In this paper, we prove the existence of self-similar algebraic spiral solutions of the 2-D incompressible Euler equations for the initial vorticity of the form <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40818_2025_203_Article_IEq1.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">\(|y|^{-\frac1\mu}\ \mathring{\omega}(\theta)\)</EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40818_2025_203_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu > \frac12\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40818_2025_203_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathring{\omega}\in L^1({\mathbb{T}})\)</EquationSource> </InlineEquation>, satisfying m-fold symmetry (<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40818_2025_203_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(m\ge 2\)</EquationSource> </InlineEquation>) and a dominant condition. As an important application, we prove the existence of weak solution when <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40818_2025_203_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathring{\omega}\)</EquationSource> </InlineEquation> is a Radon measure on <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40818_2025_203_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb{T}}\)</EquationSource> </InlineEquation> with m-fold symmetry, which is related to the vortex sheet solution.

Self-Similar Algebraic Spiral Solution of 2-D Incompressible Euler Equations

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发表日期：

2025年5月6日

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Feng Shao,
Dongyi Wei,
Zhifei Zhang

摘要

In this paper, we prove the existence of self-similar algebraic spiral solutions of the 2-D incompressible Euler equations for the initial vorticity of the form \(|y|^{-\frac1\mu}\ \mathring{\omega}(\theta)\) with \(\mu > \frac12\) and \(\mathring{\omega}\in L^1({\mathbb{T}})\) , satisfying m-fold symmetry ( \(m\ge 2\) ) and a dominant condition. As an important application, we prove the existence of weak solution when \(\mathring{\omega}\) is a Radon measure on \({\mathbb{T}}\) with m-fold symmetry, which is related to the vortex sheet solution.