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arXiv · 2207.06056

Existence of nonsymmetric logarithmic spiral vortex sheet solutions to the 2D Euler equations

T. Cieślak·P. Kokocki·W. S. Ożański
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Abstract

We consider solutions of the 2D incompressible Euler equation in the form of $M\geq 1$ cocentric logarithmic spirals. We prove the existence of a generic family of spirals that are nonsymmetric in the sense that the angles of the individual spirals are not uniformly distributed over the unit circle. Namely, we show that if $M=2$ or $M\geq 3 $ is an odd integer such that certain non-degeneracy conditions hold, then, for each $n \in \{ 1,2 \}$, there exists a logarithmic spiral with $M$ branches of relative angles arbitrarily close to $\barθ_{k} = knπ/M$ for $k=0,1,\ldots , M-1$, which include halves of the angles of the Alexander spirals. We show that the non-degeneracy conditions are satisfied if $M\in \{ 2, 3,5,7,9 \}$, and that the conditions hold for all odd $M>9$ given a certain gradient matrix is invertible, which appears to be true by numerical computations.

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T. Cieślak, P. Kokocki, W. S. Ożański. 2025-02-19. Existence of nonsymmetric logarithmic spiral vortex sheet solutions to the 2D Euler equations. https://arxiv.org/abs/2207.06056

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