Existence and Asymptotic Behavior of Large Axisymmetric Solutions for Steady Navier–Stokes System in a Pipe

Abstract

In this paper, the existence and uniqueness of strong axisymmetric solutions with large flux for the steady Navier–Stokes system in a pipe are established even when the external force is also suitably large in $L^2$. Furthermore, the exponential convergence rate at far fields for the arbitrary steady solutions with finite $H^2$ distance to the Hagen–Poiseuille flows is established as long as the external forces converge exponentially at far fields. The results can be regarded as a key step toward a Liouville-type theorem for steady solutions of Navier–Stokes system in a pipe and global existence for Leray’s problem on steady solutions of Navier–Stokes system in general infinitely long nozzles. The key point to get the existence of these large solutions is the refined estimate for the derivatives in the axial direction of the stream function and the swirl velocity, which exploits the good effect of the convection term. An important observation for the asymptotic behavior of general solutions is that the solutions are actually small at far fields when they have finite $H^2$ distance to the Hagen–Poiseuille flows. This makes the estimate for the linearized problem play a crucial role in studying the convergence of general solutions at far fields.

Appendices

Appendix A: Some Elementary Lemmas

In this appendix, we collect some basic lemmas which play important roles in the paper. The proof of Lemmas A.1–A.3 can be found in [27, Appendix A], so we omit the details here.

The following lemma is about Poincaré-type inequalities:

Lemma A.1

For a function $g\in C^2([0,1])$ it holds that

$$\begin{aligned} \int _0^1 |g |^2 r \, dr \le \int _0^1 \left| \frac{d}{dr} (r g ) \right| ^2 \frac{1}{r} \, dr. \end{aligned}$$

(142)

If, in addition, $g(0)=g(1)=0$, then one has

$$\begin{aligned} \int _0^1 \left| \frac{d}{dr} (r g ) \right| ^2 \frac{1}{r} \, dr \le \left( \int _0^1 | {\mathcal {L}}g|^2 r \, dr \right) ^{\frac{1}{2}} \left( \int _0^1 |g|^2 r\, dr \right) ^{\frac{1}{2}} \le \int _0^1 | {\mathcal {L}}g|^2 r \, dr.\nonumber \\ \end{aligned}$$

(143)

The following lemma is a variant of Hardy–Littlewood–Pólya-type inequality [11], which plays an important role in many estimates in the paper:

Lemma A.2

Let $g\in C^1([0,1])$ satisfy $g(0)=0$, one has

$$\begin{aligned} \int _0^1|g(r)|^2 dr \le \frac{1}{2} \int _0^1 |g^{\prime }(r)|^2 (1-r^2) \, dr, \end{aligned}$$

(144)

and

$$\begin{aligned} \int _0^1 |g|^2 r \, dr \le C \int _0^1 \left| \frac{d(r g) }{dr} \right| ^2 \frac{1-r^2}{r} \, dr . \end{aligned}$$

(145)

The following lemma is about a weighted interpolation inequality, which is quite similar to [9, (3.28)].

Lemma A.3

Let $g \in C^2[0, 1]$, then one has

$$\begin{aligned} \begin{aligned} \int _0^1 |g|^2r \, dr \le&C \left( \int _0^1 (1-r^2)|g|^2 r \, dr\right) ^{\frac{2}{3}} \left( \int _0^1 \left| \frac{d}{dr}(rg)\right| ^2 \frac{1}{r} \, dr\right) ^{\frac{1}{3}} \\&\ \ \ \ + C \int _0^1 (1-r^2)|g|^2 r\, dr , \end{aligned} \end{aligned}$$

(146)

and

$$\begin{aligned} \begin{aligned} \int _0^1 \left| \frac{d}{dr}(rg) \right| ^2 \frac{1}{r} \, dr&\le C \left[ \int _0^1 \frac{1-r^2}{r} \left| \frac{d}{dr} ( rg) \right| ^2 \, dr \right] ^{\frac{2}{3}} \left( \int _0^1 |{\mathcal {L}} g|^2 r \, dr \right) ^{\frac{1}{3}} \\&\quad + C \int _0^1 \frac{1-r^2}{r} \left| \frac{d}{dr} ( rg) \right| ^2 \, dr . \end{aligned} \end{aligned}$$

(147)

The following lemma is about the interpolation inequality which was used frequently in the paper.

Lemma A.4

[1, Theorem 5.2] Let $\Omega$ be a domain in ${\mathbb {R}}^n$ satisfying the cone condition. There exists a constant C, depending on n, m, p, and $\Omega$, such that if $0\le j\le m$ and $u \in W^{m, p}(\Omega )$, then

$$\begin{aligned} \Vert u\Vert _{W^{j, p}(\Omega )} \le C \Vert u\Vert _{L^p(\Omega )}^{1-\frac{j}{m}} \Vert u\Vert _{W^{m, p}(\Omega )}^{\frac{j}{m}}. \end{aligned}$$

(148)

The following Gagliardo–Nirenberg inequality can be regarded as a more general interpolation inequality:

Lemma A.5

[8, Lemma II.3.3, Theorem II.3.3] Let $\Omega$ be a locally Lipschitz domain in ${\mathbb {R}}^n$. There exists a constant C, depending on n, m, r, q, j, $\alpha$, and a, such that if $u \in W^{m, r}(\Omega )\cap L^q(\Omega )$, then

$$\begin{aligned} \Vert u\Vert _{W^{j, s} (\Omega )} \le C \Vert u\Vert _{W^{m, r}(\Omega )}^a \Vert u\Vert _{L^q(\Omega )}^{1-a}, \end{aligned}$$

(149)

where

$$\begin{aligned} \frac{1}{s} = \frac{j}{n} + a\left( \frac{1}{r} - \frac{m}{n} \right) + (1-a) \frac{1}{q}, \end{aligned}$$

for all a in the interval

$$\begin{aligned} \frac{j}{m} \le a \le 1, \end{aligned}$$

with the following exceptional cases: (1). If $j=0$, $rm<n$, $q=\infty$; (2). if $1<r< \infty$, and $m- j - n/r$ is a nonnegative integer, then (149) holds only for a satisfying $j/m \le a <1$.

Appendix B: Uniform Estimate for the Solutions

In this appendix, we give a sketch of the proof for Propositions 2.1 and 2.2.

1.1 Basic estimate

Multiplying the both sides of the Eq. (34) with $\bar{{\hat{\psi }}} r$ and integrating over (0, 1) yield

$$\begin{aligned} \begin{aligned}&\int _0^1 | {\mathcal {L}} {\hat{\psi }} |^2 r \, dr + 2\xi ^2 \int _0^1 \left| \frac{d}{dr} (r {\hat{\psi }}) \right| ^2 \frac{1}{r} \, dr + \xi ^4 \int _0^1 |{\hat{\psi }}|^2 r \, dr \\&\quad =\,\, - \Re \int _0^1 ( \widehat{F^r}(i\xi \bar{{\hat{\psi }}})r+\widehat{F^z}\partial _r(r\bar{{\hat{\psi }}}))dr - \frac{4 \Phi }{\pi } \xi \Im \int _0^1 \left[ \frac{d}{dr} (r {\hat{\psi }} ) r \overline{{\hat{\psi }}} \right] \, dr \end{aligned} \end{aligned}$$

(150)

and

$$\begin{aligned} \begin{aligned} \xi \int _0^1 \frac{{{\bar{U}}}(r) }{ r } \left| \frac{d}{dr} ( r {\hat{\psi }} ) \right| ^2 \, dr + \xi ^3 \int _0^1 {{\bar{U}}}(r) |{\hat{\psi }}|^2 r\, dr = - \Im \int _0^1 ( \widehat{F^r}(i\xi \bar{{\hat{\psi }}})r+\widehat{F^z}\partial _r(r\bar{{\hat{\psi }}}))dr. \end{aligned}\nonumber \\ \end{aligned}$$

(151)

Applying Lemma A.2 for (151) gives

$$\begin{aligned} \Phi |\xi | \int _0^1 |{\hat{\psi }}|^2 r \, dr \le C \left( \int _0^1 | \widehat{F^r} | | \xi {\hat{\psi }}| r \, dr + \int _0^1 |\widehat{F^z} | \left| \frac{d}{dr} ( r {\hat{\psi }} ) \right| \, dr \right) . \end{aligned}$$

(152)

This, together with Cauchy–Schwarz inequality and Lemma A.1, gives

$$\begin{aligned} \begin{aligned}&\frac{4\Phi }{\pi } |\xi | \int _0^1 \left| \frac{d}{dr} (r {\hat{\psi }}) \right| |r {\hat{\psi }}| \, dr \\&\quad \le C \Phi ^{\frac{1}{2}} \left[ |\xi | \int _0^1 \left| \frac{d}{dr}(r {\hat{\psi }} ) \right| ^2 \frac{1}{r} \, dr \right] ^{\frac{1}{2}} \left( \Phi |\xi | \int _0^1 |{\hat{\psi }}|^2 r \, dr \right) ^{\frac{1}{2}} \\&\quad \le \frac{1}{4} ( 1 + \xi ^2) \int _0^1 \left| \frac{d}{dr}(r {\hat{\psi }} ) \right| ^2 \frac{1}{r} \, dr + C \Phi \left( \int _0^1 | \widehat{F^r} | | \xi {\hat{\psi }}| r \, dr + \int _0^1 |\widehat{F^z} | \left| \frac{d}{dr} ( r {\hat{\psi }} ) \right| \, dr \right) \end{aligned}\nonumber \\ \end{aligned}$$

(153)

By virtue of Lemma A.1 and Cauchy–Schwarz inequality, one has

$$\begin{aligned} \int _0^1\left( |{\mathcal {L}} {\hat{\psi }}|^2 r \, + \xi ^2 \left| \frac{d}{dr}(r {\hat{\psi }} ) \right| ^2 \frac{1}{r} \, + \xi ^4 |{\hat{\psi }}|^2 r \right) \, dr \le C (1+ \Phi ^2) \int _0^1 (|\widehat{F^r} |^2 + |\widehat{F^z}|^2 ) r \, dr .\nonumber \\ \end{aligned}$$

(154)

Integrating (154) with respect to $\xi$ yields

$$\begin{aligned} \int _{-\infty }^{+\infty } \int _0^1 \left\{ \left( |{\mathcal {L}} {\hat{\psi }}|^2 + \xi ^4 |{\hat{\psi }}|^2 \right) r \, + \xi ^2 \left| \frac{d}{dr}( r {\hat{\psi }}) \right| ^2 \frac{1}{r} \right\} \, dr d\xi \le C(1 + \Phi ^2) \Vert {{\varvec{F}}}^* \Vert _{L^2(\Omega )}^2 ,\nonumber \\ \end{aligned}$$

(155)

where ${{\varvec{F}}}^*= F^r{{\varvec{e}}}_r+F^z{{\varvec{e}}}_z$. Note that ${\varvec{\omega }}^\theta = ( {\mathcal {L}} + \partial _z^2) \psi {{\varvec{e}}}_\theta$, one has

$$\begin{aligned} \begin{aligned} \Vert {{\varvec{v}}}^*\Vert _{H^1(\Omega )} \le C \Vert \nabla {{\varvec{v}}}^* \Vert _{L^2(\Omega )} = C\Vert {\varvec{\omega }}^\theta \Vert _{L^2(\Omega )} \le C (1 + \Phi ) \Vert {{\varvec{F}}}^* \Vert _{L^2(\Omega )}, \end{aligned} \end{aligned}$$

where ${{\varvec{v}}}^*= v^r{{\varvec{e}}}_r+v^z{{\varvec{e}}}_z$. Applying the regularity theory for Stokes equations ( [8]) gives

$$\begin{aligned} \begin{aligned} \Vert {{\varvec{v}}}^*\Vert _{H^2 (\Omega )}&\le C[\Vert {{\varvec{F}}}^*\Vert _{L^2(\Omega )} + \Phi \Vert \partial _z {{\varvec{v}}}^*\Vert _{L^2(\Omega )} + \Phi \Vert v^r\Vert _{L^2(\Omega )} + \Vert {{\varvec{v}}}^*\Vert _{H^1(\Omega )}] \\&\le C (1 + \Phi ^2 ) \Vert {{\varvec{F}}}^*\Vert _{L^2(\Omega )}. \end{aligned}\nonumber \\ \end{aligned}$$

(156)

1.2 Case with large flux and low frequency ($|\xi | \le \frac{1}{\epsilon _1 \Phi }$)

It follows from the energy estimate as (154) gives

$$\begin{aligned} \begin{aligned} \int _0^1 |{\mathcal {L}} {\hat{\psi }} |^2 r \, dr + 2 \xi ^2 \int _0^1 \left| \frac{d}{dr}(r {\hat{\psi }}) \right| ^2 \frac{1}{r}\, dr + \xi ^4 \int _0^1 |{\hat{\psi }}|^2 r\, dr \le C (\epsilon _1) \int _0^1 |\widehat{{{\varvec{F}}}^*}|^2 r \, dr. \end{aligned}\nonumber \\ \end{aligned}$$

(157)

Let

$$\begin{aligned} \chi _1 (\xi ) = \left\{ \begin{aligned}&1 , \ \ \ |\xi | \le \frac{1}{\epsilon _1 \Phi } , \\&0,\ \ \ \ \text {otherwise}, \end{aligned} \right. \end{aligned}$$

and $\psi _{low}$ be the function such that $\widehat{\psi _{low}} = \chi _1 (\xi ) {\hat{\psi }}.$ Define

$$\begin{aligned} v^r_{low} = \partial _z \psi _{low},\ \ \ \ v^z_{low} = - \frac{\partial _r ( r \psi _{low} )}{r}, \ \ \ \ \text {and}\ \ \ {{\varvec{v}}}^*_{low} = v^r_{low}{{\varvec{e}}}_r + v^z_{low}{{\varvec{e}}}_z. \end{aligned}$$

Similarly, one can define $F^r_{low}, F^z_{low}, {{\varvec{F}}}^*_{low}$, ${\varvec{\omega }}^\theta _{low}$. It follows from the regularity estimate for the Stokes equations ([8]) that one has

$$\begin{aligned} \Vert {{\varvec{v}}}^*_{low} \Vert _{H^2(\Omega )} \le C \Vert {{\varvec{F}}}^*_{low}\Vert _{L^2(\Omega )}, \end{aligned}$$

(158)

where C is a uniform constant independent of $\Phi$ and ${{\varvec{F}}}$.

1.3 Case with large flux and high frequency ($|\xi | \ge \epsilon _1 \sqrt{\Phi }$)

The energy estimate as (154) yields

$$\begin{aligned} \begin{aligned} \int _0^1 |{\mathcal {L}} {\hat{\psi }}|^2 r \, dr + \xi ^2 \int _0^1 \left| \frac{d}{dr} ( r {\hat{\psi }}) \right| ^2 \frac{1}{r} \, dr + \xi ^4 \int _0^1 |{\hat{\psi }}|^2 r \, dr \le C(\epsilon _1) |\xi |^{-2} \int _0^1 |\widehat{{{\varvec{F}}}^*}|^2 r \, dr \end{aligned}\nonumber \\ \end{aligned}$$

(159)

and

$$\begin{aligned} \begin{aligned} \Phi |\xi | \int _0^1 \frac{1 - r^2}{r} \left| \frac{d}{dr} ( r {\hat{\psi }}) \right| ^2 \, dr + \Phi |\xi |^3 \int _0^1 (1 - r^2) |{\hat{\psi }}|^2 r \, dr \le C (\epsilon _1) |\xi |^{-2} \int _0^1 |\widehat{{{\varvec{F}}}^*}|^2 r \, dr . \end{aligned}\nonumber \\ \end{aligned}$$

(160)

Thus, one has

$$\begin{aligned} \Vert {{\varvec{v}}}^*_{high} \Vert _{H^1(\Omega )} \le C \Vert {\varvec{\omega }}^\theta _{high}\Vert _{L^2(\Omega )} \le C \Phi ^{-\frac{1}{2}} \Vert {{\varvec{F}}}^*_{high}\Vert _{L^2(\Omega )}, \end{aligned}$$

(161)

where

$$\begin{aligned} {{\varvec{v}}}^*_{high} = v^r_{high}{{\varvec{e}}}_r + v^z_{high}{{\varvec{e}}}_z \ \ \ \ \text {with}\,\,v^r_{high} = \partial _z \psi _{high},\ \ \ \ v^z_{high} = - \frac{\partial _r ( r \psi _{high} )}{r}, \end{aligned}$$

and $\widehat{\psi _{high}} = \chi _2 (\xi ) {\hat{\psi }}$ with

$$\begin{aligned} \chi _2 (\xi ) = \left\{ \begin{aligned}&1 , \ \ \ |\xi | \ge \epsilon _1 \sqrt{\Phi } , \\&0,\ \ \ \ \text {otherwise}. \end{aligned} \right. \end{aligned}$$

Note that one can define $F^r_{high}, F^z_{high}, {{\varvec{F}}}^*_{high}$, and ${\varvec{\omega }}_{high}^\theta$ in the similar way. Hence, the regularity estimate for the Stokes equations ( [8]) gives

$$\begin{aligned} \Vert {{\varvec{v}}}^*_{high} \Vert _{H^2(\Omega )} \le C (1 + \Phi ^{\frac{1}{4}} ) \Vert {{\varvec{F}}}^*_{high} \Vert _{L^2(\Omega )}. \end{aligned}$$

(162)

Using the interpolation for (161) and (162) gives

$$\begin{aligned} \Vert {{\varvec{v}}}^*_{high} \Vert _{H^{\frac{5}{3}} (\Omega )} \le C \Vert {{\varvec{F}}}^*_{high} \Vert _{L^2(\Omega )}. \end{aligned}$$

(163)

1.4 Large flux and intermediate frequency ($\frac{1}{\epsilon _1 \Phi } \le |\xi | \le \epsilon _1 \sqrt{\Phi }$)

When the frequency belongs to the intermediate regime, i.e., $\frac{1}{\epsilon _1\Phi } \le |\xi | \le \epsilon _1 \sqrt{\Phi }$, inspired by the analysis in [9], $\psi$ can be decomposed into four parts,

$$\begin{aligned} {\hat{\psi }}(r) = \widehat{\psi _s}(r) + a I_1(|\xi |r) + b(\chi \widehat{\psi _{BL}} + \widehat{\psi _e} ). \end{aligned}$$

(164)

In what follows, we give the detailed explanations for $\widehat{\psi _s}$, $\widehat{\psi _{BL}}$, $\widehat{\psi _e}$, and $I_1$.

$\widehat{\psi _s}$ is a solution to the problem

$$\begin{aligned} \left\{ \begin{aligned}&i \xi {\bar{U}}(r) ( {\mathcal {L}} - \xi ^2) \widehat{\psi _s} - ({\mathcal {L}} - \xi ^2)^2 \widehat{\psi _s} = {\hat{f}}, \\&\widehat{\psi _s}(0) = \widehat{\psi _s}(1) = {\mathcal {L}} \widehat{\psi _s} (0) = {\mathcal {L}} \widehat{\psi _s} (1) = 0. \end{aligned} \right. \end{aligned}$$

(165)

Multiplying the both sides of the equation in (165) with $r\overline{\widehat{\psi _s}}$ and $r\overline{{\mathcal {L}}\widehat{\psi _s}}$, respectively, and using the integration by parts, yields

$$\begin{aligned}&\int _0^1 |\widehat{\psi _s}|^2 r \, dr \le C (\Phi |\xi |)^{-\frac{5}{3}} \int _0^1 |\widehat{{{\varvec{F}}}^*}|^2 r \, dr , \end{aligned}$$

(166)

$$\begin{aligned}&\int _0^1 \left| \frac{d}{dr}(r \widehat{\psi }_s) \right| ^2 \frac{1}{r}\, dr + \xi ^2 \int _0^1 \left| \widehat{\psi }_s \right| ^2 r \, dr \le C (\Phi |\xi |)^{- \frac{4}{3} } \int _0^1 |\widehat{{{\varvec{F}}}^*}|^2 r \, dr , \end{aligned}$$

(167)

$$\begin{aligned}&\int _0^1 | {\mathcal {L}} \widehat{\psi _s}|^2 r \, dr + \xi ^2 \int _0^1 \left| \frac{d}{dr} ( r \widehat{\psi _s} )\right| ^2 \frac{1}{r} \, dr + \xi ^4 \int _0^1 |\widehat{\psi _s}|^2 r \, dr \le C ( \Phi |\xi |)^{-\frac{2}{3}} \int _0^1 |\widehat{{{\varvec{F}}}^*}|^2 r \, dr ,\nonumber \\ \end{aligned}$$

(168)

$$\begin{aligned}&\int _0^1 \left| \frac{d}{dr}( r {\mathcal {L}} \widehat{\psi _s} )\right| ^2 \frac{1}{r} + \xi ^2 |{\mathcal {L}} \widehat{\psi _s} |^2 r + \xi ^4 \left| \frac{d}{dr}( r \widehat{\psi _s}) \right| ^2 \frac{1}{r} +\xi ^6 |\widehat{\psi _s} |^2 r\, dr \le C \int _0^1 |\widehat{{{\varvec{F}}}^*}|^2 r\, dr .\nonumber \\ \end{aligned}$$

(169)

$\widehat{\psi _{BL}}$ is the boundary layer profile, which is the exact solution (exponentially decay away from $r=1$) of the equation

$$\begin{aligned} \left( i \frac{\xi \Phi }{\pi } 4 (1 - r) - \frac{d^2}{dr^2} + \xi ^2\right) \left( \frac{d^2}{dr^2} - \xi ^2\right) \widehat{\psi _{BL}}=0. \end{aligned}$$

(170)

In fact, one can represent $\widehat{\psi _{BL}}$ in terms of the Airy function. Let $|\beta | = \left( \frac{ 4 |\xi | \Phi }{\pi } \right) ^{\frac{1}{3}}$ and

$$\begin{aligned} {\widetilde{G}}_{\xi , \Phi } ( \rho ) = \left\{ \begin{aligned}&Ai \left( C_{+} (\rho + \frac{\pi |\beta | \xi }{4 i \Phi }) \right) , \ \ \ \text{ when }\ \xi >0 , \\&Ai \left( C_{-} (\rho + \frac{\pi |\beta | \xi }{4 i \Phi }) \right) ,\ \ \ \ \text{ when }\ \xi < 0 , \end{aligned} \right. \end{aligned}$$

(171)

where $C_{+} = e^{ i \frac{\pi }{6}}$, $C_{-} = e^{-i \frac{\pi }{6}}$, and Ai(z) denotes the Airy function satisfying

$$\begin{aligned} \frac{d^2 Ai}{dz^2} - z Ai =0 \ \ \ \text{ in }\ {\mathbb {C}}. \end{aligned}$$

Define

$$\begin{aligned} G_{\xi , \Phi } (\rho ) = \int _{\rho }^{+\infty } e^{- \frac{|\xi |}{|\beta |} ( \rho - \tau ) } \int _{\tau }^{+ \infty } e^{- \frac{|\xi |}{|\beta |} (\sigma - \tau )} {\widetilde{G}}_{\xi , \Phi } (\sigma ) \, d\sigma d\tau \end{aligned}$$

(172)

and

$$\begin{aligned} C_{0, \xi , \Phi } = \left\{ \begin{aligned}&\frac{1}{G_{\xi , \Phi } (0) }, \ \ \ \text{ if }\ |G_{\xi , \Phi } (0) | \ge 1, \\&1,\ \ \ \ \ \ \ \ \ \ \ \ \text{ otherwise }. \end{aligned} \right. \end{aligned}$$

(173)

Then,

$$\begin{aligned} \widehat{\psi _{BL}} (r) : = C_{0, \xi , \Phi } G_{\xi , \Phi } (|\beta | ( 1 - r) ). \end{aligned}$$

(174)

satisfies $|\widehat{\psi _{BL}}(1)| \le 1$ and solves the equation (170).

$\widehat{\psi _e}$ is a remainder term, which satisfies the problem

$$\begin{aligned} \left\{ \begin{aligned}&i\xi {{\bar{U}}}(r) ( {\mathcal {L}} - \xi ^2) (\chi \widehat{\psi _{BL}} + \widehat{\psi _e}) - ({\mathcal {L}} - \xi ^2)^2 (\chi \widehat{\psi _{BL}}+ \widehat{\psi _e} ) = 0, \\&\widehat{\psi _e}(0) = \widehat{\psi _e}(1) = {\mathcal {L}}\widehat{\psi _{e}}(0) = {\mathcal {L}} \widehat{\psi _e}(1) = 0, \end{aligned} \right. \end{aligned}$$

where $\chi$ is a smooth cut-off function satisfying

$$\begin{aligned} \chi (r) = 1 \ \ \text {if} \ \ r\ge \frac{1}{2}\ \ \text {and}\ \ \chi (r)= 0\ \ \text {if}\ \ r \le \frac{1}{4}. \end{aligned}$$

Then, one can have the estimates

$$\begin{aligned}&\int _0^1 |\widehat{\psi _e}|^2 r \, dr \le C |\beta |^5 ( \Phi |\xi |)^{-2} \le C ( \Phi |\xi | )^{-\frac{1}{3}}, \end{aligned}$$

(175)

$$\begin{aligned}&\int _0^1 | {\mathcal {L}} \widehat{\psi _e} |^2 r + \xi ^2 \left| \frac{d}{dr} ( r \widehat{\psi _e}) \right| ^2 \frac{1}{r} + \xi ^4 |\widehat{\psi _e}|^2 r\, dr \le C |\beta |^5 (\Phi |\xi | )^{-\frac{6}{7}} \le C ( \Phi |\xi | )^{\frac{17}{21}},\nonumber \\ \end{aligned}$$

(176)

$$\begin{aligned}&\int _0^1 \left| \frac{d}{dr} ( r {\mathcal {L}} \widehat{\psi _e} ) \right| ^2 \frac{1}{r} + \xi ^2 |{\mathcal {L}} \widehat{\psi _e} |^2 r + \xi ^4 \int _0^1 \left| \frac{d}{dr} ( r \widehat{\psi _e}) \right| ^2 \frac{1}{r} + \xi ^6 |\widehat{\psi _e}|^2 r\, dr \le C (\Phi |\xi | )^{\frac{5}{3}}.\nonumber \\ \end{aligned}$$

(177)

$I_1(z)$ is the modified Bessel function of the first kind, which satisfies

$$\begin{aligned} \left\{ \begin{aligned}&z^2 \frac{d^2}{dz^2} I_1 (z) + z \frac{d}{dz} I_1 (z) - (z^2 + 1) I_1 (z) = 0, \\&I_1 (0) = 0,\,\, I_1(z)>0 \,\, \text {for}\,\, z>0. \end{aligned} \right. \end{aligned}$$

This implies

$$\begin{aligned} ({\mathcal {L}}-\xi ^2)I_1(|\xi |r)=0. \end{aligned}$$

The no-slip boundary conditions (35) can be recovered if a and b satisfy

$$\begin{aligned} \left\{ \begin{aligned}&a I_1 (|\xi |) + b \widehat{\psi _{BL}} (1) = 0, \\&a |\xi | I_1^{\prime } (|\xi | ) + b \frac{d}{dr} \widehat{\psi _{BL}} (1) + b \frac{d}{dr} \widehat{\psi _e} (1) = - \frac{d}{dr} \widehat{\psi _s} (1) . \end{aligned} \right. \end{aligned}$$

(178)

With the aid of the estimate for $\widehat{\psi _s}$, $\widehat{\psi _{BL}}$, and $\widehat{\psi _e}$, and the properties of $I_1$, one has

$$\begin{aligned} \begin{aligned} |b| \le&C ( \Phi |\xi |)^{-\frac{5}{6} } \left( \int _0^1 |\widehat{{{\varvec{F}}}^*}|^2 r \, dr \right) ^{\frac{1}{2}} \end{aligned} \end{aligned}$$

(179)

and

$$\begin{aligned} |a| \le C ( \Phi |\xi |)^{-\frac{5}{6} } |I_1 (|\xi | )|^{-1} \left( \int _0^1 |\widehat{{{\varvec{F}}}^*}|^2 r \, dr \right) ^{\frac{1}{2}}. \end{aligned}$$

(180)

Therefore, ${\hat{\psi }}$ defined in (164) satisfies

$$\begin{aligned} \begin{aligned}&\int _0^1 |{\mathcal {L}} {\hat{\psi }}|^2 r + \left| \frac{d}{dr}( r{\hat{\psi }}) \right| ^2 \frac{1}{r} + |{\hat{\psi }}|^2 r \,dr\\&\qquad + \int _0^1 \left| \frac{d}{dr}( r {\mathcal {L}} {\hat{\psi }}) \right| ^2 \frac{1}{r} + \xi ^2 |{\mathcal {L}}{\hat{\psi }}|^2 r + \xi ^4 \left| \frac{d}{dr}( r {\hat{\psi }}) \right| ^2 \frac{1}{r} + \xi ^6 |{\hat{\psi }}|^2 r\, dr \\&\quad \le C \int _0^1 |\widehat{{{\varvec{F}}}^*}|^2 \, dr . \end{aligned}\nonumber \\ \end{aligned}$$

(181)

Combining the estimates (154), (157), (159), and (181) together gives Proposition 3.1.

Let

$$\begin{aligned} \chi _3 (\xi ) = \left\{ \begin{aligned}&1 , \ \ \ \frac{1}{\epsilon _1 \Phi } \le |\xi | \le \epsilon _1 \sqrt{\Phi } , \\&0,\ \ \ \ \text {otherwise}, \end{aligned} \right. \end{aligned}$$

and $\psi _{med}$ be the function such that $\widehat{\psi _{med}} = \chi _3 (\xi ) {\hat{\psi }}.$ Define

$$\begin{aligned} v^r_{med} = \partial _z \psi _{med},\ \ \ \ v^z_{med} = - \frac{\partial _r ( r \psi _{med} )}{r}, \ \ \ \ \text {and}\ \ \ \ {{\varvec{v}}}^*_{med} = v^r_{med}{{\varvec{e}}}_r + v^z_{med}{{\varvec{e}}}_z. \end{aligned}$$

Similarly, we define $F^r_{med}$, $F^z_{med}$, and ${{\varvec{F}}}^*_{med}$. The solution ${{\varvec{v}}}^*_{med}$ satisfies

$$\begin{aligned} \Vert {{\varvec{v}}}^*_{med} \Vert _{H^2(\Omega )} \le C \Vert {{\varvec{F}}}^*_{med} \Vert _{L^2(\Omega )}, \end{aligned}$$

(182)

where C is a uniform constant independent of $\Phi$ and ${{\varvec{F}}}$. Combining the estimates (156), (158), (162), (163), and (182) together gives the estimates (38) and (39) in Proposition 2.1.

1.5 Estimate for the swirl velocity

Multiplying the equation in (36) by $r\overline{\widehat{v^\theta }}$ yields

$$\begin{aligned} \int _0^1 \left| \frac{d}{dr} ( r \widehat{v^\theta } ) \right| ^2 \frac{1}{r} \, dr + \xi ^2 \int _0^1 |\widehat{v^\theta }|^2 r\,dr = \Re \int _0^1 \widehat{F^\theta } \overline{\widehat{v^\theta }} r \, dr \end{aligned}$$

and

$$\begin{aligned} \frac{2\Phi }{\pi } \xi \int _0^1 (1 - r^2) |\widehat{v^\theta } |^2 r \, dr = \Im \int _0^1 \widehat{F^\theta } \overline{\widehat{v^\theta }} r\, dr . \end{aligned}$$

It follows from Lemma A.2 that one has

$$\begin{aligned} \int _0^1 \left| \frac{d}{dr}( r \widehat{v^\theta } )\right| ^2 \frac{1}{r} \, dr + \xi ^2 \int _0^1 |\widehat{v^\theta }|^2 r \, dr \le \int _0^1 |\widehat{F^\theta }|^2 r \, dr. \end{aligned}$$

(183)

Using Lemma A.3 gives

$$\begin{aligned} \begin{aligned} \int _0^1 |\widehat{v^\theta }|^2 rdr \le C (\Phi |\xi |)^{-\frac{4}{3}} \int _0^1 |\widehat{F^\theta }|^2 r \, dr. \end{aligned} \end{aligned}$$

(184)

Multiplying the equation in (36) by $r ( {\mathcal {L}} - \xi ^2) \overline{\widehat{v^\theta }}$ yields

$$\begin{aligned} {-} \int _0^1 | ( {\mathcal {L}} - \xi ^2) \widehat{v^\theta } |^2 r \, dr = \Re \int _0^1 \widehat{F^\theta } ( {\mathcal {L}} - \xi ^2) \overline{\widehat{v^\theta }} r \, dr + \frac{2 \Phi }{\pi } \xi \Im \int _0^1 2r \widehat{v^\theta } \frac{d}{dr} ( r \overline{\widehat{v^\theta }}) \, dr .\nonumber \\ \end{aligned}$$

(185)

Thus, one has

$$\begin{aligned} \int _0^1 \left( | {\mathcal {L}} \widehat{v^\theta } |^2 + 2\xi ^2 \left| \frac{1}{r} \frac{d}{dr} ( r {\widehat{v^\theta }} ) \right| ^2 + \xi ^4 |\widehat{v^\theta } |^2 \right) r\, dr \le C \int _0^1 |\widehat{F^\theta } |^2 r \, dr . \end{aligned}$$

This proves the estimate (62) in Proposition 3.4. Then, the estimate (40) can be easily obtained via the estimate for elliptic equation.

With the aid of these estimates, the existence of the solutions for the problems (34)–(35) and (36) can be established via the Galerkin method. The detailed construction for the bases needed for the Galerkin method is given in [27].