymptotics in the Navier–Stokes Equation

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • P. Topalov

摘要

Abstract

We discuss the occurrence of spatial asymptotic expansions of solutions to the Navier–Stokes equation on \( {\mathbb{R}} ^d\) . In particular, we prove that the Navier–Stokes equation is locally well-posed in a class of weighted Sobolev and asymptotic spaces. The solutions depend analytically on the initial data and time and (generically) develop nontrivial asymptotic terms as \(|x|\to\infty\) . In addition, the solutions have a spatial smoothing property that depends on the order of the asymptotic expansion.

DOI 10.1134/S1061920824601812