<p>In this paper, we study weak solutions to the steady (time-independent) fractional Navier-Stokes system in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_952_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. We offer a novel perspective to study the partial regularity of steady problem, and show that if <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_952_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \in (\frac{n+1}{6},\frac{n+2}{6})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>6</mn> </mfrac> <mo>,</mo> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> <mn>6</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the Hausdorff dimension of singular set for the steady weak solution is at most <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_952_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$n+2-6\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo>-</mo> <mn>6</mn> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation>. Our approach is inspired by the ideas of Katz and Pavlović (Geom. Funct. Anal. 12:2 (2002), 355-379) and Ożański (Anal. PDE 16:3 (2023)). This is the first attempt to apply the method of Katz and Pavlović to a steady setting.</p>

Partial Regularity for the Steady Fractional Navier-Stokes Equations in Dimension $\mathbf{{n}}$

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2025年6月18日

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Jiaqi Yang

摘要

In this paper, we study weak solutions to the steady (time-independent) fractional Navier-Stokes system in $\mathbb {R}^n$ $R^{n}$ . We offer a novel perspective to study the partial regularity of steady problem, and show that if $\alpha \in (\frac{n+1}{6},\frac{n+2}{6})$ $α \in (\frac{n + 1}{6}, \frac{n + 2}{6})$ , the Hausdorff dimension of singular set for the steady weak solution is at most $n+2-6\alpha $ $n + 2 - 6 α$ . Our approach is inspired by the ideas of Katz and Pavlović (Geom. Funct. Anal. 12:2 (2002), 355-379) and Ożański (Anal. PDE 16:3 (2023)). This is the first attempt to apply the method of Katz and Pavlović to a steady setting.

Partial Regularity for the Steady Fractional Navier-Stokes Equations in Dimension \(\mathbf{{n}}\)

摘要