Existence of homogeneous Euler flows of degree $-\alpha \notin [-2,0]$

Abstract

We consider ($-\alpha$)-homogeneous solutions to the stationary incompressible Euler equations in ${\mathbb{R}}^3\backslash \{0\}$ for $\alpha \ge 0$ and in ${\mathbb{R}}^3$ for $\alpha <0$. Shvydkoy (2018) demonstrated the nonexistence of ($-1$)-homogeneous solutions and ($-\alpha$)-homogeneous solutions in the range $0\le \alpha \le 2$ for the Beltrami and axisymmetric flows. The nonexistence result of the Beltrami ($-\alpha$)-homogeneous solutions holds for all $\alpha <1$. We show the nonexistence of axisymmetric ($-\alpha$)-homogeneous solutions without swirls for $-2\le \alpha <0$. The main result of this study is the existence of axisymmetric ($-\alpha$)-homogeneous solutions in the complementary range $\alpha \in \mathbb{R}\backslash [0,2]$. More specifically, we show the existence of axisymmetric Beltrami ($-\alpha$)-homogeneous solutions for $\alpha \in \mathbb{R}\backslash [0,2]$ and axisymmetric ($-\alpha$)-homogeneous solutions with a nonconstant Bernoulli function for $\alpha \in \mathbb{R}\backslash [-2,2]$. This is the first existence result on ($-\alpha$)-homogeneous solutions with no explicit forms. For $2<\alpha <3$, constructed ($-\alpha$)-homogeneous solutions provide new examples of the Beltrami/Euler flows in ${\mathbb{R}}^3\backslash \{0\}$ whose level sets of the proportionality factor/Bernoulli surfaces are nested surfaces created by the rotation of the sign $‘‘\infty "$.

69 pages