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<p>We introduce a novel mechanism that reveals finite time singularities within the 1D De Gregorio model and the 3D incompressible Euler equations. Remarkably, we do not construct our blow up using self-similar coordinates, but build it from infinitely many regions with vorticity, separated by vortex-free regions in between. It yields solutions of the 3D incompressible Euler equations in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb{R}^3\times [-T,0]\)</EquationSource> </InlineEquation> such that the velocity is in the space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(C^{\infty}(\mathbb{R}^3 \setminus \{0\})\cap C^{1,\alpha}\cap L^2\)</EquationSource> </InlineEquation> where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(0 &lt; \alpha \ll 1\)</EquationSource> </InlineEquation> for times <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(t\in (-T,0)\)</EquationSource> </InlineEquation> and is not <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(C^1\)</EquationSource> </InlineEquation> at time 0.</p>

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Finite Time Singularities to the 3D Incompressible Euler Equations for Solutions in\(\:C^{\infty}(\mathbb{R}^3 \setminus \{0\})\cap C^{1,\alpha}\cap L^2\)

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年7月7日

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  • Diego Córdoba,
  • Luis Martinez-Zoroa,
  • Fan Zheng

摘要

We introduce a novel mechanism that reveals finite time singularities within the 1D De Gregorio model and the 3D incompressible Euler equations. Remarkably, we do not construct our blow up using self-similar coordinates, but build it from infinitely many regions with vorticity, separated by vortex-free regions in between. It yields solutions of the 3D incompressible Euler equations in \(\mathbb{R}^3\times [-T,0]\) such that the velocity is in the space \(C^{\infty}(\mathbb{R}^3 \setminus \{0\})\cap C^{1,\alpha}\cap L^2\) where \(0 < \alpha \ll 1\) for times \(t\in (-T,0)\) and is not \(C^1\) at time 0.