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<p>In this paper, we prove the first existence result of weak solutions to the 3D Euler equation with initial vorticity concentrated in a circle and velocity field in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C([0,T],L^{2^-})\)</EquationSource> </InlineEquation>. The energy becomes finite and decreasing for positive times, with vorticity concentrated in a ring that thickens and moves in the direction of the symmetry axis. With our approach, there is no need to mollify the initial data or to rescale the time variable. We overcome the singularity of the initial data by applying convex integration within the appropriate time-weighted space.</p>

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Dissipative Euler Flows Originating from Circular Vortex Filaments

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月30日

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  • Francisco Gancedo,
  • Antonio Hidalgo-Torné,
  • Francisco Mengual

摘要

In this paper, we prove the first existence result of weak solutions to the 3D Euler equation with initial vorticity concentrated in a circle and velocity field in \(C([0,T],L^{2^-})\) . The energy becomes finite and decreasing for positive times, with vorticity concentrated in a ring that thickens and moves in the direction of the symmetry axis. With our approach, there is no need to mollify the initial data or to rescale the time variable. We overcome the singularity of the initial data by applying convex integration within the appropriate time-weighted space.