Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. II. Classification of axisymmetric no-swirl solutions

TL;DR

This paper classifies all smooth, axisymmetric, no-swirl solutions to the stationary Navier-Stokes equations with a specific homogeneity, parameterizing them in a four-dimensional space and analyzing their smoothness properties.

Contribution

It provides a complete classification of (-1)-homogeneous axisymmetric no-swirl solutions on the sphere minus poles, forming a foundation for studying solutions with non-zero swirl.

Findings

Classified solutions form a four-dimensional parameter space.

Established smoothness properties of the solution surface.

Set the stage for analyzing solutions with non-zero swirl.

Abstract

We classify all (-1)-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles, parameterizing them as a four dimensional surface with boundary in appropriate function spaces. Then we establish smoothness properties of the solution surface in the four parameters. The smoothness properties will be used in a subsequent paper where we study the existence of (-1)-homogeneous axisymmetric solutions with non-zero swirl on $\mathbb{S}^2\setminus\{S,N\}$, emanating from the four dimensional solution surface.

Equations

$$\{\begin{array}{rl}& -△u+u\cdot \nabla u+\nabla p=0,\\ & \text{mbox}divu=0.\end{array}$$

$$u=u_r{e}_r+u_{\theta }{e}_{\theta }+u_{\varphi }{e}_{\varphi },$$

$$e_r=\left(\begin{array}{c}\sin \theta \cos \varphi \\ \sin \theta \sin \varphi \\ \cos \theta \end{array}\right),e_{\theta }=\left(\begin{array}{c}\cos \theta \cos \varphi \\ \cos \theta \sin \varphi \\ -\sin \theta \end{array}\right),e_{\varphi }=\left(\begin{array}{c}-\sin \varphi \\ \cos \varphi \\ 0\end{array}\right).$$

$$u_r=-\frac{d{u}_{\theta }}{d\theta }-u_{\theta }\cot \theta .$$

$$x:=\cos \theta ,U_{\theta }:=u_{\theta }\sin \theta .$$

$$p=-\frac{1}{2}\left(\frac{d^2{u}_r}{d{\theta }^2}+(\cot \theta -u_{\theta })\frac{d{u}_r}{d\theta }+u_r^2+u_{\theta }^2\right),$$

$$(1-x^2)U_{\theta }^{\prime }+2x{U}_{\theta }+\frac{1}{2}{U}_{\theta }^2=P_c(x):=c_1(1-x)+c_2(1+x)+c_3(1-x^2),$$

$${\bar{c}}_3(c_1,c_2):=-\frac{1}{2}\left(\sqrt{1+c_1}+\sqrt{1+c_2}\right)\left(\sqrt{1+c_1}+\sqrt{1+c_2}+2\right).$$

$$J:=\{c\in {\mathbb{R}}^3\mid c_1\ge -1,c_2\ge -1,c_3\ge {\bar{c}}_3(c_1,c_2)\}.$$

$$U_{\theta }^{+}(c)=U_{\theta }^{-}(c)=U_{\theta }^{\ast }(c_1,c_2):=(1+\sqrt{1+c_1})(1-x)+(-1-\sqrt{1+c_2})(1+x).$$

$${\gamma }^{+}(c):=U_{\theta }^{+}(c)(0),{\gamma }^{-}(c):=U_{\theta }^{-}(c)(0).$$

$$I:=\{(c,\gamma )\in {\mathbb{R}}^4\mid c_1\ge -1,c_2\ge -1,c_3\ge \overline{c_3}(c_1,c_2).{\gamma }^{-}(c)\le \gamma \le {\gamma }^{+}(c)\}.$$

$$U_{\theta }(x)=\frac{2(1-x^2)}{x+\lambda },|\lambda |>1,$$

$${\tau }_1(c_1):=2-2\sqrt{1+c_1},{\tau }_2(c_1):=2+2\sqrt{1+c_1},$$

$${\tau }_1^{\prime }(c_2):=-2-2\sqrt{1+c_2},{\tau }_2^{\prime }(c_2):=-2+2\sqrt{1+c_2}.$$

$$\begin{array}{rl}& \{(x,y)\mid -1<x<1,U_{\theta }^{c,{\gamma }^{-}(c)}(x)\le y\le U_{\theta }^{c,{\gamma }^{+}(c)}(x)\}\\ & =\bigcup _{\gamma \in [{\gamma }^{-}(c),{\gamma }^{+}(c)]}\{(x,U_{\theta }^{c,\gamma }(x))\mid -1<x<1\}.\end{array}$$

$$U_{\theta}^{c,\gamma}(-1):=\left\{\begin{array}[]{ll}\tau_{2}(c_{1}),&\mbox{when }\gamma=\gamma^{+}(c),\\ \tau_{1}(c_{1}),&\mbox{otherwise},\end{array}\right.\quad U_{\theta}^{c,\gamma}(1):=\left\{\begin{array}[]{ll}\tau_{1}^{\prime}(c_{2}),&\mbox{when }\gamma=\gamma^{-}(c),\\ \tau_{2}^{\prime}(c_{2}),&\mbox{otherwise}.\end{array}\right.$$

$$\begin{array}{rl}& J_1:=\{c\in J\mid c_1>-1,c_2>-1,c_3>{\bar{c}}_3\},J_2:=\{c\in J\mid c_1=-1,c_2>-1,c_3>{\bar{c}}_3\},\\ & J_3:=\{c\in J\mid c_1>-1,c_2=-1,c_3>{\bar{c}}_3\},J_4:=\{c\in J\mid c_1=-1,c_2=-1,c_3>{\bar{c}}_3\}.\end{array}$$

$$\begin{array}{rl}& I_{k,1}:=\{(c,\gamma )\in I\mid c\in J_k,{\gamma }^{-}(c)<\gamma <{\gamma }^{+}(c)\},\\ & I_{k,2}:=\{(c,\gamma )\in I\mid c\in J_k,\gamma ={\gamma }^{+}(c)\},\\ & I_{k,3}:=\{(c,\gamma )\in I\mid c\in J_k,\gamma ={\gamma }^{-}(c)\}.\end{array}$$

$$|{\partial }_c^{\alpha }{\partial }_{\gamma }^j{U}_{\theta }^{c,\gamma }(x)|\le C(m,K),\text{mbox}forany0\le |\alpha |+j\le m,$$

$${\left(\ln \frac{1+x}{3}\right)}^2|{\partial }_c^{\alpha }{\partial }_{\gamma }^j{U}_{\theta }^{c,\gamma }(x)|\le C(m,K),\text{mbox}forany1\le |\alpha |+j\le m,{\alpha }_1=0,$$

$${\left(\ln \frac{1-x}{3}\right)}^2|{\partial }_c^{\alpha }{\partial }_{\gamma }^j{U}_{\theta }^{c,\gamma }(x)|\le C(m,K),\text{mbox}forany1\le |\alpha |+j\le m,{\alpha }_2=0,$$

$${\left(\ln \frac{1+x}{3}\right)}^2{\left(\ln \frac{1-x}{3}\right)}^2|{\partial }_c^{\alpha }{\partial }_{\gamma }^j{U}_{\theta }^{c,\gamma }(x)|\le C(m,K).$$

$$U_{\theta }(-1)={\tau }_1(c_1)\text{mbox}or{\tau }_2(c_1).$$

$$\underset{x\to -1^{+}}{\lim }(1+x)U_{\theta }^{\prime }(x)=0.$$

$$-2U_{\theta }(-1)+\frac{1}{2}{U}_{\theta }(-1{)}^2=2c_1.$$

$$c_1=\frac{1}{4}{\left[U_{\theta }(-1)-2\right]}^2-1\ge -1,$$

$$U_{\theta }(1)={\tau }_1^{\prime }(c_2)\text{mbox}or{\tau }_2^{\prime }(c_2).$$

$$|U_{\theta }(x)|\le C,-1<x<1.$$

$$-U_{\theta }(\bar{x})+\frac{1}{8}{U}_{\theta }^2(\bar{x})\le 2\tilde{x}{U}_{\theta }(\tilde{x})+\frac{1}{2}{U}_{\theta }^2(\tilde{x})\le P_c(\tilde{x})\le C_2(A).$$

Full text

1 Introduction

Consider the incompressible stationary Navier-Stokes equations (NSE) in $\mathbb{R}^{3}$:

[TABLE]

The equations are invariant under the scaling $u(x)\to\lambda u(\lambda x)$ and $p(x)\to\lambda^{2}p(\lambda x)$, $\lambda>0$. We study solutions which are invariant under the scaling. For such solutions $u$ is (-1)-homogeneous and $p$ is (-2)-homogeneous. We call them (-1)-homogeneous solutions according to the homogeneity of $u$.

We will write the NSE (1) in spherical coordinates $(r,\theta,\phi)$. A vector field $u$ can be written as

[TABLE]

where

[TABLE]

A vector field $u$ is called axisymmetric if $u_{r}$, $u_{\theta}$ and $u_{\phi}$ are independent of $\phi$, and is called no-swirl if $u_{\phi}=0$.

Landau discovered in [3] a three parameter family of explicit (-1)-homogeneous solutions of the stationary NSE (1), which are axisymmetric and with no swirl. These solutions are now called Landau solutions. The NSE (1) in the axisymmetric no-swirl case was converted earlier to an equation of Riccati type by Slezkin in [11]. The Riccati type equation was later independently derived by Yatseyev using a different method in [17], where various exact solutions were given. The Landau solutions were also independently found by Squire in [13]. Tian and Xin proved in [15] that all (-1)-homogeneous, axisymmetric nonzero solutions of (1) in $C^{2}(\mathbb{R}^{3}\setminus\{0\})$ are Landau solutions. A classification of all (-1)-homogeneous solutions was given by Šverák in [14]: all (-1)-homogeneous nonzero solutions of (1) in $C^{2}(\mathbb{R}^{3}\setminus\{0\})$ are Landau solutions. He also proved in the same paper that there is no nonzero (-1)-homogeneous solution of the stationary NSE in $C^{2}(\mathbb{R}^{n}\setminus\{0\})$ for $n\geq 4$. In dimension $n=2$, he characterized all such solutions satisfying a zero flux condition.

In [10], Serrin modeled the tornado by (-1)-homogeneous axisymmetric solutions of the three dimensional incompressible stationary Navier-Stokes equations in the half space with zero boundary conditions and one singularity on the unit sphere.

More recently, Karch and Pilarczyk showed in [2] that Landau solutions are asymptotically stable under any $L^{2}$ perturbations. Classifications of homogeneous solutions to the $2$-dimensional and $3$-dimensional stationary Euler equations are studied respectively in [6] by Luo and Shvydkoy, and in [11] by Shvydkoy. More studies on (-1)-homogeneous axisymmetric solutions of the stationary NSE (1) can be found in [1], [7], [8], [9], [10] and [16].

We are interested in analyzing solutions which are smooth on $\mathbb{S}^{2}$ minus finite points. We have classified in [4] all axisymmetric no-swirl solutions with one singularity at the south pole. They form a two dimensional surface with boundary in appropriate function spaces. These solutions are among the solutions found in [17], where the solutions were obtained by a different method. It was proved in [4] that there are no other solutions with precisely one singularity at the south pole. It was also proved there that there exists a curve of axisymmetric solutions with nonzero swirl emanating from every point in the interior and one part of the boundary of the surface of no-swirl solutions, while there is no such curve from any point on the other part of the boundary. Uniqueness results of nonzero swirl solutions near the no-swirl solution surface were also given in [4]. Our main result in this paper is the classification of all (-1)-homogeneous, axisymmetric no-swirl solutions of (1) which are smooth on $\mathbb{S}^{2}\setminus\{S,N\}$, where $S$ is the south pole and $N$ is the north pole. They are identified as a 4-dimensional surface with boundary in appropriate function spaces. We have established smoothness properties of the solutions surface in the four parameters. These properties are used in a subsequent paper [5] where we study the existence of (-1)-homogeneous axisymmetric solutions with non-zero swirl on $\mathbb{S}^{2}\setminus\{S,N\}$, emanating from the 4-dimensional solution surface.

A (-1)-homogeneous axisymmetric vector field $u$ is divergence free if and only if

[TABLE]

We work with a new unknown function and a different independent variable:

[TABLE]

As explained in [4], $(u,p)$ is a (-1)-homogeneous axisymmetric no-swirl solution of (1) if and only if $u_{\phi}=0$, $u_{r}$ is given by (2), $p$ is given by

[TABLE]

and $U_{\theta}$ satisfies, for some constants $c_{1},c_{2},c_{3}\in\mathbb{R}$,

[TABLE]

where ” ′ ” denotes differentiation in $x$, and $c:=(c_{1},c_{2},c_{3})$.

For each $c_{1}\geq-1$ and $c_{2}\geq-1$, define

[TABLE]

Define

[TABLE]

Theorem 1.1**.**

There exist $U^{\pm}_{\theta}(c)(x)\in C^{0}(J\times[-1,1])$, such that for every $c\in J$, $U^{\pm}_{\theta}(c)\in C^{\infty}(-1,1)$ satisfy (4) in $(-1,1)$, and $U^{-}_{\theta}(c)\leq U_{\theta}\leq U^{+}_{\theta}(c)$ for any solution $U_{\theta}$ of (4) in $(-1,1)$. Moreover, if $c_{3}>\bar{c}_{3}(c_{1},c_{2})$, $U^{-}_{\theta}<U^{+}_{\theta}$in $(-1,1)$, and if $c_{3}=\bar{c}_{3}(c_{1},c_{2})$,

[TABLE]

Next, for $c\in J$, introduce

[TABLE]

Define

[TABLE]

Theorem 1.2**.**

For each $(c,\gamma)$ in $I$, equation (4) has a unique solution $U^{c,\gamma}_{\theta}$ in $C^{\infty}(-1,1)\cap C^{0}[-1,1]$ satisfying $U^{c,\gamma}_{\theta}(0)=\gamma$. Moreover, these are all (-1)-homogeneous axisymmetric no-swirl solutions of the Navier-Stokes equations (1) on $\mathbb{S}^{2}\setminus\{S,N\}$.

Clearly, $U^{c,\gamma^{\pm}(c)}_{\theta}=U^{\pm}_{\theta}(c)$ for $c\in J$. Theorem 1.1 and Theorem 1.2 give a classification of all (-1)-homogeneous axisymmetric, no-swirl solutions of Navier-Stokes equations in $C^{2}(\mathbb{S}^{2}\setminus\{S,N\})$. There is a 1-1 correspondence between $U_{\theta}^{c,\gamma}$ and points in the four dimensional surface $I$.

Recall that Landau solutions are

[TABLE]

and they correspond to $U_{\theta}^{c,\gamma}$ with $c=0$ and $\gamma\in(-2,2)\setminus\{0\}$.

The solutions in $C^{\infty}(\mathbb{S}^{2}\setminus\{S\})$ correspond to $U_{\theta}^{c,\gamma}$ with $c_{2}=0$, $c_{1}=-2c_{3}$ and $\gamma^{-}(c)<\gamma\leq\gamma^{+}(c)$.

Define

[TABLE]

[TABLE]

Theorem 1.3**.**

Suppose $(c,\gamma)\in I$, then

(i) If $c_{3}>\bar{c}_{3}(c_{1},c_{2})$, then $\gamma^{-}(c)<\gamma^{+}(c)$, and for any $\gamma^{-}(c)\leq\gamma<\gamma^{\prime}\leq\gamma^{+}(c)$, $U_{\theta}^{c,\gamma}<U_{\theta}^{c,\gamma^{\prime}}$ in $(-1,1)$. Moreover,

[TABLE]

(ii)

[TABLE]

In addition to the continuity of $\gamma^{+}(c)$ and $\gamma^{-}(c)$ in $J$, they have further smoothness properties.

Theorem 1.4**.**

$\gamma^{+}(c)$* is in $C^{\infty}(J\setminus\{c\mid c_{1}=-1\})$, and $\gamma^{+}(-1,c_{2},c_{3})$ is in $C^{\infty}(J\cap\{c\mid c_{1}=-1\})$ as a function of $(c_{2},c_{3})$. $\gamma^{-}(c)$ is in $C^{\infty}(J\setminus\{c\mid c_{2}=-1\})$, and $\gamma^{-}(c_{1},-1,c_{3})$ is in $C^{\infty}(J\cap\{c\mid c_{2}=-1\})$ as a function of $(c_{1},c_{3})$.*

We also have the smoothness properties of $U^{c,\gamma}_{\theta}$ in $(c,\gamma)$. Let the subsets $J_{k}$, $1\leq k\leq 4$, of $J$ be defined as

[TABLE]

We define the following subsets of $I$: for $1\leq k\leq 4$, let

[TABLE]

As mentioned earlier, the following estimates of $U^{c,\gamma}_{\theta}$ are needed in our next paper on the existence of (-1)-homogeneous axisymmetric solutions of (1) with nonzero swirl on $\mathbb{S}^{2}\setminus\{S,N\}$.

Theorem 1.5**.**

Let $K$ be a compact set contained in one of $I_{k,l}$, $1\leq k\leq 4$, $l=1,2,3$. Then $U^{c,\gamma}_{\theta}$ is in $C^{\infty}(K\times(-1,1))$. Moreover,

(i) If $k=1$ and $l=1,2,3$, or $(k,l)=(2,2)$ or $(3,3)$, then for $-1<x<1$,

[TABLE]

where $j=0$ if $l=2,3$; $\alpha_{1}=0$ if $k=2$; and $\alpha_{2}=0$ if $k=3$.

(ii) If $(k,l)=(2,1)$ or $(2,3)$ or $(4,3)$, then for $-1<x<1$,

[TABLE]

where $j=0$ if $l=3$, and $\alpha_{2}=0$ if $k=4$.

(iii) If $(k,l)=(3,1),(3,2)$ or $(4,2)$, then for $-1<x<1$,

[TABLE]

where $j=0$ if $l=2$, and $\alpha_{1}=0$ if $k=4$.

(iv) If $(k,l)=(4,1)$, then for $-1<x<1$, and for any $1\leq|\alpha|+j\leq m$, $\alpha_{1}=\alpha_{2}=0$,

[TABLE]

To make the above notations clear, we point out that if $(k,l)=(1,2)$, estimate (9) means that for any compact set $K_{1}\subset J_{1}$, $\left|\partial_{c}^{\alpha}\left(U_{\theta}^{c,\gamma^{+}(c)}\right)\right|\leq C(m,K_{1})$. For other $I_{k,l}$ with $l=2$ or $3$, the left hand sides in (9)-(11) are interpreted analogously.

Remark 1.1**.**

The estimates in Theorem 1.5 are optimal in each $I_{k,l}$, see examples in Theorem 3.1 in [4].

Acknowledgment. The work of the second named author is partially supported by NSF grant DMS-1501004.

2 Proof of Theorems

2.1 Proof of Theorem 1.1, Theorem 1.2 and Theorem 1.3

As mentioned in Section 1, we work with the function $U_{\theta}$ and the variable $x$ given in (3). As explained in [4], the stationary NSE (1) of (-1)-homogeneous axisymmetric no-swirl solutions can be reduced to (4) for some constants $c_{1},c_{2}$ and $c_{3}$. We will show that the existence of solutions of (4) in $C^{1}(-1,1)$ depends on the constants $c_{1},c_{2}$ and $c_{3}$.

Recall the definitions in (7) and (8).

Lemma 2.1**.**

Let $\delta>0$, $U_{\theta}\in C^{1}(-1,-1+\delta)$ satisfy (4) with $c_{1},c_{2},c_{3}\in\mathbb{R}$. Then $c_{1}\geq-1$ and $U_{\theta}(-1):=\lim_{x\to-1^{+}}U_{\theta}(x)$ exists and is finite. Moreover,

[TABLE]

Proof.

By Proposition 7.1 in [4], $\lim_{x\to-1^{+}}U_{\theta}(x)$ exists and is finite and

[TABLE]

Sending $x$ to $-1$ in (4) leads to

[TABLE]

Thus,

[TABLE]

and $U_{\theta}(-1)=\tau_{1}(c_{1})$ or $\tau_{2}(c_{1})$. ∎

Lemma 2.1’****.

Let $\delta>0$, $U_{\theta}\in C^{1}(1-\delta,1)$ satisfy (4) with $c_{1},c_{2},c_{3}\in\mathbb{R}$. Then $c_{2}\geq-1$ and $U_{\theta}(1):=\lim_{x\to 1^{-}}U_{\theta}(x)$ exists and is finite. Moreover,

[TABLE]

Proof.

Consider $\tilde{U}_{\theta}(x):=-U_{\theta}(-x)$, and apply Lemma 2.1 to $\tilde{U}_{\theta}$. ∎

Lemma 2.2**.**

If $|c|\leq A$ for some constant $A>0$, then there exists some constant $C$, depending only on $A$, such that all $C^{1}$ solutions $U_{\theta}$ of (4) in $(-1,1)$ satisfy

[TABLE]

Proof.

By Lemma 2.1, there is some $C_{1}(A)>0$, such that $|U_{\theta}(\pm 1)|\leq C_{1}(A)$ for all solutions $U_{\theta}$ of (4) in $(-1,1)$.

If $\sup_{-1<x<1}|U_{\theta}(x)|\leq 8C_{1}(A)$, the proof is finished. Otherwise, there exists some $\bar{x}\in(-1,1)$ such that $|U_{\theta}(\bar{x})|=\max_{-1\leq x\leq 1}|U_{\theta}(x)|>8C_{1}(A)$. We may assume that $U_{\theta}(\bar{x})>8C_{1}(A)$, since the other case can be handled similarly. Then there exists some $-1<\tilde{x}<\bar{x}$ such that $U_{\theta}(\tilde{x})=\frac{U_{\theta}(\bar{x})}{2}$ and $U_{\theta}^{\prime}(\tilde{x})\geq 0$. By equation (4), we have

[TABLE]

It follows that $U_{\theta}(\bar{x})\leq C_{3}(A)$. The proof is finished. ∎

Lemma 2.3**.**

Let $c_{1}\geq-1$, $\tau=\tau_{2}(c_{1})$ or $\tau=\tau_{1}(c_{1})\not\in\{0,-2,-4,-6,\cdots\}$. Then for every $c_{2},c_{3}\in\mathbb{R}$, there exist $\delta>0$ depending only on an upper bound of $\sum_{i=1}^{3}|c_{i}|$ and a positive lower bound of $\inf_{k\in\mathbb{N}}|\tau+2k|$, and a sequence $\{a_{n}\}_{n=1}^{\infty}$ such that

[TABLE]

and

[TABLE]

is a real analytic solution of (4) in $(-1,-1+\delta)$. Moreover, $U_{\theta}$ is the unique real analytic solution of (4) in $(-1,-1+\delta^{\prime})$ satisfying $U_{\theta}(-1)=\tau$ for any $0<\delta^{\prime}\leq\delta$.

Proof of Lemma 2.3. Let $s=1+x$. Rewrite

[TABLE]

Suppose that $U_{\theta}=\tau+\sum_{n=1}^{\infty}a_{n}s^{n}$, then $U_{\theta}^{\prime}=\sum_{n=1}^{\infty}na_{n}s^{n-1}$. Plug them into (4),

[TABLE]

Compare coefficients,

[TABLE]

For $n\geq 3$,

[TABLE]

Since for any $n\geq 1$, $\tau\neq-2(n-1)$,

[TABLE]

it can be seen that $a_{n}$ is determined by $a_{1},...,a_{n-1}$, thus determined by $c_{1},c_{2},c_{3}$ and $\tau$.

Claim: there exists some $a>0$ large, depending only on an upper bound of $\sum_{i=1}^{3}|c_{i}|$ and a positive lower bound of $\inf_{k\in\mathbb{N}}|\tau+2k|$, such that

[TABLE]

Proof of Claim: Choose $a>1$ large such that for $1\leq n\leq 100|\tau|+100$, $|a_{n}|\leq a^{n}$.

Now for $n>100|\tau|+100$, suppose that for $1\leq k\leq n-1$, $|a_{k}|\leq a^{k}$, then by induction and the recurrence formula (13),

[TABLE]

The claim is proved.

So for $\delta<\frac{1}{a}$, $U_{\theta}=\tau+\sum_{n=1}^{\infty}a_{n}s^{n}$, with $s=1+x$, is a real analytic solution of (4) in $(-1,-1+\delta)$. The uniqueness of $U_{\theta}$ is clear from the proof above. ∎

Lemma 2.3’****.

Let $c_{2}\geq-1$, $\tau^{\prime}=\tau_{1}^{\prime}(c_{2})$ or $\tau^{\prime}=\tau_{2}^{\prime}(c_{2})\not\in\{0,2,4,6,\cdots\}$. Then for every $c_{1},c_{3}\in\mathbb{R}$, there exist $\delta>0$, depending only on an upper bound of $\sum_{i=1}^{3}|c_{i}|$ and a positive lower bound of $\inf_{k\in\mathbb{N}}|\tau^{\prime}-2k|$, and a sequence $\{a_{n}\}_{n=1}^{\infty}$ such that

[TABLE]

and

[TABLE]

is a real analytic solution of (4) in $(1-\delta,1)$. Moreover, $U_{\theta}$ is the unique real analytic solution of (4) in $(1-\delta^{\prime},1)$ satisfying $U_{\theta}(1)=\tau^{\prime}$ for any $0<\delta^{\prime}\leq\delta$.

The following two lemmas give some local comparison results.

Lemma 2.4**.**

Suppose $0<\delta<2$, $U_{\theta},\tilde{U}_{\theta}\in C^{1}(-1,-1+\delta]\cap C^{0}[-1,-1+\delta]$ satisfy

[TABLE]

Suppose also that one of the following two conditions holds.

(i) $U_{\theta}(-1)\geq\tilde{U}_{\theta}(-1)>2$.

(ii) $U_{\theta}(-1)=\tilde{U}_{\theta}(-1)=2$, and

[TABLE]

Then either

[TABLE]

or there exists $\delta^{\prime}\in(0,\delta)$ such that

[TABLE]

Proof.

Let $g=U_{\theta}-\tilde{U}_{\theta}$, then $g(-1)\geq 0$ and $g$ satisfies

[TABLE]

where $b(x)$ is given by

[TABLE]

Let

[TABLE]

Then $w$ satisfies, using (15), that

[TABLE]

Under condition either (i) or (ii), we have

[TABLE]

Using this and the fact that $g(-1)\geq 0$, we have $\liminf_{x\to-1^{+}}w(x)\geq 0$. Therefore, using (17), we have either $w>0$ in $(-1,-1+\delta)$ or there exists a constant $\delta^{\prime}$, $0<\delta^{\prime}<\delta$ such that $w\equiv 0$ in $(-1,-1+\delta^{\prime})$. The lemma is proved. ∎

Corollary 2.1**.**

For $c_{1}>-1$, $c_{2},c_{3}\in\mathbb{R}$ and $0<\delta<2$, there exists at most one solution $U_{\theta}$ of (4) in $C^{1}(-1,-1+\delta)$ satisfying

[TABLE]

Proof.

Since $\tau_{2}(c_{1})>2$ for $c_{1}>-1$, the uniqueness follows from (i) of Lemma 2.4. ∎

Similarly, we have

Lemma 2.4’****.

Suppose $0<\delta<2$, $U_{\theta},\tilde{U}_{\theta}\in C^{1}[1-\delta,1)\cap C^{0}[1-\delta,1]$ satisfy

[TABLE]

Suppose also that one of the following two conditions holds.

(i) $U_{\theta}(1)\leq\tilde{U}_{\theta}(1)<-2$,

(ii) $U_{\theta}(1)=\tilde{U}_{\theta}(1)=-2$, and

[TABLE]

Then either

[TABLE]

or there exists $\delta^{\prime}\in(0,\delta)$ such that

[TABLE]

Corollary 2.1’****.

For $c_{2}>-1$, $c_{1},c_{3}\in\mathbb{R}$ and $0<\delta<2$, there exists at most one solution $U_{\theta}$ of (4) in $C^{1}(1-\delta,1)$ satisfying

[TABLE]

Now we are ready to analyze the global behavior of axisymmetric, no-swirl solutions of NSE (4) in $(-1,1)$. The behavior of solutions depends closely on parameters $c_{1},c_{2},c_{3}\in\mathbb{R}$.

Recall the definition of $\bar{c}_{3}(c_{1},c_{2})$ given by (5), we have

Lemma 2.5**.**

*Suppose $c_{1}\geq-1$, $c_{2}\geq-1$, $c_{3}=\bar{c}_{3}(c_{1},c_{2})$, then $U_{\theta}^{*}(c_{1},c_{2})$ given by (6) is the unique $C^{1}$ solution of (4) in $(-1,1)$. In particular, *

[TABLE]

Proof.

A direct calculation shows that $U_{\theta}^{*}:=U_{\theta}^{*}(c_{1},c_{2})$ is a $C^{1}$ solution of (4) in $(-1,1)$. It remains to prove the uniqueness.

Let $U_{\theta}$ be a $C^{1}$ solution of (4) in $(-1,1)$, $U_{\theta}\not\equiv U_{\theta}^{*}$. By Lemma 2.1 and Lemma 2.1’, $U_{\theta}$ can be extended as a function in $C^{0}[-1,1]$, $U_{\theta}(-1)\in\{\tau_{1}(c_{1}),\tau_{2}(c_{1})\}$, $U_{\theta}(1)\in\{\tau_{1}^{\prime}(c_{2}),\tau_{2}^{\prime}(c_{2})\}$.

By Corollary 2.1 and (ii) of Lemma 2.4, we know that there exists a constant $0<\delta_{1}<\frac{1}{2}$ such that $U_{\theta}<U_{\theta}^{*}$ in $(-1,-1+\delta_{1})$. Similarly, by Corollary 2.1’ and (ii) of Lemma 2.4’, we know that there exists a constant $0<\delta_{2}<\frac{1}{2}$ such that $U_{\theta}>U_{\theta}^{*}$ in $(1-\delta_{2},1)$.

Therefore, there exists a point $\bar{x}\in(-1+\delta_{1},1-\delta_{2})$ such that $U_{\theta}(\bar{x})=U_{\theta}^{*}(\bar{x})$. Standard uniqueness theory of ODE implies that $U_{\theta}\equiv U_{\theta}^{*}$ in $(-1,1)$. This is a contradiction. ∎

Lemma 2.6**.**

Suppose $c_{1}\geq-1$, $c_{2}\geq-1$, $c_{3}<\bar{c}_{3}(c_{1},c_{2})$, then (4) has no solution in $C^{1}(-1,1)$.

Proof.

If $U_{\theta}$ is a $C^{1}$ solution of (4) in $(-1,1)$. By Lemma 2.1 and Lemma 2.1’, $U_{\theta}$ can be extended as a function in $C^{0}[-1,1]$, $U_{\theta}(-1)\in\{\tau_{1}(c_{1}),\tau_{2}(c_{1})\}$, $U_{\theta}(1)\in\{\tau_{1}^{\prime}(c_{2}),\tau_{2}^{\prime}(c_{2})\}$.

By Lemma 2.5, $U_{\theta}^{*}:=U_{\theta}^{*}(c_{1},c_{2})$ is the unique solution of (4) with $c_{3}=\bar{c}_{3}(c_{1},c_{2})$. Since $c_{3}<\bar{c}_{3}(c_{1},c_{2})$, $U_{\theta}\not\equiv U_{\theta}^{*}$ in any open interval in $(-1,1)$. We first assume that $U_{\theta}(\bar{x})>U_{\theta}^{*}(\bar{x})$ at some point $\bar{x}\in(-1,1)$. Since $c_{3}<\bar{c}_{3}(c_{1},c_{2})$ we have

[TABLE]

Since $U_{\theta}(-1)\leq U_{\theta}^{*}(-1)$, we have, in view of Lemma 2.4, there exists $\delta>0$ such that $U_{\theta}<U_{\theta}^{*}$ in $(-1,-1+\delta)$.

Now with $U_{\theta}(\bar{x})>U_{\theta}^{*}(\bar{x})$ and $U_{\theta}<U_{\theta}^{*}$ in $(-1,-1+\delta)$, there exist a point $\xi\in(-1+\delta,\bar{x})$ such that

[TABLE]

which contradicts inequality (18) at $\xi$.

Similar arguments lead to a contradiction when $U_{\theta}(\bar{x})<U_{\theta}^{*}(\bar{x})$ for some $\bar{x}\in(-1,1)$ by showing $U_{\theta}>U_{\theta}^{*}$ near $x=1$. The lemma is proved. ∎

Lemma 2.7**.**

Suppose $c_{1}\geq-1$, $c_{2}\geq-1$, $c_{3}>\bar{c}_{3}(c_{1},c_{2})$. Let $U_{\theta}^{+}(c)$ be the power series solution, obtained in Lemma 2.3 with $U_{\theta}^{+}(c)(-1)=\tau_{2}(c_{1})$, of (4) in $(-1,-1+\delta)$, then $U_{\theta}^{+}(c)$ can be extended to be a solution of (4) in $(-1,1)$, and $U_{\theta}^{+}(c)(1)=\tau_{2}^{\prime}(c_{2})$.

Let $U_{\theta}^{-}(c)$ be the power series solution, obtained in Lemma 2.3’ with $U_{\theta}^{-}(c)(1)=\tau_{1}^{\prime}(c_{2})$, of (4) in $(1-\delta,1)$, then $U_{\theta}^{-}(c)$ can be extended to be a solution of (4) in $(-1,1)$, and $U_{\theta}^{-}(c)(-1)=\tau_{1}(c_{1})$. Moreover, $U_{\theta}^{-}(c)<U_{\theta}^{+}(c)$ in $(-1,1)$.

Proof.

We only need to prove that $U_{\theta}^{+}:=U_{\theta}^{+}(c)$ can be extended to be a solution of (4) in $(-1,1)$ and $U_{\theta}^{+}(1)=\tau_{2}^{\prime}(c_{2})$, since similar arguments work for $U_{\theta}^{-}(c)$.

Standard existence theory of ODE implies that $U_{\theta}^{+}$ can be extended to the maximal interval of existence, say $(-1,\xi)$, $\xi\in(-1+\delta,1]$. Since $c_{3}>\bar{c}_{3}(c_{1},c_{2})$, we have, with $U_{\theta}^{*}:=U_{\theta}^{*}(c_{1},c_{2})$,

[TABLE]

Since $U_{\theta}^{+}(-1)=U_{\theta}^{*}(-1)=\tau_{2}(c_{1})\geq 2$, by Lemma 2.4 and the fact that $U_{\theta}^{+},U_{\theta}^{*}$ can not coincide in any open interval, we have $U_{\theta}^{+}>U_{\theta}^{*}$ in $(-1,\xi)$.

If $\xi<1$, since $U_{\theta}^{+}$ is bounded from below by $U_{\theta}^{*}$, there exists a sequence of points $\{x_{i}\}$ satisfying

[TABLE]

Then, in each interval $(x_{i},x_{i+1})$, we can find a point $y_{i}$ such that

[TABLE]

Taking $x=y_{i}$ in equation (4), and sending $i$ to infinity, we obtain a contradiction. So $\xi=1$. By Lemma 2.1, $\lim_{x\to 1^{+}}U_{\theta}^{+}(x)$ exists and is finite.

We have extended $U_{\theta}^{+}$ to be a solution of (4) in $C^{1}(-1,1)\cap C^{0}[-1,1]$ and $U_{\theta}^{+}>U_{\theta}^{*}$ in $(-1,1)$.

Similarly, $U_{\theta}^{-}$ can be extended to $C^{0}[-1,1]$, and $U_{\theta}^{-}<U_{\theta}^{*}<U_{\theta}^{+}$ in $(-1,1)$.

By Lemma 2.1’, $U_{\theta}^{+}(1)\in\{\tau_{1}^{\prime}(c_{2}),\tau_{2}^{\prime}(c_{2})\}$. If $c_{2}=-1$, $\tau_{1}^{\prime}(c_{2})=\tau_{2}^{\prime}(c_{2})$, so $U_{\theta}^{+}(1)=\tau_{2}^{\prime}(c_{2})$. If $c_{2}>-1$, since $U_{\theta}^{-}(1)=\tau_{1}^{\prime}(c_{2})$ and $U_{\theta}^{+}>U_{\theta}^{-}$ in $(-1,1)$, by Corollary 2.1’, we have $U_{\theta}^{+}(1)=\tau_{2}^{\prime}(c_{2})$. Similarly, $U_{\theta}^{-}(-1)=\tau_{1}(c_{1})$. Lemma 2.7 is proved. ∎

Lemma 2.8**.**

Suppose $c_{1}\geq-1$, $c_{2}\geq-1$, $c_{3}>\bar{c}_{3}(c_{1},c_{2})$, then any $C^{1}$ solution $U_{\theta}$ of (4) in $(-1,1)$ other than $U_{\theta}^{\pm}(c)$ satisfies

[TABLE]

[TABLE]

Proof.

By Lemma 2.1 and Lemma 2.1’, $U_{\theta}$ can be extended to $C^{0}[-1,1]$ with $U_{\theta}(-1)=\tau_{1}(c_{1})$ or $\tau_{2}(c_{1})$, and $U_{\theta}(1)=\tau_{1}^{\prime}(c_{2})$ or $\tau^{\prime}_{2}(c_{2})$.

We only need to prove $U_{\theta}<U_{\theta}^{+}(c)$ in $(-1,1)$ and $U_{\theta}(-1)=\tau_{1}(c_{1})$, since similar arguments imply that $U_{\theta}>U_{\theta}^{-}(c)$ in $(-1,1)$ and $U_{\theta}(1)=\tau_{2}^{\prime}(c_{2})$.

From the standard uniqueness theory of ODE, we know that the graph of $U_{\theta}$ and $U_{\theta}^{+}(c)$ can not intersect in $(-1,1)$. So we either have $U_{\theta}<U_{\theta}^{+}(c)$ in $(-1,1)$ or $U_{\theta}>U_{\theta}^{+}(c)$ in $(-1,1)$.

If $U_{\theta}>U_{\theta}^{+}(c)$ in $(-1,-1+\delta)$, then, by Lemma 2.1, $U_{\theta}(-1)=U_{\theta}^{+}(c)(-1)=\tau_{2}(c_{1})\geq 2$. Note that $U_{\theta}^{+}(c)$ satisfies (14), we can apply Lemma 2.4 to obtain $U_{\theta}\leq U_{\theta}^{+}(c)$, a contradiction. So $U_{\theta}<U_{\theta}^{+}(c)$ in $(-1,1)$.

If $\tau_{1}(c_{1})<\tau_{2}(c_{1})$, the uniqueness result Corollary 2.1 implies that $U_{\theta}(-1)=\tau_{1}(c_{1})$. If $\tau_{1}(c_{1})=\tau_{2}(c_{1})$, we again have $U_{\theta}(-1)=\tau_{1}(c_{1})$. Lemma 2.8 is proved. ∎

Proof of Theorem 1.1: For $c\in J$, if $c_{3}=\bar{c}_{3}$, by Lemma 2.5, $U^{*}_{\theta}(c_{1},c_{2})$ in (6) is the unique solution of (4) in $(-1,1)$.

If $c_{3}>\bar{c}_{3}$, let $U^{+}_{\theta}(c)$ and $U^{-}_{\theta}(c)$ be the functions in Lemma 2.7. By Lemma 2.3, Lemma 2.3’, Lemma 2.7 and Lemma 2.8, $U^{\pm}_{\theta}(c)\in C^{\infty}(-1,1)\cap C^{0}[-1,1]$ satisfy (4) in $(-1,1)$, and $U^{-}_{\theta}(c)<U^{+}_{\theta}(c)$. Moreover, $U^{-}_{\theta}(c)\leq U_{\theta}\leq U^{+}_{\theta}(c)$ for any solution $U_{\theta}$ of (4) in $(-1,1)$.

Now we prove the continuity of $U^{+}_{\theta}(c)(x)$ in $(c,x)$, the same arguments applies to $U^{-}_{\theta}$.

For every $(\hat{c},\hat{x})\in J\times[-1,1]$, we prove the continuity of $U_{\theta}^{+}$ at $(\hat{c},\hat{x})$. By Lemma 2.3, there exists some $\delta>0$, such that $U^{+}_{\theta}(c)(x)$ is continuous in $(B_{1}(\hat{c})\cap J)\times[-1,-1+\delta]$, where $B_{1}(\hat{c})$ is the unit ball in $\mathbb{R}^{3}$ centered at $\hat{c}$.

Consider

[TABLE]

for $a$ close to $a_{0}:=U_{\theta}^{+}(c)(-1+\frac{\delta}{2})$.

By standard ODE theories, for any $0<\epsilon<2-\delta$, there exists some positive constants $\mu$, such that $U^{+}_{\theta}\in C((a_{0}-\mu,a_{0}+\mu)\times(B_{1}(\hat{c})\cap J)\times[-1+\frac{\delta}{4},1-\epsilon])$.

The continuity of $U^{+}_{\theta}(c)(x)$ at $\hat{x}=1$ follows from Lemma 2.11’, which will be given later.∎

Proof of Theorem 1.2: Let $(c,\gamma)\in I$. If $c_{3}=\bar{c}_{3}$, then $\gamma=\gamma^{+}(c)=\gamma^{-}(c)$ by Theorem 1.1, $U^{c,\gamma}_{\theta}:=U^{\pm}_{\theta}(c)$ given by (6) is the unique solution of (4) satisfying $U^{c,\gamma}_{\theta}(0)=\gamma$.

If $c_{3}>\bar{c}_{3}(c_{1},c_{2})$, and $\gamma=\gamma^{\pm}(c)$, then $U^{c,\gamma}_{\theta}:=U^{\pm}_{\theta}(c)$ is the unique solution of (4) satisfying $U^{c,\gamma}_{\theta}(0)=\gamma$.

For $c_{3}>\bar{c}_{3}(c_{1},c_{2})$, and $\gamma^{-}(c)<\gamma<\gamma^{+}(c)$, let $U_{\theta}^{c,\gamma}$ be the unique local solution of (4) satisfying $U_{\theta}^{c,\gamma}(0)=\gamma$. By standard ODE theory, $U_{\theta}^{c,\gamma}$ can be extended to a $C^{\infty}$ solution in $(-1,1)$ satisfying $U_{\theta}^{-}(c)<U_{\theta}^{c,\gamma}<U_{\theta}^{+}(c)$.

By Lemma 2.1 and Lemma 2.1’, $U_{\theta}^{c,\gamma}$ can be extend as a function in $C^{0}[-1,1]$.

To complete the proof of Theorem 1.2, it remains to show that $\{U^{c,\gamma}_{\theta}|(c,\gamma)\in J\}$ are all the solutions.

For $c\in\mathbb{R}^{3}$, let $U_{\theta}$ be a solution of (4) in $(-1,1)$, By Lemma 2.1 and Lemma 2.1’, $c_{1}\geq-1$ and $c_{2}\geq-1$. Then by Lemma 2.6, $c_{3}\geq\bar{c}_{3}$. So $c\in J$. By Theorem 1.1, we have $U^{-}_{\theta}(c)\leq U_{\theta}\leq U_{\theta}^{+}(c)$. So $\gamma:=U_{\theta}(0)$ satisfies $\gamma^{-}(c)\leq\gamma\leq\gamma^{+}(c)$, and $U_{\theta}=U^{c,\gamma}_{\theta}$.

∎

Lemma 2.9**.**

Suppose $c_{1}\geq-1$, $c_{2}\geq-1$, $c_{3}>\bar{c}_{3}(c_{1},c_{2})$, then $\gamma^{-}(c)<\gamma^{+}(c)$, and the graphs

[TABLE]

foliate the set

[TABLE]

in the sense that for any $\gamma,\gamma^{\prime}\in\mathbb{R}$, $\gamma^{-}(c)\leq\gamma<\gamma^{\prime}\leq\gamma^{+}(c)$, $U_{\theta}^{c,\gamma}<U_{\theta}^{c,\gamma^{\prime}}$ in $(-1,1)$ and $K_{2}=\bigcup_{\gamma^{-}(c)\leq\gamma\leq\gamma^{+}(c)}K_{1}(\gamma)$. Moreover, $U_{\theta}^{c,\gamma}$ is a continuous function of $(c,\gamma,x)$ in $J\times[\gamma^{-}(c),\gamma^{+}(c)]\times(-1,1)$.

Proof.

By standard uniqueness theories of ODE,

[TABLE]

It is obvious that $K_{1}(\gamma)\subseteq K_{2}$. On the other hand, let $(x_{0},y_{0})\in K_{2}$, so $-1<x_{0}<1$ and $U_{\theta}^{c,\gamma^{-}}(x_{0})<y_{0}<U_{\theta}^{c,\gamma^{+}}(x_{0})$. By standard existence and uniqueness theories of ODE, there exists a $C^{1}$ solution $U_{\theta}$ of (4) in $(-1,1)$ satisfying $U_{\theta}(x_{0})=y_{0}$ and $U_{\theta}^{c,\gamma^{-}}<U_{\theta}<U_{\theta}^{c,\gamma^{+}}$ in $(-1,1)$. In particular,

[TABLE]

$U_{\theta}=U_{\theta}^{c,\gamma}$ with $\gamma=U_{\theta}(0)$ and therefore $x_{0},y_{0}\in K_{1}(\gamma)$. We have proved that $K_{2}=\bigcup_{\gamma^{-}\leq\gamma\leq\gamma^{+}}K_{1}(\gamma)$.

The continuity of $U_{\theta}^{c,\gamma}$ for $(c,\gamma,x)$ in $J\times[\gamma^{-}(c),\gamma^{+}(c)]\times(-1,1)$ can be derived from (4), and the continuous dependence of ODE on its boundary conditions. ∎

Proof of Theorem 1.3: Theorem 1.3 follows from Lemma 2.5 - Lemma 2.9. ∎

2.2 Proof of Theorem 1.4 and Theorem 1.5

In the following context, in $J\cap\{c\in J\mid c_{1}=-1\}$, $U_{\theta}^{+}(c)=U_{\theta}^{+}(-1,c_{2},c_{3})$ is viewed as a function of $(c_{2},c_{3})$, and $\partial_{c}^{\alpha}U_{\theta}^{+}(c)(x)$ means $\partial_{(c_{2},c_{3})}^{\alpha}U_{\theta}^{+}(c)(x)$. In $J\cap\{c\mid c_{2}=-1\}$, $U_{\theta}^{-}(c)=U_{\theta}^{-}(c_{1},-1,c_{3})$ is viewed as a function of $(c_{1},c_{3})$, and $\partial_{c}^{\alpha}U_{\theta}^{-}(c)(x)$ means $\partial_{(c_{1},c_{3})}^{\alpha}U_{\theta}^{-}(c)(x)$.

Lemma 2.10**.**

For any integer $m\geq 0$, and any compact subset $K$ contained in either $J\setminus\{c\mid c_{1}=-1\}$ or $J\cap\{c\in J\mid c_{1}=-1\}$, there exist some positive constants $\delta$ and $C$, depending only on $m$ and $K$, such that $U_{\theta}^{+}(c)\in C^{m}(K\times(-1,-1+\delta))$, and

[TABLE]

Proof.

Let $\alpha=(\alpha^{1},\alpha^{2},\alpha^{3})$ denote a multi-index where $\alpha^{i}\geq 0$, $i=1,2,3$. The partial derivative $\partial^{\alpha}=\partial^{\alpha_{1}}_{c_{1}}\partial^{\alpha_{2}}_{c_{2}}\partial^{\alpha_{3}}_{c_{3}}$ and the absolute value $|\alpha|=\sum_{i=1}^{3}\alpha^{i}$.

By Lemma 2.3 and its proof, there exists $\delta>0$, depending only on $K$, such that for $c\in K$, $U_{\theta}^{+}(c)$ can be expressed as

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Estimate (23) guarantees that the power series expansion of $U^{+}_{\theta}(c)(x)$ is uniformly convergent in $(-1,-1+\delta)$.

By the above expressions and relations it can be seen that $\tau(c)$ and $a_{n}(c)$ are all $C^{\infty}$ functions of $c$ in $J$. So to prove the lemma, we just need to show that there exists some $\delta^{\prime}>0$, depending only on $m$ and $K$, such that for any multi-index $\alpha$ satisfying $1\leq|\alpha|\leq m$, the series

[TABLE]

is absolutely convergent in $(-1,-1+\delta^{\prime})$ uniformly for $c\in K$.

Case 1: $K\subset J\setminus\{c\mid c_{1}=-1\}$.

Let $C(m,K)$ be a constant depending only on $m$ and $K$ which may vary from line to line. If $K$ is a compact set in $J\setminus\{c\mid c_{1}=-1\}$, there exists some constant $\delta_{1}(K)>0$, such that $4+4c_{1}\geq\delta_{1}(K)$. Using this, (21), (22), and the fact that $\tau>2$, we have

[TABLE]

Next, let $g_{n}(c):=\frac{1}{2n-2+\tau}$. By the above estimates and the fact that $\tau>2$, we have

[TABLE]

To prove the existence of $\delta^{\prime}$ such that the series in (24) is convergent for all $1\leq|\alpha|\leq m$ uniformly in $K$, we will only need to show the following:

Claim: there exists some $a>0$, depending only on $m$ and $K$, such that

[TABLE]

holds for all $n\geq 1$.

Proof of Claim: We prove it by induction on $n$. Let $a$ be a constant to be determined in the proof.

By estimate (25), there exists some constant $\bar{a}$, depending only on $m$ and $K$, such that for all $a\geq\bar{a}$, $(P_{1})$ and $(P_{2})$ hold. We may assume that $\bar{a}\geq\frac{1}{2\delta}$ so that we know from (23) that

[TABLE]

for all $c\in K$ and $n\geq 1$.

Now for $n\geq 3$, suppose that for some $a\geq\bar{a}$, $(P_{k})$ holds for all $1\leq k\leq n-1$.

Let $Q_{n}(c):=\sum_{k+l=n,k,l\geq 1}a_{k}a_{l}$. Then (22) can be written as

[TABLE]

So

[TABLE]

Using (26), by computation we have

[TABLE]

Let $a\geq\bar{a}$, using the definition of $Q_{n}(c)$, by induction we have that,

[TABLE]

Similarly, by (26), (27) and the induction hypothesis, we have

[TABLE]

Plug (29) and (30) in (28), we have that for $|\alpha|\geq 1$,

[TABLE]

If from the beginning we use $a=\max\{\bar{a},C(m,K)\}$ for the induction hypothesis, we have

[TABLE]

So the claim is true for all $n$. The lemma is proved for $K\subset J\setminus\{c\mid c_{1}=-1\}$.

Case 2: $K\subset J\cap\{c\mid c_{1}=-1\}$.

In this case $\tau=2$ and $g_{n}(c)$ is a constant in $K$. By similar arguments as in Case 1, we have the same estimate for $a_{n}$ and the proof is finished. ∎

Corollary 2.2**.**

For any $K\subset J\setminus\{c\mid c_{1}=-1\}$ or $J\cap\{c\mid c_{1}=-1\}$, $U_{\theta}^{+}(c)\in C^{\infty}(K\times(-1,1))$. Moreover, for any $\epsilon>0$, $m\in\mathbb{N}$, there exists some positive constant $C$, depending only on $m$, $K$, and $\epsilon$, such that

[TABLE]

Proof.

We know that $U^{+}_{\theta}(c)$ satisfies (4) in $(-1,1)$ and $||U^{+}_{\theta}||_{L^{\infty}(-1,1)}\leq C$, where $C$ depends only on $K$. By Lemma 2.10, for any positive integer $m$, there exist some positive constants $\delta$ and $C$, depending only on $m$ and $K$, such that $U_{\theta}^{+}(c)\in C^{m}(K\times(-1,-1+\delta))$ and (20) holds.

Consider (19) for $a$ close to $a_{0}:=U_{\theta}^{+}(c)(-1+\frac{\delta}{2})$. By standard ODE theories, for any $0<\epsilon<2-\delta$, there exist some positive constants $\mu$ and $C$, depending on $m$, $K$ and $\epsilon$, such that if $|a-a_{0}|<\mu$, then there exists a solution $U_{\theta}\in C^{m}((a_{0}-\mu,a_{0}+\mu)\times K\times[-1+\frac{\delta}{4},1-\epsilon])$ of (19), and

[TABLE]

It follows, also in view of (20), that $U^{+}_{\theta}(c)=U_{\theta}|_{a=a_{0}}$ satisfies (31). ∎

Similarly to Lemma 2.10 and Corollary 2.2 we have

Lemma 2.10’****.

For any integer $m\geq 0$, and any compact set $K$ contained in either $J\setminus\{c\in J\mid c_{2}=-1\}$ or $J\cap\{c\in J\mid c_{2}=-1\}$, there exist some positive constants $\delta$ and $C$, depending only on $m$ and $K$, such that $U_{\theta}^{-}(c)\in C^{m}(K\times(1-\delta,1))$, and

[TABLE]

Corollary 2.2’****.

For any $K\subset J\setminus\{c\in J\mid c_{2}=-1\}$ or $J\cap\{c\in J\mid c_{2}=-1\}$, $U_{\theta}^{-}(c)\in C^{\infty}(K\times(-1,1))$. Moreover, for any $\epsilon>0$, $m\in\mathbb{N}$, there exists some positive constant $C$, depending only on $m$, $K$, and $\epsilon$, such that

[TABLE]

Theorem 1.4 can be obtained from Corollary 2.2 and Corollary 2.2’.

To prove Theorem 1.5, we make the following observations.

By Corollary 2.2 and Corollary 2.2’, we know that for $1\leq k\leq 4$ and $l=2,3$, $U_{\theta}^{+}(c)$ and $U_{\theta}^{-}(c)$ are smooth in $I_{k,l}$. Here the smoothness means $U_{\theta}^{+}(c)$ and $U_{\theta}^{-}(c)$ are smooth restricted to each $I_{k,l}$.

By standard ODE theory, since $U_{\theta}$ satisfies (4), it is smooth in $I_{k,1}$ for each $1\leq k\leq 4$. So a solution $U_{\theta}$ of the initial value problem

[TABLE]

is smooth with respect to $(c,\gamma)$ in each $I_{k,l}$, $1\leq k\leq 4$, $1\leq l\leq 3$. It remains to prove the estimates (i)-(iv) in Theorem 1.5.

We first make some estimates about the solutions $U_{\theta}$ of (32).

Recall that for each $(c,\gamma)\in I$, there is a solution $U_{\theta}=U_{\theta}^{c,\gamma}$ satisfying (32).

Lemma 2.11**.**

Let $K$ be a compact subset of $I\setminus\{(c,\gamma)\mid\gamma=\gamma^{+}(c)\}$. Then for any $\epsilon>0$, there exists some $\delta>0$, depending only on $\epsilon$ and $K$, such that for any $(c,\gamma)\in K$,

[TABLE]

Proof.

We prove it by contradiction. Suppose the contrary, there exist some $\epsilon>0$ and a sequence $(c^{i},\gamma^{i})\in K$ and $-1<x_{i}<-1+\frac{1}{i}$, such that

[TABLE]

Since $K$ is compact, there exist a subsequence, still denoted as $(c^{i},\gamma^{i})$, and some $(c,\gamma)\in K$, such that $(c^{i},\gamma^{i})\to(c,\gamma)$ as $i\to\infty$.

Denote $U_{\theta}^{i}=U_{\theta}^{c^{i},\gamma^{i}}$. By standard ODE theory, we have that $U_{\theta}^{i}\to U_{\theta}:=U_{\theta}^{c,\gamma}$ in $C^{1}_{loc}(-1,1)$. We first assume that

[TABLE]

Since $(c,\gamma),(c^{i},\gamma^{i})\in K$, we have $\gamma<\gamma^{+}(c)$ and $\gamma^{i}<\gamma^{+}(c^{i})$. Then, by Theorem 1.3 (ii), $U_{\theta}^{i}(-1)=2-2\sqrt{1+c_{1}^{i}}$ and $U_{\theta}(-1)=2-2\sqrt{1+c_{1}}$.

Since $c^{i}\to c$, we have $U_{\theta}^{i}(-1)\to U_{\theta}(-1)$, and therefore for sufficiently large $i$,

[TABLE]

Case 1: $U_{\theta}(-1)<2$.

There exists some $\epsilon_{1}>0$, such that $U_{\theta}(-1)+3\epsilon_{1}<\min\{2,U_{\theta}(-1)+\frac{\epsilon}{4}\}$. For sufficiently large $i$ we have $U_{\theta}^{i}(-1)<2-\epsilon_{1}$. Since $U_{\theta}^{i}\to U_{\theta}$ in $C^{1}_{loc}(-1,1)$, we have

[TABLE]

By the continuity of $U_{\theta}$,

[TABLE]

Thus for large $j$, there exists $i_{j}\geq j$, such that $-1<x_{i_{j}}<-1+\frac{1}{j}$ and

[TABLE]

By (34), $U_{\theta}^{i_{j}}(x_{i_{j}})>U_{\theta}(-1)+2\epsilon_{1}$.

Choose $\tilde{x}_{i_{j}}\in(x_{i_{j}},-1+\frac{1}{j})$, satisfying

[TABLE]

Plugging $U^{i_{j}}_{\theta}$ and $\tilde{x}_{i_{j}}$ in (4), using the above, we have

[TABLE]

Sending $j\to\infty$ in (35) leads to

[TABLE]

where $\xi:=U_{\theta}(-1)+2\epsilon_{1}\in(U_{\theta}(-1),2)$.

Since $h(s)$ is a decreasing function when $s\leq 2$, we have

[TABLE]

a contradiction.

Case 2: $U_{\theta}(-1)=2$.

By (34) and the convergence of $U_{\theta}^{i}(-1)$ to $U_{\theta}(-1)$, we may choose $\tilde{x}_{i}\in(-1,x_{i})$ satisfying

[TABLE]

Plugging $U^{i}_{\theta}$ and $\tilde{x}_{i}$ in (4), using the above, we have

[TABLE]

Sending $i\to\infty$, the above leads to

[TABLE]

a contradiction.

Now, if instead of (33),

[TABLE]

then for sufficiently large $i$, we have

[TABLE]

As in the proof of Case 1, there exists $\tilde{x}_{i_{j}}\to-1$, such that

[TABLE]

Plugging $U^{i_{j}}_{\theta}$ and $\tilde{x}_{i_{j}}$ in (4), using the above, we have

[TABLE]

Sending $j\to\infty$ in the above leads to

[TABLE]

Since $h(s)$ is a decreasing function when $s\leq 2$, we have

[TABLE]

a contradiction. ∎

Similarly we have

Lemma 2.11’****.

Let $K$ be a compact subset of $I\setminus\{(c,\gamma)\mid\gamma=\gamma^{-}(c)\}$. Then for any $\epsilon>0$, there exists some $\delta>0$, depending only on $\epsilon$ and $K$, such that for any $(c,\gamma)\in K$,

[TABLE]

Lemma 2.12**.**

Let $K$ be a compact subset of $I\setminus\{(c,\gamma)\mid c_{1}=-1\textrm{ or }\gamma=\gamma^{+}(c)\}$. Then for any $\epsilon>0$, there exist some positive constants $\delta$ and $C$, depending only on $\epsilon$ and $K$, such that for any $(c,\gamma)\in K$,

[TABLE]

Proof.

For convenience, let us denote $U_{\theta}=U_{\theta}^{c,\gamma}$, $\alpha_{0}=\sqrt{1+c_{1}}$, $\tau_{1}=\tau_{1}(c_{1})=2-2\sqrt{1+c_{1}}$, and $\tau_{2}=\tau_{2}(c_{1})=2+2\sqrt{1+c_{1}}$. Since $\gamma<\gamma^{+}(c)$, $U_{\theta}^{c,\gamma}(-1)=\tau_{1}$.

Since $U_{\theta}$ satisfies (32), we have

[TABLE]

Let $w:=\int_{0}^{x}\frac{U_{\theta}-\tau_{2}}{2(1-s^{2})}ds$. We have

[TABLE]

By Lemma 2.11, there exists some $\delta=\delta(\epsilon,K)>0$, such that $|U_{\theta}(x)-\tau_{1}|<\epsilon$ for all $x\in(-1,-1+\delta)$. By Lemma 2.2, $|U_{\theta}(x)|\leq C(K)$ and therefore $|h(x)|\leq C(K)(1+x)$ for $x\in(-1,1)$. So for all $x\in(-1,-1+\delta)$,

[TABLE]

and

[TABLE]

Thus

[TABLE]

Plugging this into (36), we have

[TABLE]

∎

Lemma 2.12’****.

Let $K$ be a compact subset of $I\setminus\{(c,\gamma)\mid c_{2}=-1\textrm{ or }\gamma=\gamma^{-}(c)\}$. Then for any $\epsilon>0$, there exists some positive constants $\delta$ and $C$, depending only on $\epsilon$ and $K$, such that for any $(c,\gamma)\in K$,

[TABLE]

Lemma 2.13**.**

Let $K$ be a compact subset of $I\cap\{(c,\gamma)\mid c_{1}=-1,\gamma<\gamma^{+}(c)\}$. Then for any $\epsilon>0$, there exists some $\delta>0$, depending only on $\epsilon$ and $K$, such that for any $(c,\gamma)\in K$,

[TABLE]

Proof.

If $U_{\theta}:=U_{\theta}^{c,\gamma}$ is a solution of (32) with $(c,\gamma)\in I$, $c_{1}=-1$, and $\gamma<\gamma^{+}(c)$, we have $U_{\theta}(-1)=2$. Denote

[TABLE]

Then by Theorem 1.3 in [4], $g(-1)=4$, $g(x)$ satisfies

[TABLE]

We prove the lemma by contradiction. Assume there exist some $\epsilon>0$ and a sequence $(c^{i},\gamma^{i})\in K$ and $-1<x_{i}<-1+\frac{1}{i}$, such that

[TABLE]

Since $K$ is compact, there exist a subsequence, still denoted as $(c^{i},\gamma^{i})$, and some $(c,\gamma)\in K$, such that $(c^{i},\gamma^{i})\to(c,\gamma)$ as $i\to\infty$.

Denote $g_{i}=g^{c^{i},\gamma^{i}}$. By standard ODE theory, we have that $g_{i}\to g:=g^{c,\gamma}$ in $C^{1}_{loc}(-1,1)$. As explained earlier, $g(-1)=4$.

We first assume that

[TABLE]

Using this and the fact that $g_{i}\to g$ in $C^{1}_{loc}(-1,1)$, by similar arguments as in the proof of Lemma 2.11, we have that there exist $x_{i}\leq\tilde{x}_{i}\to-1$, such that

[TABLE]

Let $h(s):=-2s+\frac{1}{2}s^{2}$. By (37) we have that

[TABLE]

Sending $i\to\infty$, we have

[TABLE]

On the other hand, since $\xi>4$, so $h(\xi)>0$. A contradiction.

Now if instead of (38), we have

[TABLE]

Without loss of generality, we assume that $0<\epsilon<1$. As in Case 1 of the proof of Lemma 2.11, there exists $\tilde{x}_{i_{j}}\to-1$, such that

[TABLE]

By (37) we have that

[TABLE]

Sending $i\to\infty$, we have

[TABLE]

On the other hand, since $3<\xi<4$, so $h(\xi)<h(4)=0$. A contradiction. ∎

Similarly, we have

Lemma 2.13’****.

Let $K$ be a compact subset of $I\cap\{(c,\gamma)\mid c_{2}=-1,\gamma>\gamma^{-}(c)\}$. Then for any $\epsilon>0$, there exists some $\delta>0$, depending only on $\epsilon$ and $K$, such that for any $(c,\gamma)\in K$,

[TABLE]

The next lemma strengthens Lemma 2.13.

Lemma 2.14**.**

Let $K$ be a compact subset of $I\cap\{(c,\gamma)\mid c_{1}=-1,\gamma<\gamma^{+}(c)\}$. Then for any $\epsilon>0$, there exists some positive constants $\delta$ and $C$, depending only on $\epsilon$ and $K$, such that for any $(c,\gamma)\in K$,

[TABLE]

Proof.

For convenience let us denote $U_{\theta}=U_{\theta}^{c,\gamma}$. Let $V:=U_{\theta}-2-\frac{4}{\ln\frac{1+x}{3}}$. Then $V$ satisfies the equation

[TABLE]

where $h:=P_{c}(x)-P_{c}(-1)-\frac{4(1+x)}{\left(\ln\frac{1+x}{3}\right)^{2}}-2(1+x)V-4(1+x)-\frac{8(1+x)}{\ln\frac{1+x}{3}}$. We have, using Lemma 2.11, that there exists some $\delta=\delta(\epsilon,K)$, such that $|h|\leq C(\epsilon,K)(1+x)$ for all $x\in(-1,-1+\delta)$.

Let $w:=\int_{-\frac{1}{2}}^{x}\frac{\frac{1}{2}V+\frac{4}{\ln\frac{1+s}{3}}}{1-s^{2}}ds$. We have

[TABLE]

Since $V\left(-\frac{1}{2}\right)=U_{\theta}\left(-\frac{1}{2}\right)-2+\frac{4}{\ln 6}$, we have $|V\left(-\frac{1}{2}\right)|\leq C(K)$. By Lemma 2.13, making $\delta=\delta(\epsilon,K)>0$ smaller if necessary, we have $|(U_{\theta}-2)\ln\frac{1+x}{3}-4|<\epsilon$, i.e. $|V|\leq\frac{\epsilon}{\left|\ln\frac{1+x}{3}\right|}$, for all $-1<x<-1+\delta$. We also have $|h|\leq C(\epsilon,K)(1+x)$ for all $x\in(-1,-1+\delta)$. Thus for all $-1<x<-1+\delta<-\frac{1}{2}$, we have

[TABLE]

and

[TABLE]

So

[TABLE]

Plugging this into (39), we have

[TABLE]

The proof is finished. ∎

Similarly we have the following strengthening of Lemma 2.13’.

Lemma 2.14’****.

Let $K$ be a compact subset of $I\cap\{(c,\gamma)\mid c_{2}=-1,\gamma>\gamma^{-}(c)\}$. Then for any $\epsilon>0$, there exists some positive constants $\delta$ and $C$, depending only on $\epsilon$ and $K$, such that for any $(c,\gamma)\in K$,

[TABLE]

Now using Lemma 2.11–Lemma 2.14’, we prove the following estimates of partial derivatives of $U_{\theta}:=U_{\theta}^{c,\gamma}$ with respect to $(c,\gamma)$ on each $I_{k,l}$.

Lemma 2.15**.**

For any $\epsilon>0$, $m\in\mathbb{N}$, and compact subset $K$ of $I\setminus\{(c,\gamma)\mid c_{1}=-1\textrm{ or }\gamma=\gamma^{+}(c)\}$, there exists some positive constant $C$, depending only on $m$, $K$, and $\epsilon$, such that

[TABLE]

Proof.

We prove the lemma by induction. We use $C(m,K,\epsilon)$ and $C$ to denote constants which may be different from line to line, and their dependence is clear from the context.

We know by (32) that

[TABLE]

[TABLE]

and $\frac{\partial U_{\theta}(0)}{\partial\gamma}=1$, $\frac{\partial U_{\theta}(0)}{\partial c_{i}}=0$, $i=1,2,3$. Denote

[TABLE]

Then

[TABLE]

and for $i=1,2,3$,

[TABLE]

By the definition of $a(x)$, Lemma 2.2 and Lemma 2.12, we have that there exists some constant $C=C(\epsilon,K)$ such that

[TABLE]

Since when $(c,\gamma)\in K$, $U_{\theta}(-1)<2$, there exists some $C(K,\epsilon)$, such that $e^{-a(x)}\leq C(K,\epsilon)$. Thus by (41) and (42) we have that for $-1<x<1-\epsilon$,

[TABLE]

Now for $m\geq 2$, suppose that $C(m_{1},K,\epsilon)$ exist for all $1\leq m_{1}\leq m-1$, then for any $|\alpha|+j=m$,

[TABLE]

This leads to

[TABLE]

where

[TABLE]

Notice that $\partial_{c}^{\alpha}\partial_{\gamma}^{j}U_{\theta}(0)=0$ for all $|\alpha|+j\geq 2$, we have

[TABLE]

By the induction assumption, $h\in L^{\infty}(-1,1-\epsilon)$ and there exists some positive constant $C$, depending only on $m$, $K$, and $\epsilon$ such that $|h|_{L^{\infty}(-1,1-\epsilon)}\leq C$. So we have

[TABLE]

The proof is finished. ∎

Similarly, using Lemma 2.2 and Lemma 2.12’ we have

Lemma 2.15’****.

For any $\epsilon>0$, $m\in\mathbb{N}$, and compact subset $K$ of $I\setminus\{(c,\gamma)\mid c_{2}=-1\textrm{ or }\gamma=\gamma^{-}(c)\}$, there exists some positive constant $C$, depending only on $m$, $K$, and $\epsilon$, such that

[TABLE]

Lemma 2.16**.**

For any $\epsilon>0$, $m\in\mathbb{N}$, and compact subset $K$ of $I\cap\{(c,\gamma)\mid c_{1}=-1,\gamma<\gamma^{+}(c)\}$,there exists some positive constant $C$, depending only on $m$, $K$, and $\epsilon$, such that

[TABLE]

Proof.

We prove the lemma by induction. Denote $C(m,K,\epsilon)$ and $C$ to be constants which may vary from line to line, and their dependence is clear from the context. Similar as the proof of Lemma 2.15, we have (41) and (42) where $a(x)$ is defined by (40). By the definition of $a(x)$, Lemma 2.2 and Lemma 2.14, there exists some constant $C=C(m,K,\epsilon)$, such that

[TABLE]

Notice in this case, $i=2$ or $3$ in (42), and $|\partial_{c_{i}}P_{c}|\leq C(1+x)$ for some constant $C$ depending only on $K$, so we have that for $-1<x<1-\epsilon$,

[TABLE]

Now suppose that $C(m_{1},K,\epsilon)$ exists for all $1\leq m_{1}\leq m-1$. As in the proof of the previous lemma we have, for all $|\alpha|+j=m$ and $\alpha_{1}=0$, that

[TABLE]

where

[TABLE]

Then, by the induction assumption, $h\in L^{\infty}(-1,1-\epsilon)$ and there is some positive constant $C$, depending only on $m$, $K$, and $\epsilon$, such that $\left(\ln\frac{1+x}{3}\right)^{4}|h(x)|\leq C$ for all $-1<x<1-\epsilon$. Using this estimate we then have

[TABLE]

The lemma is proved. ∎

Similarly, using Lemma 2.14’ we have

Lemma 2.16’****.

For any $\epsilon>0$, $m\in\mathbb{N}$, and compact subset $K$ of $I\cap\{c,\gamma)\mid c_{2}=-1,\gamma>\gamma^{-}(c)\}$, there exists some positive constant $C$, depending only on $m$, $K$, and $\epsilon$, such that

[TABLE]

Theorem 1.5 follows from Corollary 2.2, Corollary 2.2’, Lemma 2.15, 2.15’, 2.16 and 2.16’.