Existence of Homogeneous Euler Flows of Degree \(-\alpha \notin [-2,0]\)

We consider ( \(-\alpha \) $-\alpha$ )-homogeneous solutions to the stationary incompressible Euler equations in \({\mathbb {R}}^{3}\backslash \{0\}\) ${\mathbb{R}}^3\backslash \{0\}$ for \(\alpha \geqq 0\) $\alpha \geqq 0$ and in \({\mathbb {R}}^{3}\) ${\mathbb{R}}^3$ for \(\alpha <0\) $\alpha <0$ . Shvydkoy (2018) demonstrated the nonexistence of ( \(-1\) $-1$ )-homogeneous solutions \((u,p)\in C^{1}({\mathbb {R}}^{3}\backslash \{0\})\) $(u,p)\in C^1({\mathbb{R}}^3\backslash \{0\})$ and ( \(-\alpha \) $-\alpha$ )-homogeneous solutions in the range \(0\leqq \alpha \leqq 2\) $0\leqq \alpha \leqq 2$ for the Beltrami and axisymmetric flow; namely, that no ( \(-\alpha \) $-\alpha$ )-homogeneous solutions \((u,p)\in C^{1}({\mathbb {R}}^{3}\backslash \{0\})\) $(u,p)\in C^1({\mathbb{R}}^3\backslash \{0\})$ for \(1\leqq \alpha \leqq 2\) $1\leqq \alpha \leqq 2$ and \((u,p)\in C^{2}({\mathbb {R}}^{3}\backslash \{0\})\) $(u,p)\in C^2({\mathbb{R}}^3\backslash \{0\})$ for \(0\leqq \alpha < 1\) $0\leqq \alpha <1$ exist among these particular classes of flows other than irrotational solutions for integers \(\alpha \) $\alpha$ . The nonexistence result of the Beltrami ( \(-\alpha \) $-\alpha$ )-homogeneous solutions \((u,p)\in C^{2}({\mathbb {R}}^{3}\backslash \{0\})\) $(u,p)\in C^2({\mathbb{R}}^3\backslash \{0\})$ holds for all \(\alpha <1\) $\alpha <1$ . We show the nonexistence of axisymmetric ( \(-\alpha \) $-\alpha$ )-homogeneous solutions without swirls \((u,p)\in C^{2}({\mathbb {R}}^{3}\backslash \{0\})\) $(u,p)\in C^2({\mathbb{R}}^3\backslash \{0\})$ for \(-2\leqq \alpha <0\) $-2\leqq \alpha <0$ . The main result of this study is the existence of axisymmetric ( \(-\alpha \) $-\alpha$ )-homogeneous solutions in the complementary range \(\alpha \in {\mathbb {R}}\backslash [0,2]\) $\alpha \in \mathbb{R}\backslash [0,2]$ . More specifically, we show the existence of axisymmetric Beltrami ( \(-\alpha \) $-\alpha$ )-homogeneous solutions \((u,p)\in C^{1}({\mathbb {R}}^{3}\backslash \{0\})\) $(u,p)\in C^1({\mathbb{R}}^3\backslash \{0\})$ for \(\alpha >2\) $\alpha >2$ and \((u,p)\in C({\mathbb {R}}^{3})\) $(u,p)\in C({\mathbb{R}}^3)$ for \(\alpha <0\) $\alpha <0$ and axisymmetric ( \(-\alpha \) $-\alpha$ )-homogeneous solutions with a nonconstant Bernoulli function \((u,p)\in C^{1}({\mathbb {R}}^{3}\backslash \{0\})\) $(u,p)\in C^1({\mathbb{R}}^3\backslash \{0\})$ for \(\alpha >2\) $\alpha >2$ and \((u,p)\in C({\mathbb {R}}^{3})\) $(u,p)\in C({\mathbb{R}}^3)$ for \(\alpha <-2\) $\alpha <-2$ , including axisymmetric ( \(-\alpha \) $-\alpha$ )-homogeneous solutions without swirls \((u,p)\in C^{2}({\mathbb {R}}^{3}\backslash \{0\})\) $(u,p)\in C^2({\mathbb{R}}^3\backslash \{0\})$ for \(\alpha >2\) $\alpha >2$ and \((u,p)\in C^{1}({\mathbb {R}}^{3}\backslash \{0\})\cap C({\mathbb {R}}^{3})\) $(u,p)\in C^1({\mathbb{R}}^3\backslash \{0\})\cap C({\mathbb{R}}^3)$ for \(\alpha <-2\) $\alpha <-2$ . This is the first existence result on ( \(-\alpha \) $-\alpha$ )-homogeneous solutions with no explicit forms. The level sets of the axisymmetric stream function of the irrotational ( \(-\alpha \) $-\alpha$ )-homogeneous solutions in the cross-section are the Jordan curves for \(\alpha =3\) $\alpha =3$ . For \(2<\alpha <3\) $2<\alpha <3$ , we show the existence of axisymmetric ( \(-\alpha \) $-\alpha$ )-homogeneous solutions whose stream function level sets are the Jordan curves. They provide new examples of the Beltrami/Euler flows in \({\mathbb {R}}^{3}\backslash \{0\}\) ${\mathbb{R}}^3\backslash \{0\}$ whose level sets of the proportionality factor/Bernoulli surfaces are nested surfaces created by the rotation of the sign \(``\infty ''\) $``{\infty }^{\prime \prime }$ .