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

Appendices

The Nonexistence of Two-Dimensional Reflection Symmetric Homogeneous Solutions

We demonstrate the nonexistence of rotational two-dimensional (2D) reflection symmetric ($-\alpha$)-homogeneous solutions to the Euler equations (1.1) for $-1\leqq \alpha \leqq 1$ in Theorem 1.7 (i) and (ii) by using the equations on a semicircle.

1.1 Equations on the Circle

We first derive the 2D homogeneous solution’s equations on a circle [103]. We use the polar coordinates $(r,\phi )$ for $x=(x_1,x_2)$ and the associated orthogonal frame

$$\begin{aligned} {\textbf{e}}_{r}=\left( \begin{array}{c} \cos \phi \\ \sin \phi \end{array} \right) ,\quad {\textbf{e}}_\phi =\left( \begin{array}{c} -\sin \phi \\ \cos \phi \end{array} \right) . \end{aligned}$$

The 2D Euler equation is expressed as

$$\begin{aligned} \begin{aligned} u\cdot \nabla u+\nabla p&=0,\\ \nabla \cdot u&=0. \end{aligned} \end{aligned}$$

(A.1)

We denote the $\pi /2$ counterclockwise rotation of $u=(u^{1},u^{2})$ by $u^{\perp }=(-u^{2},u^{1})$ and express (A.1) with the rotation $\omega =\partial _{x_1}u^{2}-\partial _{x_2}u^{1}$ and the Bernoulli function $\Pi =p+|u|^{2}/2$ as

$$\begin{aligned} \begin{aligned} \omega u^{\perp }+\nabla \Pi&=0,\\ \nabla \cdot u&=0. \end{aligned} \end{aligned}$$

(A.2)

By multiplying u by (A.2$)_1$ and by taking rotation to (A.2$)_1$, respectively,

$$\begin{aligned} u\cdot \nabla \Pi =0,\quad u\cdot \nabla \omega =0. \end{aligned}$$

(A.3)

We denote ($-\alpha$)-homogeneous solutions u to (A.2) by

$$\begin{aligned} u&=\frac{1}{r^{\alpha }}(v+f{\textbf{e}}_r),\quad v=a{\textbf{e}}_{\phi }, \end{aligned}$$

(A.4)

and the functions $a(\phi )$ and $f(\phi )$. The rotation of u is expressed as

$$\begin{aligned} \omega =\frac{1}{r^{\alpha +1}}\left( (1-\alpha )a-f' \right) . \end{aligned}$$

(A.5)

Substituting u into (A.2) implies the equations for (a, f, p),

$$\begin{aligned}&\omega a+2\alpha \Pi =0,\\&\omega f+ \Pi '=0,\\&(1-\alpha )f+a'=0. \end{aligned}$$

The second and third equations imply that p is constant. Thus (a, f) satisfy the equations on ${\mathbb {S}}^{1}$:

$$\begin{aligned} \begin{aligned}&af'=a^{2}+\alpha f^{2}+2\alpha p,\\&(1-\alpha )f+a'=0. \end{aligned} \end{aligned}$$

(A.6)

The conditions (A.3) imply

$$\begin{aligned} a\Pi '=2\alpha f \Pi ,\quad a\omega '=(\alpha +1) f \omega . \end{aligned}$$

(A.7)

1.2 Radially Symmetric Solutions

The simplest homogeneous solutions to (A.1) are radially symmetric solutions.

Theorem A.1

All radially symmetric ($-\alpha$)-homogeneous solutions $u\in C^{1}({\mathbb {R}}^{2}\backslash \{0\})$ to (A.1) for $\alpha \in {\mathbb {R}}$ are expressed by (A.4) with the constants

$$\begin{aligned} \begin{aligned}&a,f\in {\mathbb {R}},\quad a^{2}+f^{2}+2p=0\quad \text {for}\ \alpha =1, \\&a\in {\mathbb {R}},\quad f=0,\quad a^{2}+2\alpha p=0\quad \text {for}\ \alpha \in {\mathbb {R}}\backslash \{1\}. \end{aligned} \end{aligned}$$

(A.8)

The solution for $\alpha =1$ is irrotational.

Proof

By radial symmetry, a and f in (A.4) are constant. By (A.6$)_2$, $(\alpha -1)f=0$. If $\alpha =1$, a satisfies (A.8$)_1$ by (A.6$)_1$. If $\alpha \ne 1$, $f=0$ and a satisfies (A.8$)_2$ by (A.6$)_1$. By (A.5), u is irrotational for $\alpha =1$. $\square$

1.3 Reflection Symmetric Solutions

The second simplest homogeneous solutions to (A.1) may be reflection symmetric solutions,

$$\begin{aligned} \begin{aligned} u^{1}(x_1,x_2)&=u^{1}(x_1,-x_2), \\ u^{2}(x_1,x_2)&=-u^{2}(x_1,-x_2),\\ p(x_1,x_2)&=p(x_1,-x_2). \end{aligned} \end{aligned}$$

(A.9)

This symmetry imposes the boundary condition,

$$\begin{aligned} u^{2}(x_1,0)=0. \end{aligned}$$

(A.10)

The rotation is an odd function for the $x_2$-variable,

$$\begin{aligned} \omega (x_1,x_2)=-\omega (x_1,-x_2). \end{aligned}$$

(A.11)

Thus continuous $\omega$ vanishes on the boundary,

$$\begin{aligned} \omega (x_1,0)=0. \end{aligned}$$

(A.12)

The functions (a, f) satisfy equations on the semicircle for $\phi \in (0,\pi )$ with constant p:

$$\begin{aligned} \begin{aligned}&af'=a^{2}+\alpha f^{2}+2\alpha p,\\&(1-\alpha )f+a'=0,\\&a(0)=a(\pi )=0. \end{aligned} \end{aligned}$$

(A.13)

Theorem A.2

Irrotational reflection symmetric ($-\alpha$)-homogeneous solutions $u\in C^{1}({\mathbb {R}}^{2}\backslash \{0\})$ to (A.1) exist if and only if $\alpha \in {\mathbb {Z}}$. They are given by (A.4) for

$$\begin{aligned} a=C\sin ((\alpha -1)\phi ),\quad f=C\cos \left( (\alpha -1)\phi \right) ,\quad C\in {\mathbb {R}},\quad \alpha (C^{2}+2p)=0. \end{aligned}$$

(A.14)

Proof

The equation (A.5) implies that $(1-\alpha )a-f'=0$. By (A.13$)_2$ and (A.13$)_3$,

$$\begin{aligned}&-f''=(\alpha -1)^{2}f\quad \text {in}\ (0,\pi ),\\&f'(0)=f'(\pi )=0. \end{aligned}$$

Thus $\alpha \in {\mathbb {Z}}$ and $f=C\cos ((\alpha -1)\phi )$ for some constant C. By $(1-\alpha )a-f'=0$ and (A.13$)_3$, $a=C\sin ((\alpha -1)\phi )$. By (A.13$)_1$, the constant C satisfies (A.14$)_4$. $\square$

Remark A.3

For $\alpha =1$, the irrotational reflection symmetric ($-1$)-homogeneous solution is $u=(C/r){\textbf{e}}_{r}$ for $C\in {\mathbb {R}}$ satisfying $C^{2}+2p=0$ (The solution $(1.4)_1$ is the case $C=1$). For $\alpha \in {\mathbb {Z}}\backslash \{1\}$, the irrotational reflection symmetric ($-\alpha$)-homogeneous solution is $u=(C/r^{n+1})(\sin (n\phi ) {\textbf{e}}_{\phi }+\cos (n\phi ) {\textbf{e}}_{r} )$ for $n=\alpha -1$ and $C\in {\mathbb {R}}$ satisfying $(n+1)(C^{2}+2p)=0$. The stream function of this solution is $\psi =(-C/n r^{n})\sin (n\phi )$ (The solution $(1.4)_2$ is the case $C=-n$).

1.4 The Nonexistence for $-1\leqq \alpha \leqq 1$

We demonstrate the nonexistence of rotational reflection symmetric ($-\alpha$)-homogeneous solutions to (A.1) for $-1\leqq \alpha \leqq 1$ by using the first integral conditions (A.7).

Proof of Theorem 1.7 (i) and (ii)

For $\alpha =1$, the equations (A.13$)_2$ and (A.13$)_3$ imply $a=0$. By (A.13$)_1$, f is constant. Thus u is irrotational by (A.5). We show that reflection symmetric ($-\alpha$)-homogeneous solutions for $-1\leqq \alpha < 1$ are irrotational by using the conditions (A.7$)_1$ for $0\leqq \alpha <1$ and (A.7$)_2$ for $-1\leqq \alpha <0$.

For $0<\alpha <1$, we first show that $u\in C^{1}({\mathbb {R}}^{2}\backslash \{0\})$ satisfies $a \Pi =0$ in $(0,\pi )$. Suppose that $a \Pi \ne 0$ on some interval $J\subset (0,\pi )$. The equations (A.13$)_1$ and (A.7$)_1$ imply

$$\begin{aligned} |\Pi |^{1-\alpha }|a|^{2\alpha }=C, \end{aligned}$$

for some constant C. If $C\ne 0$, J is extendable to $(0,\pi )$. However, the condition $0<\alpha <1$ and the boundary condition (A.13$)_3$ imply that the left-hand side vanishes as $\phi \rightarrow 0$. Thus $C=0$. This contradicts $a\Pi \ne 0$ on J. Hence $a\Pi =0$ in $(0,\pi )$.

We show that $\Pi =0$ in $(0,\pi )$. Suppose that $\Pi \ne 0$ on some interval $J\subset (0,\pi )$. Then, $a=0$ and $f=0$ by (A.13$)_2$. Since $p=0$ by (A.13$)_1$, $\Pi =0$. This is a contradiction and hence $\Pi =0$ in $(0,\pi )$. By (A.2), $\omega u^{\perp }=0$. Thus $\omega =0$ in $(0,\pi )$ and u is irrotational.

For $\alpha =0$, the condition (A.7$)_1$ implies $a\Pi '=0$. Suppose that $\Pi '\ne 0$ on some interval $J\subset (0,\pi )$. Then, $a=0$ and $f=0$ by (A.13$)_2$. Hence $\Pi '=0$. This is a contradiction and hence $\Pi '=0$ in $(0,\pi )$. By (A.2), $\omega =0$ in $(0,\pi )$ and u is irrotational.

For $-1< \alpha <0$, we first show that $u\in C^{2}({\mathbb {R}}^{2}\backslash \{0\})$ satisfies $a \omega =0$ in $(0,\pi )$. Suppose that $a\omega \ne 0$ on some interval $J\subset (0,\pi )$. The equations (A.13)$_2$ and (A.7)$_2$ imply

$$\begin{aligned} |\omega |^{1-\alpha }|a|^{1+\alpha }=C, \end{aligned}$$

for some constant C. By applying the same argument as that for $\Pi$ and $0<\alpha <1$, we conclude that $C=0$. This contradicts $a\omega \ne 0$ on J. Hence $a\omega =0$ in $(0,\pi )$.

We show that $\omega =0$ in $(0,\pi )$. Suppose that $\omega \ne 0$ on some interval $J\subset (0,\pi )$. Then, $a=0$ and $f=0$ by (A.13$)_2$. Hence $\omega =0$ by (A.5). This is a contradiction and hence $\omega =0$ in $(0,\pi )$. Thus u is irrotational.

For $\alpha =-1$, the condition (A.7$)_2$ implies that $a\omega '=0$. If $\omega '\ne 0$ on some interval $J\subset (0,\pi )$, $a=0$ and $f=0$ by (A.13$)_2$ and hence $\omega =0$ by (A.5). This is a contradiction. Thus $\omega '=0$ in $(0,\pi )$. Since $\omega \in C^{1}({\mathbb {R}}^{2}\backslash \{0\})$, the boundary condition (A.12) implies that $\omega =0$. Thus u is irrotational. $\square$

The Existence of Two-and-a-Half-Dimensional Reflection Symmetric Homogeneous Solutions

We demonstrate the existence of rotational two-dimensional (2D) reflection symmetric ($-\alpha$)-homogeneous solutions to (1.1) in Theorem 1.7 (iii) and (iv). We consider more general two-and-a-half-dimensional (D) reflection symmetric solutions $u=(u^{1},u^{2},u^{3})$ to (1.1) satisfying

$$\begin{aligned} \begin{aligned} u^{1}(x_1,x_2)&=u^{1}(x_1,-x_2), \\ u^{2}(x_1,x_2)&=-u^{2}(x_1,-x_2), \\ u^{3}(x_1,x_2)&=-u^{3}(x_1,-x_2). \end{aligned} \end{aligned}$$

(B1)

Theorem B. 1

The following holds for rotational D reflection symmetric ($-\alpha$)-homogeneous solutions $u=u^{H}+u^{3}e_z$ and $r^{2\alpha }p=\text {const.}$ to (1.1):

• (i) For $\alpha >1$, solutions $u\in C^{1}({\mathbb {R}}^{2}\backslash \{0\})$ such that $u^{H}\in C^{2}({\mathbb {R}}^{2}\backslash \{0\})$ and $u^{3}\in C^{1}({\mathbb {R}}^{2}\backslash \{0\})$ exist.

• (ii) For $\alpha <-1$, solutions $u\in C({\mathbb {R}}^{2})$ such that $u^{H}\in C^{1}({\mathbb {R}}^{2}\backslash \{0\})\cap C({\mathbb {R}}^{2})$ and $u^{3}\in C({\mathbb {R}}^{2})$ exist.

1.1 The Existence of Solutions to the Autonomous Dirichlet Problem

We construct solutions in Theorem B. 1 by solutions of the automonous Dirichlet problem. We consider symmetric solenoidal vector fields (B1) expressed by the Clebsch representation

$$\begin{aligned} u=\nabla \times (\psi \nabla z)+G \nabla z. \end{aligned}$$

(B2)

here, $\nabla =\nabla _{{\mathbb {R}}^{3}}$ is the gradient in ${\mathbb {R}}^{3}$ and $(r,\phi ,z)$ are the cylindrical coordinates. We denote the function $u^{3}$ by G. We assume that the stream function $\psi$ is an odd function for the $x_2$-variable and vanishes on the $x_1$-axis,

$$\begin{aligned} \psi (x_1,0)=0. \end{aligned}$$

the rotation of u is expressed as

$$\begin{aligned} \nabla \times u=\nabla \times (G\nabla z)+(-\Delta \psi ) \nabla z. \end{aligned}$$

For translation-reflection symmetric solutions to the Euler equations (2.1), the Bernoulli function $\Pi =p+|u|^{2}/2$ and the function G are first integrals of u, i.e.,

$$\begin{aligned} u\cdot \nabla \Pi =0, \quad u\cdot \nabla G=0. \end{aligned}$$

We assume that they are globally functions of $\psi$, i.e., $\Pi =\Pi (\psi )$, $G=G(\psi )$. By using the triple product,

$$\begin{aligned} (\nabla \times u)\times u&=-\frac{1}{2}\nabla G^{2}+(-\Delta \psi )\nabla \psi ,\\ \nabla \Pi&=\Pi '(\psi )\nabla \psi . \end{aligned}$$

Here, $\Pi '(\psi )$ denotes the differentiation for the variable $\psi$. Thus $\psi$ satisfies the semilinear Dirichlet problem:

$$\begin{aligned} \begin{aligned} -\Delta \psi&=-\Pi '(\psi )+G'(\psi )G(\psi ),\quad (x_1,x_2)\in {\mathbb {R}}^{2}_{+}, \\ \psi (x_1,0)&=0,\quad x_1\in {\mathbb {R}}. \end{aligned} \end{aligned}$$

(B3)

Solutions to the above 2D Dirichlet problem for prescribed $\Pi (\psi )$ and $G(\psi )$ provide D reflection symmetric solutions to (1.1).

We choose particular $\Pi (\psi )$ and $G(\psi )$ to construct D reflection symmetric homogeneous solutions to (1.1). For ($-\alpha$)-homogeneous solutions to (1.1), stream functions are ($-\alpha +1$)-homogeneous and expressed as

$$\begin{aligned} \psi (x_1,x_2)=\frac{w(\phi )}{r^{\beta }},\quad \beta =\alpha -1. \end{aligned}$$

The left-hand side of (B.3$)_1$ is expressed as

$$\begin{aligned} -\Delta \psi =-\frac{1}{r^{\beta +2}}(\beta ^{2}w+w''). \end{aligned}$$

We choose the functions $\Pi (\psi )$ and $G(\psi )$ by

$$\begin{aligned} \begin{aligned} \Pi (\psi )&=C_1|\psi |^{2+\frac{2}{\beta }}=C_1\frac{|w|^{2+\frac{2}{\beta }}}{r^{2\beta +2}},\\ G(\psi )&=C_2\psi |\psi |^{\frac{1}{\beta }}=C_2\frac{w|w|^{\frac{1}{\beta }}}{r^{\beta +1}}, \end{aligned} \end{aligned}$$

for constants $C_1$, $C_2\in {\mathbb {R}}$. The right-hand side of (B.3$)_1$ is is expressed as

$$\begin{aligned} -\Pi '(\psi )+G'(\psi )G(\psi )=c\frac{w|w|^{\frac{2}{\beta }}}{r^{\beta +2}} \end{aligned}$$

for the constant

$$\begin{aligned} c=\left( -2C_1+C_{2}^{2}\right) \left( 1+\frac{1}{\beta }\right) . \end{aligned}$$

Then function w satisfies the autonomous Dirichlet problem:

$$\begin{aligned} \begin{aligned} -w''&=\beta ^{2}w+cw|w|^{\frac{2}{\beta }},\quad \phi \in (0,\pi ),\\ w(0)&=w(\pi )=0. \end{aligned} \end{aligned}$$

(B4)

Theorem A.4

For $\beta \in {\mathbb {R}}\backslash [-2,0]$ and $c>0$, there exists a solution $w\in C^{2}[0,\pi ]$ to (B4). For $\beta >0$, $w\in C^{3}[0,\pi ]$. For $0<\beta <1$, there exists a positive solution to (B4).

Remark B. 2

• (i) Positive solutions to (B4) for $0<\beta <1$ are symmetric with respect to $\phi =\pi /2$ and decreasing for $\phi >\pi /2$ since (B.4$)_1$ is an autonomous equation [67, Theorem 1], cf. [67, Theorem $1'$].

• (ii) No positive solutions to (B4) exist for $\beta \in {\mathbb {R}}\backslash [-2,1)$ and $c>0$, cf. [121, Remarks 6.3 (ii)]. In fact, the principal eigenvalue of the operator $-\partial _{\phi }^{2}$ is one and its associated eigenfunction is $\sin \phi$. By multiplying $\sin \phi$ by (B.4$)_1$ and integration by parts,

  $$\begin{aligned} (1-\beta ^{2})\int _{0}^{\pi }w \sin \phi \text {d}\phi =c\int _{0}^{\pi }w |w|^{\frac{2}{\beta }}\sin \phi \text {d}\phi . \end{aligned}$$

  Thus $0<\beta <1$ for positive w.

• (iii) The number of zero points are finite for solutions to (B4) for $\beta >0$ by the uniqueness of the ODE as observed for the nonautonomous case in Remarks 3.2 (iii).

Proof

The construction of solutions to (B4) is easier than that of solutions to (3.4). In fact, the problem (B4) is expressed as

$$\begin{aligned} \begin{aligned} -L_{\beta }w&=g(w)\quad \text {in}\ \Omega ,\\ w&=0\qquad \text {on}\ \partial \Omega , \end{aligned} \end{aligned}$$

(B5)

with the symbols $L_{\beta }=\partial _{\phi }^{2}+\beta ^{2}$, $g(w)= c w|w|^{\frac{2}{\beta }}$, and $\Omega =(0,\pi )$. The eigenvalues of the operator $-L_{\beta }$ are explicitly given by $\mu _{n}=n^{2}-\beta ^{2}$ for $n=1,2,\dots$. Hence $\mu _1>0$ for $0<\beta <1$ and $\mu _1\leqq 0$ for $\beta \in {\mathbb {R}}\backslash [-2,1)$. The orthonormal basis on $L^{2}(\Omega )$ are $e_n=\sqrt{2/\pi }\sin {n\phi }$. We take $N\in {\mathbb {N}}$ such that $\mu _1<\mu _2<\cdots<\mu _N\leqq 0<\mu _{N+1}<\cdots$ and consider the direct sum decomposition $H^{1}_{0}(\Omega )=Y\oplus Z$ for $Y=\text {span}(e_1,\cdots ,e_N)$ and $Z=\{z\in H^{1}_{0}(\Omega )\ |\ (z,y)_{L^{2}}=0,\ y \in Y\ \}$. The functional associated with (B5) is the following:

$$\begin{aligned} I[w]&=\frac{1}{2}\int _{\Omega }(|w'|^{2}-\beta ^{2}|w|^{2})\text {d}\phi -\int _{\Omega }G(w)\text {d}\phi ,\\ G(w)&=\int _{0}^{w}g(s)\text {d}s=\frac{c \beta }{2(\beta +1)}|w|^{2(1+\frac{1}{\beta })}. \end{aligned}$$

This functional satisfies the (PS$)_c$ condition for any $c\in {\mathbb {R}}$ and desired estimates on subsets. We apply the mountain pass theorem (Lemma 3.4) for $0<\beta <1$, the linking theorem (Lemma 3.6) for $1\leqq \beta$, and the saddle point theorem (Lemma 3.7) for $\beta <-2$. Since the equation (B4) is autonomous, critical points $w\in H^{1}_{0}(\Omega )$ belong to $w\in C^{2}[0,\pi ]$. For $\beta >0$, differentiating (B.4$)_1$ by $\phi$ implies that $w\in C^{3}[0,\pi ]$. The existence of positive solutions follows from an application of the mountain pass theorem to a modified problem for $0<\beta <1$ as we observed for the nonautonomous equation in (3.14). $\square$

1.2 Regularity of Two-and-a-Half-Dimensional Reflection Symmetric Homogeneous Solutions

We construct D reflection symmetric homogeneous solutions by solutions of (B4) in Theorem A.4. We choose constants $\alpha$, $C_1$, and $C_2$ satisfying

$$\begin{aligned} \alpha \in {\mathbb {R}}\backslash [-1,1],\quad C_1,C_2\in {\mathbb {R}},\quad -2C_1+C_2^{2}>0, \end{aligned}$$

(B6)

so that

$$\begin{aligned} \beta =\alpha -1\in {\mathbb {R}}\backslash [-2,0],\quad c=(-2C_1+C_2^{2})\left( 1+\frac{1}{\beta }\right) >0. \end{aligned}$$

(B7)

Lemma B. 3

Let $\alpha$, $C_1$, and $C_2$ satisfy (B6). Let $w\in C^{2}[0,\pi ]$ be a solution to (B4) for $\beta$ and c in (B7). For the odd extension of w to $[-\pi ,0]$, set

$$\begin{aligned} \psi&=\frac{w(\phi )}{r^{\beta }},\quad \Pi =C_1|\psi |^{2+\frac{2}{\beta }},\quad G=C_2\psi |\psi |^{\frac{1}{\beta }},\\ u&=\nabla \times (\psi \nabla z)+G\nabla z=u^{H}+u^{3}e_{z},\\ p&=\Pi -\frac{1}{2}|u|^{2}. \end{aligned}$$

Then u is ($-\alpha$)-homogeneous and $r^{2\alpha }p$ is constant. They satisfy the following regularity properties:

• (i) For $\alpha <-1$,

  $$\begin{aligned} u^{H}\in C^{1}({\mathbb {R}}^{2}\backslash \{0\})\cap C({\mathbb {R}}^{2}),\quad u^{3}\in C({\mathbb {R}}^{2}). \end{aligned}$$

  (B8)

• (ii) For $\alpha >1$,

  $$\begin{aligned} u^{H}\in C^{2}({\mathbb {R}}^{2}\backslash \{0\}),\quad u^{3}\in C^{1}({\mathbb {R}}^{2}\backslash \{0\}). \end{aligned}$$

  (B9)

Proof

The functions u and p are expressed in terms of w as

$$\begin{aligned} \begin{aligned} r^{\beta +1}u&=w'e_r+\beta we_{\phi }+C_2w|w|^{\frac{1}{\beta }}e_{z}, \\ r^{2\beta +2}p&=-\frac{c\beta }{2(\beta +1)}|w|^{2+\frac{2}{\beta }}-\frac{1}{2}\left( |w'|^{2}+\beta ^{2}w^{2} \right) . \end{aligned} \end{aligned}$$

Differentiating the second equation by $\phi$ implies that $r^{2\beta +2}p$ is independent of $\phi$. For $\beta <-2$, $w\in C^{2}[-\pi ,\pi ]$ implies that $u^{H}\in C^{1}({\mathbb {R}}^{2}\backslash \{0\})$. Since u vanishes at the origin, $u\in C({\mathbb {R}}^{2})$. Thus (B8) holds. For $\beta >0$, $w\in C^{3}[-\pi ,\pi ]$ implies (B9). $\square$

We show that (u, p) in Lemma B. 3 are homogeneous solutions to (1.1). In the translation reflection symmetric setting, the equations (1.1) for $u=u^{H}+u^{3}e_{z}$ are equivalent to

$$\begin{aligned} \begin{aligned} u^{H}\cdot \nabla _{{\mathbb {R}}^{2}} u^{H}+\nabla _{{\mathbb {R}}^{2}} p&=0,\\ \nabla _{{\mathbb {R}}^{2}} \cdot u^{H}&=0,\\ u^{H}\cdot \nabla _{{\mathbb {R}}^{2}} u^{3}&=0. \end{aligned} \end{aligned}$$

(B10)

Proposition B. 4

The functions (u, p) in Lemma B. 3 are ($-\alpha$)-homogeneous solutions to (B10) in ${\mathbb {R}}^{2}\backslash \{0\}$ for $\alpha >1$ and in ${\mathbb {R}}^{2}$ for $\alpha <-1$.

Proof

Since $w\in C^{2}[-\pi ,\pi ]$, the stream function $\psi \in C^{2}({\mathbb {R}}^{2}\backslash \{0\})$ is a reflection symmetric solution to the elliptic equation (B.3$)_1$ in ${\mathbb {R}}^{2}\backslash \{0\}$ for

$$\begin{aligned} \Pi '(\psi )&=2C_1\left( 1+\frac{1}{\beta }\right) \frac{w|w|^{\frac{2}{\beta }}}{r^{\beta +2}},\\ \left( \frac{1}{2}G^{2}(\psi ) \right) '&=C_2^{2}\left( 1+\frac{1}{\beta }\right) \frac{w|w|^{\frac{2}{\beta }}}{r^{\beta +2}}. \end{aligned}$$

We suppress the subscript for $\nabla =\nabla _{{\mathbb {R}}^{2}}$ and use the symbol $\nabla ^{\perp }=(-\partial _2,\partial _1)$. By multiplying $\nabla \psi$ by (B.3$)_1$,

$$\begin{aligned} -\Delta \psi \nabla \psi +\nabla \left( \Pi -\frac{1}{2}G^{2}\right) =0. \end{aligned}$$

Since $u^{H}=-\nabla ^{\perp } \psi$, $-\Delta \psi =\nabla ^{\perp } \cdot u^{H}$, and $p=\Pi -|u^{H}|^{2}/2-|G|^{2}/2$,

$$\begin{aligned} \nabla ^{\perp } \cdot u^{H}(u^{H})^{\perp }+\nabla \left( p+\frac{1}{2}|u^{H}|^{2}\right) =0. \end{aligned}$$

By the identity,

$$\begin{aligned} u^{H}\cdot \nabla u^{H}=\nabla ^{\perp } \cdot u^{H}(u^{H})^{\perp }+\nabla \frac{1}{2}|u^{H}|^{2}, \end{aligned}$$

(B.10$)_1$ and (B.10$)_2$ hold in ${\mathbb {R}}^{2}\backslash \{0\}$. For $\alpha >1$, taking divergence to

$$\begin{aligned} u^{H}u^{3}=-\nabla ^{\perp } \left( C_2\frac{\beta }{2\beta +1}|\psi |^{2+\frac{1}{\beta }}\right) , \end{aligned}$$

implies that (B.10$)_3$ holds in ${\mathbb {R}}^{2}\backslash \{0\}$. Thus (u, p) is a ($-\alpha$)-homogeneous solution to (B10) in ${\mathbb {R}}^{2}\backslash \{0\}$ for $\alpha >1$.

For $\alpha <-1$, (B.10$)_1$ and (B.10$)_2$ in ${\mathbb {R}}^{2}\backslash \{0\}$ hold for $u^{H}\in C^{1}({\mathbb {R}}^{2}\backslash \{0\})\cap C({\mathbb {R}}^{2})$ and (B.10$)_3$ in ${\mathbb {R}}^{2}\backslash \{0\}$ holds for $u^{3}\in C({\mathbb {R}}^{2})$ in the distributional sense. Since (u, p) vanishes at the origin, by the cut-off function argument around the origin, (u, p) satisfies (B10) in ${\mathbb {R}}^{2}$ in the distributional sense. $\square$

Proof of Theorems B. 1, 1.7 (iii) and (iv) and 1.10

For the constants $\alpha$, $C_1$ and $C_2$ satisfying (B6), Lemma B. 3 and Proposition B. 4 imply the existence of rotational reflection symmetric D ($-\alpha$)-homogeneous solutions to (1.1) in Theorem B. 1. In particular, taking $C_2=0$ implies the existence of rotational 2D reflection symmetric ($-\alpha$)-homogeneous solutions in Theorem 1.7 (iii) and (iv). For $1<\alpha <2$, the existence of positive solutions to (B4) implies the existence of 2D rotational reflection symmetric ($-\alpha$)-homogeneous solutions whose stream function level sets are homeomorphic to those of the irrotational ($-2$)-homogeneous solution in Theorem 1.8. $\square$

Remark B. 5

For $\alpha \in {\mathbb {R}}\backslash [-1,1]$, $C_1=0$ and $C_2\ne 0$, the solutions in Theorem B. 1 are D reflection symmetric Beltrami ($-\alpha$)-homogeneous solutions to (1.1). By $-\Delta \psi =G'G$ and the Clebsch representation for $\nabla =\nabla _{{\mathbb {R}}^{3}}$,

$$\begin{aligned} \nabla \times u=G'(\nabla \times (\psi \nabla z)+G\nabla z)=G'u. \end{aligned}$$

For $\alpha >1$, u admits the proportionality factor $\varphi =G'=C_2(1+1/\beta )|\psi |^{1/\beta }$.