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Existence of steady Navier-Stokes flows
exterior to an infinite cylinder

Mitsuo Higaki, Ryoma Horiuchi

Abstract. We consider the 3D steady Navier-Stokes system on the exterior of an infinite cylinder under the action of an external force. We are concerned with the class of solutions in which the velocity field is vertically uniform and at rest at horizontal infinity. This configuration includes the 2D exterior problem for the steady Navier-Stokes system known to have characteristic difficulties. We prove the existence of solutions in the above class for a given nonsymmetric external force with suitable decay. The proof can be adapted to the 2D problem and gives a generalization of H. (2023) in view of the regularity of solutions.

Keywords. Navier-Stokes system; Two-dimensional exterior domains; Scale-critical decay.

2020 MSC. 35Q30, 35B35, 76D05, 76D17.

1 Introduction

We consider the steady Navier-Stokes system on Ω×ℝ\Omega\times\mathbb{R}

{−Δ​u+∇p=−u⋅∇u+fin​Ω×ℝdiv⁡u=0in​Ω×ℝu=bon​∂Ω×ℝu​(x,z)→0as​|x|→∞.\left\{\begin{array}[]{ll}-\Delta u+\nabla p=-u\cdot\nabla u+f&\mbox{in}\ \Omega\times\mathbb{R}\\ \operatorname{div}u=0&\mbox{in}\ \Omega\times\mathbb{R}\\ u=b&\mbox{on}\ \partial\Omega\times\mathbb{R}\\ u(x,z)\to 0&\mbox{as}\ |x|\to\infty.\end{array}\right. (NS)

Here Ω={x=(x1,x2)∈ℝ2||x|>1}\Omega=\{x=(x_{1},x_{2})\in\mathbb{R}^{2}~{}|~{}|x|>1\} denotes a exterior disk, and thus Ω×ℝ\Omega\times\mathbb{R} is the exterior of an infinite cylinder. The velocity field u=(u1,u2,u3)u=(u_{1},u_{2},u_{3}) and the pressure field pp are unknown functions, while the external force ff and the boundary condition bb are given. We use standard notation for derivatives: ∂j=∂/∂xj\partial_{j}=\partial/\partial x_{j}, ∂z=∂/∂z\partial_{z}=\partial/\partial z, Δ=∑j=12∂j2+∂z2\Delta=\sum_{j=1}^{2}\partial^{2}_{j}+\partial_{z}^{2}, ∇=(∂1,∂2,∂z)\nabla=(\partial_{1},\partial_{2},\partial_{z}), div⁡u=∑j=12∂juj+∂zu3\operatorname{div}u=\sum_{j=1}^{2}\partial_{j}u_{j}+\partial_{z}u_{3} and u⋅∇u=(∑j=12uj​∂j+u3​∂z)​uu\cdot\nabla u=(\sum_{j=1}^{2}u_{j}\partial_{j}+u_{3}\partial_{z})u.

Our interest is to construct a solution for ff that does not decay in the vertical direction. This configuration includes the 2D exterior problem for the steady Navier-Stokes system if one takes f=(f1​(x),f2​(x),0)f=(f_{1}(x),f_{2}(x),0). It is well-known that this problem has characteristic difficulties due to the lack of certain embeddings in 2D unbounded domains and to the famous Stokes paradox [1, 10, 2, 15, 3]. There are many open problems although mathematically fundamental. For an overview of the field, see the recent survey [9] by Korobkov-Ren.

Because of the absence of general theory, it is an important subject to construct 2D steady Navier-Stokes flows for given data Ω,f,b\Omega,f,b belonging to an appropriate class, and to study their properties at infinity. There are two approaches in this context. One is based on symmetry. We impose symmetry on the given data to improve decay of solutions by cancellation in integrals; see [2, 14, 3, 16, 13, 17, 18] for the work in this direction.

The other approach is more constructive and is based on perturbation. We perturb the Navier-Stokes system around an exact solution invariant under the scaling symmetry

u​(x)↦λ​u​(λ​x),p​(x)↦λ2​p​(λ​x)u(x)\mapsto\lambda u(\lambda x),\quad p(x)\mapsto\lambda^{2}p(\lambda x)

and construct a solution, regarded as a remainder at infinity, to the perturbed system. The underlying idea is that the transport by a scale-invariant flow improves decay structure of solutions to the perturbed system. We typically use an exact solution found by Hamel [6]:

𝒱​(x)=α​x⟂|x|2−γ​x|x|2,𝒬​(x)=−|𝒱​(x)|22,γ,α∈ℝ,{\mathcal{V}}(x)=\alpha\frac{x^{\perp}}{|x|^{2}}-\gamma\frac{x}{|x|^{2}},\quad{\mathcal{Q}}(x)=-\frac{|{\mathcal{V}}(x)|^{2}}{2},\quad\gamma,\alpha\in\mathbb{R},

where x⊥=(−x2,x1)x^{\bot}=(-x_{2},x_{1}) and we remark that 𝒱{\mathcal{V}} is a linear combination of the rotating flow x⟂/|x|2x^{\perp}/|x|^{2} and the flow x/|x|2x/|x|^{2} carrying flux −2​π​γ-2\pi\gamma. Indeed, solutions around (𝒱,𝒬)({\mathcal{V}},{\mathcal{Q}}) are constructed by Hillairet-Wittwer [8] for |α|>48|\alpha|>\sqrt{48} when γ=0\gamma=0 and by [7] for any α\alpha when γ>2\gamma>2. As related work, we refer to [4] analyzing the exterior problem around the fast rotating flows and Maekawa-Tsurumi [12] constructing the Navier-Stokes flows in ℝ2\mathbb{R}^{2} around rotating flows. Besides, Guillod-Wittwer [5] generalizes the Hamel solutions.

In this paper, we are interested in using the perturbation method to construct solutions around (𝒱,𝒬)({\mathcal{V}},{\mathcal{Q}}) to the three-dimensional system (NS). There are at least two difficulties. First, since the equations are three-dimensional, the vorticity-streamfunction formulation useful in 2D settings [8, 7, 12] cannot be applied directly. Second, it is not trivial whether the transport by (𝒱,𝒬)({\mathcal{V}},{\mathcal{Q}}) is also present in the three-dimensional part of the flows.

Let us introduce notation for the main result. For ρ≥0\rho\geq 0, we define a Banach space by

Lρ∞​(Ω)={f∈L∞​(Ω)|‖f‖Ls∞<∞},‖f‖Ls∞:=ess​supx∈Ω⁡|x|s​|f​(x)|.\displaystyle\begin{split}L^{\infty}_{\rho}(\Omega)&=\{f\in L^{\infty}(\Omega)~{}|~{}\|f\|_{L^{\infty}_{s}}<\infty\},\qquad\|f\|_{L^{\infty}_{s}}:=\operatorname*{ess\,sup}_{x\in\Omega}\,|x|^{s}|f(x)|.\end{split} (1.1)

In the cylindrical coordinates on Ω×ℝ\Omega\times\mathbb{R}

x1=r​cos⁡θ,x2=r​sin⁡θ,r=|x|≥1,θ∈[0,2​π),𝐞r=(x|x|,0),𝐞θ=(x⊥|x|,0),𝐞3=(0,0,1),\displaystyle\begin{split}&x_{1}=r\cos\theta,\quad x_{2}=r\sin\theta,\quad r=|x|\geq 1,\quad\theta\in[0,2\pi),\\ &{\bf e}_{r}=\Big{(}\frac{x}{|x|},0\Big{)},\quad{\bf e}_{\theta}=\Big{(}\frac{x^{\bot}}{|x|},0\Big{)},\quad{\bf e}_{3}=(0,0,1),\end{split} (1.2)

we denote a three-dimensional vector field v=(v1,v2,v3)v=(v_{1},v_{2},v_{3}) by

v=vr𝐞r+vθ𝐞θ+v3𝐞3,(vr,vθ):=((v1,v2)⋅𝐞r,(v1,v2)⋅𝐞θ).v=v_{r}{\bf e}_{r}+v_{\theta}{\bf e}_{\theta}+v_{3}{\bf e}_{3},\qquad(v_{r},v_{\theta}):=\big{(}(v_{1},v_{2})\cdot{\bf e}_{r},(v_{1},v_{2})\cdot{\bf e}_{\theta}\big{)}.

For a scalar function s=s​(r,θ,z)s=s(r,\theta,z) on Ω×ℝ\Omega\times\mathbb{R} and n∈ℤn\in\mathbb{Z}, we set

sn​(r,z)=12​π​∫02​πs​(r,t,z)​e−i​n​t​dt.s_{n}(r,z)=\frac{1}{2\pi}\int_{0}^{2\pi}s(r,t,z)e^{-int}\,{\rm d}t.

Then, for a vector field vv on Ω×ℝ\Omega\times\mathbb{R}, we define the operator 𝒫n{\mathcal{P}}_{n} by

(𝒫n​v)​(r,θ,z)=vr,n​(r,z)​ei​n​θ​𝐞r+vθ,n​(r,z)​ei​n​θ​𝐞θ+vz,n​(r,z)​ei​n​θ​𝐞3.({\mathcal{P}}_{n}v)(r,\theta,z)=v_{r,n}(r,z)e^{in\theta}{\bf e}_{r}+v_{\theta,n}(r,z)e^{in\theta}{\bf e}_{\theta}+v_{z,n}(r,z)e^{in\theta}{\bf e}_{3}.

We also define 𝒫n{\mathcal{P}}_{n} acting on scalar- and tensor-valued mappings in an obvious manner.

Our main result states the existence of solutions to (NS) which are vertically uniform.

Theorem 1.1

For α∈ℝ\alpha\in\mathbb{R}, γ>2\gamma>2 and 2<ρ<32<\rho<3 with ρ≤γ\rho\leq\gamma, there is a constant ε=ε​(α,γ,ρ)\varepsilon=\varepsilon(\alpha,\gamma,\rho) such that the following holds. Suppose that the boundary data bb is given by

b=b​(x)=(α​x⊥−γ​x,0)b=b(x)=(\alpha x^{\bot}-\gamma x,0)

and that the external force ff is a distribution on Ω\Omega and given by

f=g+div⁡Ff=g+\operatorname{div}F

where g∈L2​ρ−1∞​(Ω)3g\in L^{\infty}_{2\rho-1}(\Omega)^{3} and F∈L2​(ρ−1)∞​(Ω)3×3F\in L^{\infty}_{2(\rho-1)}(\Omega)^{3\times 3} satisfy

∑n∈ℤ(‖𝒫n​g‖L2​ρ−1∞+‖𝒫n​F‖L2​(ρ−2)∞)≤ε.\displaystyle\sum_{n\in\mathbb{Z}}\big{(}\|\mathcal{P}_{n}g\|_{L^{\infty}_{2\rho-1}}+\|\mathcal{P}_{n}F\|_{L^{\infty}_{2(\rho-2)}}\big{)}\leq\varepsilon.

Then there is a weak solution u=u​(x)u=u(x) of (NS) unique in a suitable set (defined in the proof in Section 3). Moreover, the solution u​(x)u(x) has the asymptotic behavior

u​(x)=α​(x⊥|x|2,0)−γ​(x|x|2,0)+O​(|x|−ρ+1)as​|x|→∞.\displaystyle u(x)=\alpha\Big{(}\frac{x^{\bot}}{|x|^{2}},0\Big{)}-\gamma\Big{(}\frac{x}{|x|^{2}},0\Big{)}+O(|x|^{-\rho+1})\quad\mbox{as}\ |x|\to\infty. (1.3)
Remark 1.2
  1. (i)

    The precise definition of weak solutions is given in Subsection 1.1.

  2. (ii)

    Theorem 1.1 contains [7, Theorem 1.1] for 2D problems as the special case when f=gf=g and g3=0g_{3}=0 in the assumption. In this case, the solutions belong to Wloc2,2​(Ω¯)W^{2,2}_{{\rm loc}}(\overline{\Omega}) by elliptic regularity, which cannot expected in Theorem 1.1 due to the regularity of ff.

  3. (iii)

    Even though the given data f,bf,b are independent of the variable zz, the solvability of (NS) cannot be reduced to that of the 2D Navier-Stokes system on Ω\Omega. Indeed, for a smooth external force f=f​(x)f=f(x), the vertical component u3=u3​(x)u_{3}=u_{3}(x) is subject to

    −(∂12+∂22)​u3=−(u1​∂1+u2​∂2)​u3+f3.-(\partial_{1}^{2}+\partial_{2}^{2})u_{3}=-(u_{1}\partial_{1}+u_{2}\partial_{2})u_{3}+f_{3}.

    If one linearizes this equation, the 2D Laplace operator appears whose fundamental solution has logarithmic growth. Thus the proof of (1.3) needs careful computation.

  4. (iv)

    It is physically more natural to consider solutions of (NS) for given data that are neither vertically uniform nor decaying. However, the problem is difficult even for the vertically periodic data. Indeed, in addition to the analysis of the linearized problem in Section 2, we need to deal with the linear problems corresponding to the non-constant periodic data, which cannot be regarded as small perturbations from the Stokes system due to γ>2\gamma>2. New machinery is needed for this family of linear problems. Let us mention Kozono-Terasawa-Wakasugi [11] providing asymptotic behavior for the axisymmetric steady Navier-Stokes system in the above vertically periodic setting.

Let us outline the proof of Theorem 1.1 whose details will be given in Section 3. To adapt the above exact solution (𝒱,𝒬)({\mathcal{V}},{\mathcal{Q}}) to a three-dimensional system (NS), we set

V​(x)=(𝒱​(x),0)=α​(x⊥|x|2,0)−γ​(x|x|2,0)\displaystyle V(x)=({\mathcal{V}}(x),0)=\alpha\Big{(}\frac{x^{\bot}}{|x|^{2}},0\Big{)}-\gamma\Big{(}\frac{x}{|x|^{2}},0\Big{)} (1.4)

and Q=𝒬Q={\mathcal{Q}}. Then the new pair (v,q):=(u−V,p−Q)(v,q):=(u-V,p-Q) solves

{−Δ​v+v⋅∇V+V⋅∇v+∇q=−v⋅∇v+fin​Ω×ℝdiv⁡v=0in​Ω×ℝv=0on​∂Ω×ℝv​(x,z)→0as​|x|→∞.\left\{\begin{array}[]{ll}-\Delta v+v\cdot\nabla V+V\cdot\nabla v+\nabla q=-v\cdot\nabla v+f&\mbox{in}\ \Omega\times\mathbb{R}\\ \operatorname{div}v=0&\mbox{in}\ \Omega\times\mathbb{R}\\ v=0&\mbox{on}\ \partial\Omega\times\mathbb{R}\\ v(x,z)\to 0&\mbox{as}\ |x|\to\infty.\end{array}\right. (1.5)

By the formula

u⋅∇v+v⋅∇u=−u×rot⁡v−v×rot⁡u+∇(|u+v|2−|u|2−|v|22),\displaystyle u\cdot\nabla v+v\cdot\nabla u=-u\times\operatorname{rot}v-v\times\operatorname{rot}u+\nabla\Big{(}\frac{|u+v|^{2}-|u|^{2}-|v|^{2}}{2}\Big{)}, (1.6)

where ×\times is the cross product in ℝ3\mathbb{R}^{3}, and by rot⁡V=0\operatorname{rot}V=0, we rewrite (1.5) as

{−Δ​v−V×rot⁡v+∇q1=−v⋅∇v+fin​Ω×ℝdiv⁡v=0in​Ω×ℝv=0on​∂Ω×ℝv​(x,z)→0as​|x|→∞\left\{\begin{array}[]{ll}-\Delta v-V\times\operatorname{rot}v+\nabla q_{1}=-v\cdot\nabla v+f&\mbox{in}\ \Omega\times\mathbb{R}\\ \operatorname{div}v=0&\mbox{in}\ \Omega\times\mathbb{R}\\ v=0&\mbox{on}\ \partial\Omega\times\mathbb{R}\\ v(x,z)\to 0&\mbox{as}\ |x|\to\infty\end{array}\right. (NP)

with the pressure

∇q1=∇(q+|V+v|2−|V|2−|v|22).\nabla q_{1}=\nabla\Big{(}q+\frac{|V+v|^{2}-|V|^{2}-|v|^{2}}{2}\Big{)}.

We analyze the perturbed nonlinear system (NP) to prove Theorem 1.1. As will be seen in Section 2, the fundamental solution for the linearized problem has better decay compared with the one for the non-perturbed case when α=γ=0\alpha=\gamma=0. Consequently, an improvement of decay can be seen similar to the 2D case [8, 7, 12]. However, it should be emphasized, that a new analysis is necessary because the system (NP) is three-dimensional.

This paper is organized as follows. In Section 2, we study the linearized problem of (NP) and prove decay estimates of the solutions. In Section 3, we prove Theorem 1.1.

1.1 Notation and Terminology

We summarize the notation and terminology used throughout this paper.

Notation. We denote by CC the constant and by C​(a,b,c,…)C(a,b,c,\ldots) the constant depending on a,b,c,…a,b,c,\ldots. Both of these may vary from line to line. We use the function spaces on Ω\Omega

W^1,2​(Ω)\displaystyle\widehat{W}^{1,2}(\Omega) ={χ∈Lloc2​(Ω¯)|∇χ∈L2​(Ω)2},\displaystyle=\{\chi\in L^{2}_{{\rm loc}}(\overline{\Omega})~{}|~{}\nabla\chi\in L^{2}(\Omega)^{2}\},
C0,σ∞​(Ω)\displaystyle C^{\infty}_{0,\sigma}(\Omega) ={ψ=(ψ1,ψ2)∈C0∞​(Ω)2|∂1ψ1+∂2ψ2=0}\displaystyle=\{\psi=(\psi_{1},\psi_{2})\in C^{\infty}_{0}(\Omega)^{2}~{}|~{}\partial_{1}\psi_{1}+\partial_{2}\psi_{2}=0\}

and Lσ2​(Ω)L^{2}_{\sigma}(\Omega) which is the completion of C0,σ∞​(Ω)C^{\infty}_{0,\sigma}(\Omega) in the L2L^{2}-norm. If there is no confusion, we use the same notation to denote the quantities concerning scalar-, vector- or tensor-valued mappings. For example, ⟨⋅,⋅⟩\langle\cdot,\cdot\rangle denotes to the inner product on L2​(Ω)L^{2}(\Omega), L2​(Ω)2L^{2}(\Omega)^{2} or L2​(Ω)2×2L^{2}(\Omega)^{2\times 2}.

Weak solutions. Let us clarify the definition of weak solutions of (NS) in Theorem 1.1. Let ff and bb satisfy the assumption in Theorem 1.1. Then a three-dimensional vector field u∈W^1,2​(Ω)3u\in\widehat{W}^{1,2}(\Omega)^{3} is called a weak solution of (NS) if uu satisfies div⁡u=∂1u1+∂2u2=0\operatorname{div}u=\partial_{1}u_{1}+\partial_{2}u_{2}=0 in the sense of distributions, (u−b)|∂Ω=0(u-b)|_{\partial\Omega}=0 in the sense of trace, and it holds that

∫Ω∇u⋅∇φ=∫Ω(u⊗u)⋅∇φ+∫Ωg⋅φ−∫ΩF⋅∇φ,\displaystyle\int_{\Omega}\nabla u\cdot\nabla\varphi=\int_{\Omega}(u\otimes u)\cdot\nabla\varphi+\int_{\Omega}g\cdot\varphi-\int_{\Omega}F\cdot\nabla\varphi,
for any φ=(φ1,φ2,φ3)∈C0,σ∞​(Ω)×C0∞​(Ω)\varphi=(\varphi_{1},\varphi_{2},\varphi_{3})\in C^{\infty}_{0,\sigma}(\Omega)\times C^{\infty}_{0}(\Omega).

Axisymmetricity. A scalar function on Ω×ℝ\Omega\times\mathbb{R} with variable (r,θ,z)(r,\theta,z) is said to be axisymmetric if it is independent of θ\theta. A vector field v=vr​𝐞r+vθ​𝐞θ+v3​𝐞3v=v_{r}{\bf e}_{r}+v_{\theta}{\bf e}_{\theta}+v_{3}{\bf e}_{3} on Ω×ℝ\Omega\times\mathbb{R} with variable (r,θ,z)(r,\theta,z) is said to be axisymmetric if the scalar functions vr,vθ,v3v_{r},v_{\theta},v_{3} are axisymmetric. The axisymmetricity of tensor fields on Ω×ℝ\Omega\times\mathbb{R} is defined in a similar manner.

2 Linearized Problem

In this section, we consider the linearized problem of (NP)

{−Δ​v−V×rot⁡v+∇q=fin​Ω×ℝdiv⁡v=0in​Ω×ℝv=0on​∂Ω×ℝv​(x,z)→0as​|x|→∞.\left\{\begin{array}[]{ll}-\Delta v-V\times\operatorname{rot}v+\nabla q=f&\mbox{in}\ \Omega\times\mathbb{R}\\ \operatorname{div}v=0&\mbox{in}\ \Omega\times\mathbb{R}\\ v=0&\mbox{on}\ \partial\Omega\times\mathbb{R}\\ v(x,z)\to 0&\mbox{as}\ |x|\to\infty.\end{array}\right. (LP)

Assume that this system does not depend on the variable zz. Set

vh=(v1,v2),rot2​D⁡vh=∂1v2−∂2v1.\displaystyle v_{\rm h}=(v_{1},v_{2}),\qquad\operatorname{rot}_{{\rm 2D}}v_{\rm h}=\partial_{1}v_{2}-\partial_{2}v_{1}. (2.1)

Then, by computation

−V×rot⁡v\displaystyle-V\times\operatorname{rot}v =−(V1V20)×(∂2v3−∂1v3rot2​D⁡vh)\displaystyle=-\left(\begin{array}[]{c}V_{1}\\ V_{2}\\ 0\end{array}\right)\times\left(\begin{array}[]{c}\partial_{2}v_{3}\\ -\partial_{1}v_{3}\\ \operatorname{rot}_{{\rm 2D}}v_{\rm h}\end{array}\right)
=(−V2V10)​rot2​D⁡vh+(00V⋅∇v3),\displaystyle=\left(\begin{array}[]{c}-V_{2}\\ V_{1}\\ 0\end{array}\right)\operatorname{rot}_{{\rm 2D}}v_{\rm h}+\left(\begin{array}[]{c}0\\ 0\\ V\cdot\nabla v_{3}\end{array}\right),

we separate (LP) into two systems; one is for the horizontal components vhv_{\rm h}

{−Δh​vh+(Vh)⊥​rot2​D⁡vh+∇hq=fhin​Ωdivh⁡vh=0in​Ωvh=0on​∂Ωvh​(x)→0as​|x|→∞\left\{\begin{array}[]{ll}-\Delta_{\rm h}v_{\rm h}+(V_{\rm h})^{\bot}\operatorname{rot}_{{\rm 2D}}v_{\rm h}+\nabla_{\rm h}q=f_{\rm h}&\mbox{in}\ \Omega\\ \operatorname{div}_{\rm h}v_{\rm h}=0&\mbox{in}\ \Omega\\ v_{\rm h}=0&\mbox{on}\ \partial\Omega\\ v_{\rm h}(x)\to 0&\mbox{as}\ |x|\to\infty\end{array}\right. (LPh)

where

−Δh=−(∂12+∂22),∇h=(∂1,∂2),divh⁡vh=∂1v1+∂2v2,\displaystyle-\Delta_{\rm h}=-(\partial_{1}^{2}+\partial_{2}^{2}),\qquad\nabla_{\rm h}=(\partial_{1},\partial_{2}),\qquad\operatorname{div}_{\rm h}v_{\rm h}=\partial_{1}v_{1}+\partial_{2}v_{2}, (2.2)

and the other is for the vertical component v3v_{3}

{−Δh​v3+Vh⋅∇hv3=f3in​Ωv3=0on​∂Ωv3​(x)→0as​|x|→∞.\left\{\begin{array}[]{ll}-\Delta_{\rm h}v_{3}+V_{\rm h}\cdot\nabla_{\rm h}v_{3}=f_{3}&\mbox{in}\ \Omega\\ v_{3}=0&\mbox{on}\ \partial\Omega\\ v_{3}(x)\to 0&\mbox{as}\ |x|\to\infty.\end{array}\right. (LPv)

Notice that both problems are imposed on the 2D exterior domain Ω\Omega. The problem (LPh) is studied in [7] for the case when external forces are more regular. We revisit the results and generalize the class of solutions in view of regularity. On the other hand, (LPv) for the vertical component requires new consideration which is not covered in [7].

2.1 Preliminaries

This subsection collects notation for mappings on Ω×ℝ\Omega\times\mathbb{R} independent of the variable zz. We denote by (r,θ)(r,\theta) the variables in the polar coordinates on Ω\Omega; see (1.2). Moreover, we identify 𝐞r{\bf e}_{r} and 𝐞θ{\bf e}_{\theta} respectively with the two-dimensional vectors x/|x|x/|x| and x⊥/|x|x^{\bot}/|x| defined on Ω\Omega if there is no confusion. Some of the notation in this subsection are special cases of those in the introduction, but are duplicated for clarity of explanation.

In the polar coordinates, we denote a two-dimensional vector field vh=(v1,v2)v_{\rm h}=(v_{1},v_{2}) by

vh=vr​𝐞r+vθ​𝐞θ,(vr,vθ):=((v1,v2)⋅𝐞r,(v1,v2)⋅𝐞θ).v_{\rm h}=v_{r}{\bf e}_{r}+v_{\theta}{\bf e}_{\theta},\qquad(v_{r},v_{\theta}):=\big{(}(v_{1},v_{2})\cdot{\bf e}_{r},(v_{1},v_{2})\cdot{\bf e}_{\theta}\big{)}.

Then the operators divh\operatorname{div}_{\rm h}, rot2​D\operatorname{rot}_{{\rm 2D}} and −Δh-\Delta_{\rm h} in (2.1)–(2.2) are represented by

divh⁡vh=1r​∂r(r​vr)+1r​∂θvθ,rot2​D⁡vh=1r​∂r(r​vθ)−1r​∂θvr,−Δh​vh={−∂r(1r​∂r(r​vr))−1r2​∂θ2vr+2r2​∂θvθ}​𝐞r+{−∂r(1r​∂r(r​vθ))−1r2​∂θ2vθ−2r2​∂θvr}​𝐞θ.\displaystyle\begin{split}\operatorname{div}_{\rm h}v_{\rm h}&=\frac{1}{r}\partial_{r}(rv_{r})+\frac{1}{r}\partial_{\theta}v_{\theta},\\ \operatorname{rot}_{{\rm 2D}}v_{\rm h}&=\frac{1}{r}\partial_{r}(rv_{\theta})-\frac{1}{r}\partial_{\theta}v_{r},\\ -\Delta_{\rm h}v_{\rm h}&=\Big{\{}-\partial_{r}\Big{(}\frac{1}{r}\partial_{r}(rv_{r})\Big{)}-\frac{1}{r^{2}}\partial_{\theta}^{2}v_{r}+\frac{2}{r^{2}}\partial_{\theta}v_{\theta}\Big{\}}{\bf e}_{r}\\ &\quad+\Big{\{}-\partial_{r}\Big{(}\frac{1}{r}\partial_{r}(rv_{\theta})\Big{)}-\frac{1}{r^{2}}\partial_{\theta}^{2}v_{\theta}-\frac{2}{r^{2}}\partial_{\theta}v_{r}\Big{\}}{\bf e}_{\theta}.\end{split} (2.3)

If a 3×33\times 3 tensor field FF on Ω×ℝ\Omega\times\mathbb{R}

F\displaystyle F =Fr​r​𝐞r⊗𝐞r+Fr​θ​𝐞r⊗𝐞θ+Fr​3​𝐞r⊗𝐞3\displaystyle=F_{rr}{\bf e}_{r}\otimes{\bf e}_{r}+F_{r\theta}{\bf e}_{r}\otimes{\bf e}_{\theta}+F_{r3}{\bf e}_{r}\otimes{\bf e}_{3}
+Fθ​r​𝐞θ⊗𝐞r+Fθ​θ​𝐞θ⊗𝐞θ+Fθ​3​𝐞θ⊗𝐞3\displaystyle\quad+F_{\theta r}{\bf e}_{\theta}\otimes{\bf e}_{r}+F_{\theta\theta}{\bf e}_{\theta}\otimes{\bf e}_{\theta}+F_{\theta 3}{\bf e}_{\theta}\otimes{\bf e}_{3}
+F3​r​𝐞3⊗𝐞r+F3​θ​𝐞3⊗𝐞θ+F33​𝐞3⊗𝐞3\displaystyle\quad+F_{3r}{\bf e}_{3}\otimes{\bf e}_{r}+F_{3\theta}{\bf e}_{3}\otimes{\bf e}_{\theta}+F_{33}{\bf e}_{3}\otimes{\bf e}_{3}

is independent of zz, then we have

div⁡F=(div⁡F)r​𝐞r+(div⁡F)θ​𝐞θ+(div⁡F)3​𝐞3,\displaystyle\operatorname{div}F=(\operatorname{div}F)_{r}{\bf e}_{r}+(\operatorname{div}F)_{\theta}{\bf e}_{\theta}+(\operatorname{div}F)_{3}{\bf e}_{3}, (2.4)

where

(div⁡F)r\displaystyle(\operatorname{div}F)_{r} =1r​∂r(r​Fr​r)+1r​(∂θFθ​r−Fθ​θ),\displaystyle=\frac{1}{r}\partial_{r}(rF_{rr})+\frac{1}{r}(\partial_{\theta}F_{\theta r}-F_{\theta\theta}),
(div⁡F)θ\displaystyle(\operatorname{div}F)_{\theta} =1r​∂r(r​Fr​θ)+1r​(∂θFθ​θ+Fθ​r),\displaystyle=\frac{1}{r}\partial_{r}(rF_{r\theta})+\frac{1}{r}(\partial_{\theta}F_{\theta\theta}+F_{\theta r}),
(div⁡F)3\displaystyle(\operatorname{div}F)_{3} =1r​∂r(r​Fr​3)+1r​∂θFθ​3.\displaystyle=\frac{1}{r}\partial_{r}(rF_{r3})+\frac{1}{r}\partial_{\theta}F_{\theta 3}.

In particular, using divh\operatorname{div}_{\rm h} in (2.2), we see that

(div⁡F)3=divh⁡(Fr​3​𝐞r+Fθ​3​𝐞θ).\displaystyle(\operatorname{div}F)_{3}=\operatorname{div}_{\rm h}(F_{r3}{\bf e}_{r}+F_{\theta 3}{\bf e}_{\theta}). (2.5)

For a 2×22\times 2 tensor field FF on Ω\Omega

F=Fr​r​𝐞r⊗𝐞r+Fr​θ​𝐞r⊗𝐞θ+Fθ​r​𝐞θ⊗𝐞r+Fθ​θ​𝐞θ⊗𝐞θ,\displaystyle F=F_{rr}{\bf e}_{r}\otimes{\bf e}_{r}+F_{r\theta}{\bf e}_{r}\otimes{\bf e}_{\theta}+F_{\theta r}{\bf e}_{\theta}\otimes{\bf e}_{r}+F_{\theta\theta}{\bf e}_{\theta}\otimes{\bf e}_{\theta},

we have

divh⁡F=(divh⁡F)r​𝐞r+(divh⁡F)θ​𝐞θ,\displaystyle\operatorname{div}_{\rm h}F=(\operatorname{div}_{\rm h}F)_{r}{\bf e}_{r}+(\operatorname{div}_{\rm h}F)_{\theta}{\bf e}_{\theta},

where

(divh⁡F)r=1r​∂r(r​Fr​r)+1r​(∂θFθ​r−Fθ​θ),(divh⁡F)θ=1r​∂r(r​Fr​θ)+1r​(∂θFθ​θ+Fθ​r).\displaystyle\begin{split}(\operatorname{div}_{\rm h}F)_{r}&=\frac{1}{r}\partial_{r}(rF_{rr})+\frac{1}{r}(\partial_{\theta}F_{\theta r}-F_{\theta\theta}),\\ (\operatorname{div}_{\rm h}F)_{\theta}&=\frac{1}{r}\partial_{r}(rF_{r\theta})+\frac{1}{r}(\partial_{\theta}F_{\theta\theta}+F_{\theta r}).\end{split}

Let n∈ℤn\in\mathbb{Z}. For a scalar function s=s​(r,θ)s=s(r,\theta) on Ω\Omega, we define the projection

sn​(r)=12​π​∫02​πs​(r,t)​e−i​n​t​dt\displaystyle s_{n}(r)=\frac{1}{2\pi}\int_{0}^{2\pi}s(r,t)e^{-int}\,{\rm d}t (2.6)

on the Fourier mode nn. For a vector field v=v​(r,θ)v=v(r,\theta) on Ω\Omega

v​(r,θ)=vr​(r,θ)​𝐞r+vθ​(r,θ)​𝐞θ+v3​(r,θ)​𝐞3,v(r,\theta)=v_{r}(r,\theta){\bf e}_{r}+v_{\theta}(r,\theta){\bf e}_{\theta}+v_{3}(r,\theta){\bf e}_{3},

using (2.6), we define the operator 𝒫n{\mathcal{P}}_{n} by

(𝒫n​v)​(r,θ)=vr,n​(r)​ei​n​θ​𝐞r+vθ,n​(r)​ei​n​θ​𝐞θ+v3,n​(r)​ei​n​θ​𝐞3.\displaystyle({\mathcal{P}}_{n}v)(r,\theta)=v_{r,n}(r)e^{in\theta}{\bf e}_{r}+v_{\theta,n}(r)e^{in\theta}{\bf e}_{\theta}+v_{3,n}(r)e^{in\theta}{\bf e}_{3}. (2.7)

Applying the operator 𝒫n{\mathcal{P}}_{n} to (2.4), we obtain

𝒫n​(div⁡F)=(div⁡F)r,n​(r)​ei​n​θ​𝐞r+(div⁡F)θ,n​(r)​ei​n​θ​𝐞θ+(div⁡F)3,n​(r)​ei​n​θ​𝐞3,\displaystyle{\mathcal{P}}_{n}(\operatorname{div}F)=(\operatorname{div}F)_{r,n}(r)e^{in\theta}{\bf e}_{r}+(\operatorname{div}F)_{\theta,n}(r)e^{in\theta}{\bf e}_{\theta}+(\operatorname{div}F)_{3,n}(r)e^{in\theta}{\bf e}_{3},

where

(div⁡F)r,n\displaystyle(\operatorname{div}F)_{r,n} =1r​dd​r​(r​Fr​r,n)+1r​(i​n​Fθ​r,n−Fθ​θ,n),\displaystyle=\frac{1}{r}\frac{\,{\rm d}}{\,{\rm d}r}(rF_{rr,n})+\frac{1}{r}(inF_{\theta r,n}-F_{\theta\theta,n}),
(div⁡F)θ,n\displaystyle(\operatorname{div}F)_{\theta,n} =1r​dd​r​(r​Fr​θ,n)+1r​(i​n​Fθ​θ,n+Fθ​r,n),\displaystyle=\frac{1}{r}\frac{\,{\rm d}}{\,{\rm d}r}(rF_{r\theta,n})+\frac{1}{r}(inF_{\theta\theta,n}+F_{\theta r,n}),
(div⁡F)3,n\displaystyle(\operatorname{div}F)_{3,n} =1r​dd​r​(r​Fr​3,n)+i​nr​Fθ​3,n.\displaystyle=\frac{1}{r}\frac{\,{\rm d}}{\,{\rm d}r}(rF_{r3,n})+\frac{in}{r}F_{\theta 3,n}.

We also define 𝒫n{\mathcal{P}}_{n} acting on scalar- and tensor-valued mappings in an obvious manner. For vector-valued ff or tensor-valued FF, we simply denote 𝒫n​f{\mathcal{P}}_{n}f by fnf_{n} or 𝒫n​F{\mathcal{P}}_{n}F by FnF_{n}. We do not identify 𝒫n​s{\mathcal{P}}_{n}s with sns_{n} when ss is scalar-valued to avoid confusion with (2.6).

2.2 Horizontal Components

Let n∈ℤn\in\mathbb{Z}. Applying 𝒫n{\mathcal{P}}_{n} to vhv_{\rm h} and fhf_{\rm h}

(𝒫n​vh)​(θ,r)\displaystyle({\mathcal{P}}_{n}v_{\rm h})(\theta,r) =vr,n​(r)​ei​n​θ​𝐞r+vθ,n​(r)​ei​n​θ​𝐞θ,\displaystyle=v_{r,n}(r)e^{in\theta}{\bf e}_{r}+v_{\theta,n}(r)e^{in\theta}{\bf e}_{\theta},
(𝒫n​fh)​(θ,r)\displaystyle({\mathcal{P}}_{n}f_{\rm h})(\theta,r) =fr,n​(r)​ei​n​θ​𝐞r+fθ,n​(r)​ei​n​θ​𝐞θ\displaystyle=f_{r,n}(r)e^{in\theta}{\bf e}_{r}+f_{\theta,n}(r)e^{in\theta}{\bf e}_{\theta}

and inserting these to (LPh), we see that (vr,n​(r),vθ,n​(r))(v_{r,n}(r),v_{\theta,n}(r)) and qn​(r)q_{n}(r) satisfy

−dd​r​(1r​dd​r​(r​vr,n))+n2r2​vr,n+2​i​nr2​vθ,n−αr2​(dd​r​(r​vθ,n)−i​n​vr,n)+∂rqn=fr,n,r>1,\displaystyle\begin{aligned} &-\frac{\,{\rm d}}{\,{\rm d}r}\Big{(}\frac{1}{r}\frac{\,{\rm d}}{\,{\rm d}r}(rv_{r,n})\Big{)}+\frac{n^{2}}{r^{2}}v_{r,n}+\frac{2in}{r^{2}}v_{\theta,n}\\ &\qquad\qquad-\frac{\alpha}{r^{2}}\Big{(}\frac{\,{\rm d}}{\,{\rm d}r}(rv_{\theta,n})-inv_{r,n}\Big{)}+\partial_{r}q_{n}=f_{r,n},\quad r>1,\\ \end{aligned} (2.8)
−dd​r​(1r​dd​r​(r​vθ,n))+n2r2​vθ,n−2​i​nr2​vr,n−γr2​(dd​r​(r​vθ,n)−i​n​vr,n)+i​nr​qn=fθ,n,r>1,\displaystyle\begin{aligned} &-\frac{\,{\rm d}}{\,{\rm d}r}\Big{(}\frac{1}{r}\frac{\,{\rm d}}{\,{\rm d}r}(rv_{\theta,n})\Big{)}+\frac{n^{2}}{r^{2}}v_{\theta,n}-\frac{2in}{r^{2}}v_{r,n}\\ &\qquad\qquad-\frac{\gamma}{r^{2}}\Big{(}\frac{\,{\rm d}}{\,{\rm d}r}(rv_{\theta,n})-inv_{r,n}\Big{)}+\frac{in}{r}q_{n}=f_{\theta,n},\quad r>1,\end{aligned} (2.9)

the divergence-free and boundary conditions

dd​r​(r​vr,n)+i​n​vθ,n=0,vr,n​(1)=vθ,n​(1)=0,\displaystyle\frac{\,{\rm d}}{\,{\rm d}r}(rv_{r,n})+inv_{\theta,n}=0,\qquad v_{r,n}(1)=v_{\theta,n}(1)=0, (2.10)

and the condition at infinity

|vr,n​(r)|+|vθ,n​(r)|→0,r→∞.\displaystyle|v_{r,n}(r)|+|v_{\theta,n}(r)|\to 0,\quad r\to\infty. (2.11)

2.2.1 Axisymmetric Part

Proposition 2.1

Let α∈ℝ\alpha\in\mathbb{R}, γ>2\gamma>2 and 2<ρ≤γ2<\rho\leq\gamma. Suppose that fh=fh,0f_{\rm h}=f_{{\rm h},0} is given by fh,0=divh⁡F0f_{{\rm h},0}=\operatorname{div}_{\rm h}F_{0} for some F0∈𝒫0​L2​(ρ−1)∞​(Ω)2×2F_{0}\in{\mathcal{P}}_{0}L^{\infty}_{2(\rho-1)}(\Omega)^{2\times 2}. Then there is a unique weak solution vh,0∈𝒫0​Lσ2​(Ω)∩W01,2​(Ω)2v_{{\rm h},0}\in{\mathcal{P}}_{0}L^{2}_{\sigma}(\Omega)\cap W^{1,2}_{0}(\Omega)^{2} of (LPh) satisfying

vh,0​(r,θ)=vθ,0​(r)​𝐞θv_{{\rm h},0}(r,\theta)=v_{\theta,0}(r){\bf e}_{\theta}

and

‖vh,0‖Lρ−1∞+1γ−1​‖∇hvh,0‖Lρ∞\displaystyle\|v_{{\rm h},0}\|_{L^{\infty}_{\rho-1}}+\frac{1}{\gamma-1}\|\nabla_{\rm h}v_{{\rm h},0}\|_{L^{\infty}_{\rho}} ≤C​(γ−1)(γ−2)​(ρ−2)​‖F0‖L2​(ρ−1)∞.\displaystyle\leq\frac{C(\gamma-1)}{(\gamma-2)(\rho-2)}\|F_{0}\|_{L^{\infty}_{2(\rho-1)}}. (2.12)

The constant CC is independent of α\alpha, γ\gamma and ρ\rho.

Proof.

To simplify, we omit the subscript “h{\rm h}” for vhv_{\rm h} and fhf_{\rm h}. Put n=0n=0 in (LABEL:eq.LPh.polar.vr)–(2.11). The radial part vr,0​(r)v_{r,0}(r) is identically zero since both (r​vr,0)′=0(rv_{r,0})^{\prime}=0 and vr,0​(1)=0v_{r,0}(1)=0 hold by (2.10). Besides, the angular part vθ,0​(r)v_{\theta,0}(r) solves the ordinary differential equation

−d2​vθ,0d​r2−1+γr​d​vθ,0d​r+1−γr2​vθ,0=fθ,0,r>1,vθ,0​(1)=0.\displaystyle-\frac{\,{\rm d}^{2}v_{\theta,0}}{\,{\rm d}r^{2}}-\frac{1+\gamma}{r}\frac{\,{\rm d}v_{\theta,0}}{\,{\rm d}r}+\frac{1-\gamma}{r^{2}}v_{\theta,0}=f_{\theta,0},\quad r>1,\qquad v_{\theta,0}(1)=0. (2.13)

As the uniqueness of solutions is clear, we may only consider the existence and estimates.

At first we assume that F0∈𝒫0​C0∞​(Ω)2×2F_{0}\in{\mathcal{P}}_{0}C^{\infty}_{0}(\Omega)^{2\times 2}, which gives

fθ,0=(divh⁡F)θ,0=1r​dd​r​(r​Fr​θ,0)+1r​Fθ​r,0.\displaystyle f_{\theta,0}=(\operatorname{div}_{\rm h}F)_{\theta,0}=\frac{1}{r}\frac{\,{\rm d}}{\,{\rm d}r}(rF_{r\theta,0})+\frac{1}{r}F_{\theta r,0}. (2.14)

As is shown in [7, Proof of Proposition 3.1], the solution of (2.13) is represented by

vθ,0​(r)=1γ−2{−(∫1∞s2fθ,0(s)ds)r−γ+1+r−γ+1∫1rsγfθ,0(s)ds+r−1∫r∞s2fθ,0(s)ds}.\begin{split}v_{\theta,0}(r)&=\frac{1}{\gamma-2}\bigg{\{}-\bigg{(}\int_{1}^{\infty}s^{2}f_{\theta,0}(s)\,{\rm d}s\bigg{)}r^{-\gamma+1}\\ &\qquad\qquad\quad+r^{-\gamma+1}\int_{1}^{r}s^{\gamma}f_{\theta,0}(s)\,{\rm d}s+r^{-1}\int_{r}^{\infty}s^{2}f_{\theta,0}(s)\,{\rm d}s\bigg{\}}.\end{split}

By integration by parts based on (2.14), we rewrite this formula as

vθ,0​(r)=1γ−2[(∫1∞sFr​θ,0(s)ds−∫1∞sFθ​r,0(s)ds)r−γ+1+r−γ+1​{−(γ−1)​∫1rsγ−1​Fr​θ,0​(s)​ds+∫1rsγ−1​Fθ​r,0​(s)​ds}+r−1(−∫r∞sFr​θ,0(s)ds+∫r∞sFθ​r,0(s)ds)].\begin{split}&v_{\theta,0}(r)\\ &=\frac{1}{\gamma-2}\bigg{[}\bigg{(}\int_{1}^{\infty}sF_{r\theta,0}(s)\,{\rm d}s-\int_{1}^{\infty}sF_{\theta r,0}(s)\,{\rm d}s\bigg{)}r^{-\gamma+1}\\ &\qquad\qquad\quad+r^{-\gamma+1}\bigg{\{}-(\gamma-1)\int_{1}^{r}s^{\gamma-1}F_{r\theta,0}(s)\,{\rm d}s+\int_{1}^{r}s^{\gamma-1}F_{\theta r,0}(s)\,{\rm d}s\bigg{\}}\\ &\qquad\qquad\quad+r^{-1}\bigg{(}-\int_{r}^{\infty}sF_{r\theta,0}(s)\,{\rm d}s+\int_{r}^{\infty}sF_{\theta r,0}(s)\,{\rm d}s\bigg{)}\bigg{]}.\end{split} (2.15)

An associated pressure 𝒫0​q{\mathcal{P}}_{0}q is obtained from the equation (LABEL:eq.LPh.polar.vr). Set v0​(r,θ)=vθ,0​(r)​𝐞θv_{0}(r,\theta)=v_{\theta,0}(r){\bf e}_{\theta}. Then the pair (v0,∇h𝒫0​q)(v_{0},\nabla_{\rm h}{\mathcal{P}}_{0}q) is smooth and satisfies (LABEL:eq.LPh.polar.vr)–(2.11) in the classical sense.

Next we let F0∈𝒫0​L2​(ρ−1)∞​(Ω)2×2F_{0}\in{\mathcal{P}}_{0}L^{\infty}_{2(\rho-1)}(\Omega)^{2\times 2}. By computation

(∫1∞s​|F0​(s)|​ds)​r−γ+1≤Cρ−2​‖F0‖L2​(ρ−1)∞​r−γ+1\displaystyle\begin{split}\bigg{(}\int_{1}^{\infty}s|F_{0}(s)|\,{\rm d}s\bigg{)}r^{-\gamma+1}\leq\frac{C}{\rho-2}\|F_{0}\|_{L^{\infty}_{2(\rho-1)}}r^{-\gamma+1}\end{split}

and

r−γ+1​∫1rsγ−1​|F0​(s)|​ds+r−1​∫r∞s​|F0​(s)|​ds≤Cρ−2​‖F0‖L2​(ρ−1)∞​r−ρ+1,\displaystyle\begin{split}&r^{-\gamma+1}\int_{1}^{r}s^{\gamma-1}|F_{0}(s)|\,{\rm d}s+r^{-1}\int_{r}^{\infty}s|F_{0}(s)|\,{\rm d}s\leq\frac{C}{\rho-2}\|F_{0}\|_{L^{\infty}_{2(\rho-1)}}r^{-\rho+1},\end{split}

one can verify that the desired estimate (2.12) follows from the formula (2.15).

Finally, we show that v0​(r,θ)=vθ,0​(r)​𝐞θv_{0}(r,\theta)=v_{\theta,0}(r){\bf e}_{\theta} with vθ,0​(r)v_{\theta,0}(r) defined in (2.15) is a weak solution of (LPh) for general F0∈𝒫0​L2​(ρ−1)∞​(Ω)2×2F_{0}\in{\mathcal{P}}_{0}L^{\infty}_{2(\rho-1)}(\Omega)^{2\times 2}. By the definition of v0v_{0}, we have v0∈Lσ2​(Ω)∩W01,2​(Ω)2v_{0}\in L^{2}_{\sigma}(\Omega)\cap W^{1,2}_{0}(\Omega)^{2}. Let {F0(m)}m=1∞⊂𝒫0​C0∞​(Ω)2×2\{F^{(m)}_{0}\}_{m=1}^{\infty}\subset{\mathcal{P}}_{0}C^{\infty}_{0}(\Omega)^{2\times 2} be such that limm→∞F0(m)=F0\displaystyle{\lim_{m\to\infty}F^{(m)}_{0}=F_{0}} in L1​(Ω)∩L2​(Ω)L^{1}(\Omega)\cap L^{2}(\Omega). Let v0(m)v^{(m)}_{0} be given by (2.15) replacing F0F_{0} by F0(m)F^{(m)}_{0}, and let 𝒫0​q(m){\mathcal{P}}_{0}q^{(m)} be the associated pressure. Then, using the estimate for G∈L1​(0,∞;s​d​s)2×2G\in L^{1}(0,\infty;s\,{\rm d}s)^{2\times 2}

r−γ​∫1rsγ−1​|G​(s)|​ds+r−2​∫r∞s​|G​(s)|​ds≤(∫1∞|G​(s)|​s​ds)​r−2\displaystyle r^{-\gamma}\int_{1}^{r}s^{\gamma-1}|G(s)|\,{\rm d}s+r^{-2}\int_{r}^{\infty}s|G(s)|\,{\rm d}s\leq\bigg{(}\int_{1}^{\infty}|G(s)|s\,{\rm d}s\bigg{)}r^{-2}

which implies

‖∇h(v0−v0(m))‖L2≤C​(‖F0−F0(m)‖L1+‖F0−F0(m)‖L2),\|\nabla_{\rm h}(v_{0}-v^{(m)}_{0})\|_{L^{2}}\leq C(\|F_{0}-F^{(m)}_{0}\|_{L^{1}}+\|F_{0}-F^{(m)}_{0}\|_{L^{2}}),

we have, by integration by parts,

⟨∇hv0,∇hφ⟩+⟨Vh⊥​rot2​D⁡v0,φ⟩+⟨F0,∇hφ⟩=limm→∞(⟨∇hv0(m),∇hφ⟩+⟨Vh⊥​rot2​D⁡v0(m),φ⟩+⟨F0(m),∇hφ⟩)=limm→∞⟨−Δh​v0(m)+Vh⊥​rot2​D⁡v0(m)−divh⁡F0(m),φ⟩=limm→∞⟨∇h𝒫0​q(m),φ⟩=0,φ∈C0,σ∞​(Ω).\displaystyle\begin{split}&\langle\nabla_{\rm h}v_{0},\nabla_{\rm h}\varphi\rangle+\langle V_{\rm h}^{\bot}\operatorname{rot}_{{\rm 2D}}v_{0},\varphi\rangle+\langle F_{0},\nabla_{\rm h}\varphi\rangle\\ &=\lim_{m\to\infty}\Big{(}\langle\nabla_{\rm h}v^{(m)}_{0},\nabla_{\rm h}\varphi\rangle+\langle V_{\rm h}^{\bot}\operatorname{rot}_{{\rm 2D}}v^{(m)}_{0},\varphi\rangle+\langle F^{(m)}_{0},\nabla_{\rm h}\varphi\rangle\Big{)}\\ &=\lim_{m\to\infty}\langle-\Delta_{\rm h}v^{(m)}_{0}+V_{\rm h}^{\bot}\operatorname{rot}_{{\rm 2D}}v^{(m)}_{0}-\operatorname{div}_{\rm h}F^{(m)}_{0},\varphi\rangle\\ &=\lim_{m\to\infty}\langle\nabla_{\rm h}{\mathcal{P}}_{0}q^{(m)},\varphi\rangle=0,\quad\varphi\in C^{\infty}_{0,\sigma}(\Omega).\end{split}

Hence we see that v0v_{0} is a weak solution of (LPh). This completes the proof. ∎

2.2.2 Non-axisymmetric Part

Let n≠0n\neq 0. We set

nγ={n2+(γ2)2}12,ζn=(nγ2+i​α​n)12\displaystyle n_{\gamma}=\Big{\{}n^{2}+\Big{(}\frac{\gamma}{2}\Big{)}^{2}\Big{\}}^{\frac{1}{2}},\qquad\zeta_{n}=(n_{\gamma}^{2}+i\alpha n)^{\frac{1}{2}} (2.16)

and

ξn=ℜ⁡(ζn)=nγ2​[{1+(α​nnγ2)2}12+1]12.\displaystyle\begin{split}\xi_{n}&=\Re(\zeta_{n})=\frac{n_{\gamma}}{\sqrt{2}}\bigg{[}\Big{\{}1+\Big{(}\frac{\alpha n}{n_{\gamma}^{2}}\Big{)}^{2}\Big{\}}^{\frac{1}{2}}+1\bigg{]}^{\frac{1}{2}}.\end{split} (2.17)

By direct computation, we have

ξn≤|ζn|≤2​ξn,C1≤ξn|n|≤(|α|12+γ),0<(ξn−γ2)−1<C2​γ|n|\displaystyle\xi_{n}\leq|\zeta_{n}|\leq\sqrt{2}\xi_{n},\qquad C_{1}\leq\frac{\xi_{n}}{|n|}\leq(|\alpha|^{\frac{1}{2}}+\gamma),\qquad 0<\Big{(}\xi_{n}-\frac{\gamma}{2}\Big{)}^{-1}<C_{2}\frac{\gamma}{|n|} (2.18)

with C1,C2C_{1},C_{2} independent of nn, α\alpha and γ\gamma.

Proposition 2.2

Let n≠0n\neq 0 and let α∈ℝ\alpha\in\mathbb{R}, γ>2\gamma>2 and 2<ρ<32<\rho<3 with ρ≤γ\rho\leq\gamma. Suppose that fh=fh,nf_{\rm h}=f_{{\rm h},n} is give by fh,n=divh⁡Fnf_{{\rm h},n}=\operatorname{div}_{\rm h}F_{n} for some Fn∈𝒫n​L2​(ρ−1)∞​(Ω)2×2F_{n}\in{\mathcal{P}}_{n}L^{\infty}_{2(\rho-1)}(\Omega)^{2\times 2}. Then there is a unique weak solution vh,n∈𝒫n​Lσ2​(Ω)∩W01,2​(Ω)2v_{{\rm h},n}\in{\mathcal{P}}_{n}L^{2}_{\sigma}(\Omega)\cap W^{1,2}_{0}(\Omega)^{2} of (LPh) satisfying

‖vh,n‖Lρ−1∞+1|n|​‖∇hvh,n‖Lρ∞≤C|n|​(|n|−ρ+2)​ξn2​(ξn−γ2)−1​‖Fn‖L2​(ρ−1)∞.\displaystyle\|v_{{\rm h},n}\|_{L^{\infty}_{\rho-1}}+\frac{1}{|n|}\|\nabla_{\rm h}v_{{\rm h},n}\|_{L^{\infty}_{\rho}}\leq\frac{C}{|n|(|n|-\rho+2)}\xi_{n}^{2}\Big{(}\xi_{n}-\frac{\gamma}{2}\Big{)}^{-1}\|F_{n}\|_{L^{\infty}_{2(\rho-1)}}. (2.19)

The constant CC is independent of nn, α\alpha, γ\gamma and ρ\rho.

Proof.

To simplify, we omit the subscript “h{\rm h}” for vhv_{\rm h} and fhf_{\rm h}. As the uniqueness of solutions can be shown as in [7, Proof of Proposition 3.2], we only prove the existence and estimates.

At first we assume that Fn∈𝒫n​C0∞​(Ω)2×2F_{n}\in{\mathcal{P}}_{n}C^{\infty}_{0}(\Omega)^{2\times 2}, which gives

fr,n=(divh⁡F)r,n=1r​dd​r​(r​Fr​r,n)+1r​(i​n​Fθ​r,n−Fθ​θ,n),fθ,n=(divh⁡F)θ,n=1r​dd​r​(r​Fr​θ,n)+1r​(i​n​Fθ​θ,n+Fθ​r,n).\displaystyle\begin{split}f_{r,n}&=(\operatorname{div}_{\rm h}F)_{r,n}=\frac{1}{r}\frac{\,{\rm d}}{\,{\rm d}r}(rF_{rr,n})+\frac{1}{r}(inF_{\theta r,n}-F_{\theta\theta,n}),\\ f_{\theta,n}&=(\operatorname{div}_{\rm h}F)_{\theta,n}=\frac{1}{r}\frac{\,{\rm d}}{\,{\rm d}r}(rF_{r\theta,n})+\frac{1}{r}(inF_{\theta\theta,n}+F_{\theta r,n}).\end{split} (2.20)

Set

ωn​(r)=((rot2​D⁡vn)​e−i​n​θ)​(r).\displaystyle\omega_{n}(r)=\big{(}(\operatorname{rot}_{{\rm 2D}}v_{n})e^{-in\theta}\big{)}(r).

From [7, Proof of Proposition 3.2], we see that ωn​(r)\omega_{n}(r) solves

−d2​ωnd​r2−1+γr​d​ωnd​r+n2+i​α​nr2​ωn=(rot​fn)n,r>1\displaystyle-\frac{\,{\rm d}^{2}\omega_{n}}{\,{\rm d}r^{2}}-\frac{1+\gamma}{r}\frac{\,{\rm d}\omega_{n}}{\,{\rm d}r}+\frac{n^{2}+i\alpha n}{r^{2}}\omega_{n}=({\rm rot}\,f_{n})_{n},\quad r>1 (2.21)

and, recalling ζn\zeta_{n} in (2.16), that it can be represented by

ωn​(r)=Φn​[fn]​(r)+cn​[fn]​r−ζn−γ2.\displaystyle\omega_{n}(r)=\Phi_{n}[f_{n}](r)+c_{n}[f_{n}]r^{-\zeta_{n}-\frac{\gamma}{2}}. (2.22)

Here 111Due to a typographical error, the denominator 2​ζn2\zeta_{n} in Φn​[fn]​(r)\Phi_{n}[f_{n}](r) is written as ζn\zeta_{n} in [7, (3.19)–(3.20)]. It should be emphasized, however, that this error does not affect the main results in [7].

Φn​[fn]​(r)=r−ζn−γ22​ζn​∫1rsζn+γ2+1​(rot​fn)n​(s)​ds+rζn−γ22​ζn​∫r∞s−ζn+γ2+1​(rot​fn)n​(s)​ds=−r−ζn−γ22​ζn​∫1rsζn+γ2​{i​n​fr,n​(s)+(ζn+γ2)​fθ,n​(s)}​ds+rζn−γ22​ζn​∫r∞s−ζn+γ2​{−i​n​fr,n​(s)+(ζn−γ2)​fθ,n​(s)}​ds\displaystyle\begin{split}\Phi_{n}[f_{n}](r)&=\frac{r^{-\zeta_{n}-\frac{\gamma}{2}}}{2\zeta_{n}}\int_{1}^{r}s^{\zeta_{n}+\frac{\gamma}{2}+1}({\rm rot}\,f_{n})_{n}(s)\,{\rm d}s\\ &\quad+\frac{r^{\zeta_{n}-\frac{\gamma}{2}}}{2\zeta_{n}}\int_{r}^{\infty}s^{-\zeta_{n}+\frac{\gamma}{2}+1}({\rm rot}\,f_{n})_{n}(s)\,{\rm d}s\\ &=-\frac{r^{-\zeta_{n}-\frac{\gamma}{2}}}{2\zeta_{n}}\int_{1}^{r}s^{\zeta_{n}+\frac{\gamma}{2}}\Big{\{}inf_{r,n}(s)+\Big{(}\zeta_{n}+\frac{\gamma}{2}\Big{)}f_{\theta,n}(s)\Big{\}}\,{\rm d}s\\ &\quad+\frac{r^{\zeta_{n}-\frac{\gamma}{2}}}{2\zeta_{n}}\int_{r}^{\infty}s^{-\zeta_{n}+\frac{\gamma}{2}}\Big{\{}-inf_{r,n}(s)+\Big{(}\zeta_{n}-\frac{\gamma}{2}\Big{)}f_{\theta,n}(s)\Big{\}}\,{\rm d}s\end{split}

and

cn​[fn]=−(ζn+|n|+γ2−2)​∫1∞s−|n|+1​Φn​[fn]​(s)​ds.\displaystyle\begin{split}c_{n}[f_{n}]&=-\Big{(}\zeta_{n}+|n|+\frac{\gamma}{2}-2\Big{)}\int_{1}^{\infty}s^{-|n|+1}\Phi_{n}[f_{n}](s)\,{\rm d}s.\end{split} (2.23)

By integration by parts based on (2.20), we rewrite the formula of Φn​[fn]​(r)\Phi_{n}[f_{n}](r) as

Φn​[fn]​(r)=−Fr​θ,n​(r)+r−ζn−γ2​∫1rsζn+γ2−1​G1​(s)​ds+rζn−γ2​∫r∞s−ζn+γ2−1​G2​(s)​ds,\displaystyle\begin{split}\Phi_{n}[f_{n}](r)&=-F_{r\theta,n}(r)+r^{-\zeta_{n}-\frac{\gamma}{2}}\int_{1}^{r}s^{\zeta_{n}+\frac{\gamma}{2}-1}G_{1}(s)\,{\rm d}s\\ &\quad+r^{\zeta_{n}-\frac{\gamma}{2}}\int_{r}^{\infty}s^{-\zeta_{n}+\frac{\gamma}{2}-1}G_{2}(s)\,{\rm d}s,\end{split} (2.24)

where

G1​(r)=i​n2​ζn​(ζn+γ2−1)​Fr​r,n​(r)+12​ζn​(ζn+γ2)​(ζn+γ2−1)​Fr​θ,n​(r)−12​ζn​(ζn+γ2−n2)​Fθ​r,n​(r)−i​n2​ζn​(ζn+γ2−1)​Fθ​θ,n​(r),G2​(r)=−i​n2​ζn​(ζn−γ2+1)​Fr​r,n​(r)+12​ζn​(ζn−γ2)​(ζn−γ2+1)​Fr​θ,n​(r)+12​ζn​(ζn−γ2+n2)​Fθ​r,n​(r)+i​n2​ζn​(ζn−γ2+1)​Fθ​θ,n​(r).\displaystyle\begin{split}G_{1}(r)&=\frac{in}{2\zeta_{n}}\Big{(}\zeta_{n}+\frac{\gamma}{2}-1\Big{)}F_{rr,n}(r)+\frac{1}{2\zeta_{n}}\Big{(}\zeta_{n}+\frac{\gamma}{2}\Big{)}\Big{(}\zeta_{n}+\frac{\gamma}{2}-1\Big{)}F_{r\theta,n}(r)\\ &\quad-\frac{1}{2\zeta_{n}}\Big{(}\zeta_{n}+\frac{\gamma}{2}-n^{2}\Big{)}F_{\theta r,n}(r)-\frac{in}{2\zeta_{n}}\Big{(}\zeta_{n}+\frac{\gamma}{2}-1\Big{)}F_{\theta\theta,n}(r),\\ G_{2}(r)&=-\frac{in}{2\zeta_{n}}\Big{(}\zeta_{n}-\frac{\gamma}{2}+1\Big{)}F_{rr,n}(r)+\frac{1}{2\zeta_{n}}\Big{(}\zeta_{n}-\frac{\gamma}{2}\Big{)}\Big{(}\zeta_{n}-\frac{\gamma}{2}+1\Big{)}F_{r\theta,n}(r)\\ &\quad+\frac{1}{2\zeta_{n}}\Big{(}\zeta_{n}-\frac{\gamma}{2}+n^{2}\Big{)}F_{\theta r,n}(r)+\frac{in}{2\zeta_{n}}\Big{(}\zeta_{n}-\frac{\gamma}{2}+1\Big{)}F_{\theta\theta,n}(r).\end{split}

Then, using ωn​(r)\omega_{n}(r) in (2.22), we see that vnv_{n} defined by the Biot-Savart law

vn​(r,θ)=vr,n​(r)​ei​n​θ​𝐞r+vr,n​(r)​ei​n​θ​𝐞θ,vr,n​(r)=i​n2​|n|​(r−|n|−1​∫1rs|n|+1​ωn​(s)​ds+r|n|−1​∫r∞s−|n|+1​ωn​(s)​ds),vθ,n​(r)=12​(r−|n|−1​∫1rs|n|+1​ωn​(s)​ds−r|n|−1​∫r∞s−|n|+1​ωn​(s)​ds)\displaystyle\begin{split}v_{n}(r,\theta)&=v_{r,n}(r)e^{in\theta}{\bf e}_{r}+v_{r,n}(r)e^{in\theta}{\bf e}_{\theta},\\ v_{r,n}(r)&=\frac{in}{2|n|}\Big{(}r^{-|n|-1}\int_{1}^{r}s^{|n|+1}\omega_{n}(s)\,{\rm d}s+r^{|n|-1}\int_{r}^{\infty}s^{-|n|+1}\omega_{n}(s)\,{\rm d}s\Big{)},\\ v_{\theta,n}(r)&=\frac{1}{2}\Big{(}r^{-|n|-1}\int_{1}^{r}s^{|n|+1}\omega_{n}(s)\,{\rm d}s-r^{|n|-1}\int_{r}^{\infty}s^{-|n|+1}\omega_{n}(s)\,{\rm d}s\Big{)}\end{split} (2.25)

is a smooth solution of (LPh) with some smooth associated pressure 𝒫n​q{\mathcal{P}}_{n}q. The reader is referred to [7, Proof of Proposition 3.2] for the details when ff is a regular function.

Next we let Fn∈𝒫n​L2​(ρ−1)∞​(Ω)2×2F_{n}\in{\mathcal{P}}_{n}L^{\infty}_{2(\rho-1)}(\Omega)^{2\times 2}. Recalling (2.17)–(2.18), we compute

|r−ζn−γ2​∫1rsζn+γ2−1​G1​(s)​ds|+|rζn−γ2​∫r∞s−ζn+γ2−1​G2​(s)​ds|≤C​ξn​(r−ξn−γ2​∫1rsξn+γ2−2​ρ+1​ds+rξn−γ2​∫r∞s−ξn+γ2−2​ρ+1​ds)≤C​ξn​(ξn−γ2)−1​‖Fn‖L2​(ρ−1)∞​r−ρ\displaystyle\begin{split}&\bigg{|}r^{-\zeta_{n}-\frac{\gamma}{2}}\int_{1}^{r}s^{\zeta_{n}+\frac{\gamma}{2}-1}G_{1}(s)\,{\rm d}s\bigg{|}+\bigg{|}r^{\zeta_{n}-\frac{\gamma}{2}}\int_{r}^{\infty}s^{-\zeta_{n}+\frac{\gamma}{2}-1}G_{2}(s)\,{\rm d}s\bigg{|}\\ &\leq C\xi_{n}\bigg{(}r^{-\xi_{n}-\frac{\gamma}{2}}\int_{1}^{r}s^{\xi_{n}+\frac{\gamma}{2}-2\rho+1}\,{\rm d}s+r^{\xi_{n}-\frac{\gamma}{2}}\int_{r}^{\infty}s^{-\xi_{n}+\frac{\gamma}{2}-2\rho+1}\,{\rm d}s\bigg{)}\\ &\leq C\xi_{n}\Big{(}\xi_{n}-\frac{\gamma}{2}\Big{)}^{-1}\|F_{n}\|_{L^{\infty}_{2(\rho-1)}}r^{-\rho}\end{split}

to estimate the terms in (2.22)–(2.24) as

rρ​|Φn​[fn]​(r)|+|n|ξn​|cn​[fn]|+|n|ξn​rρ​|ωn​(r)|≤C​ξn​(ξn−γ2)−1​‖Fn‖L2​(ρ−1)∞.\displaystyle\begin{split}r^{\rho}|\Phi_{n}[f_{n}](r)|+\frac{|n|}{\xi_{n}}|c_{n}[f_{n}]|+\frac{|n|}{\xi_{n}}r^{\rho}|\omega_{n}(r)|\leq C\xi_{n}\Big{(}\xi_{n}-\frac{\gamma}{2}\Big{)}^{-1}\|F_{n}\|_{L^{\infty}_{2(\rho-1)}}.\end{split}

Hence the desired estimate (2.19) is obtained by (2.25) and easy computation using

|vn​(r,θ)|+r|n|​|∇vn​(r,θ)|≤C|n|​ξn2​(ξn−γ2)−1​‖F‖L2​(ρ−1)∞×(r−|n|−1​∫1rs|n|−ρ+1​ds+r|n|−1​∫r∞s−|n|−ρ+1​ds).\displaystyle\begin{split}&|v_{n}(r,\theta)|+\frac{r}{|n|}|\nabla v_{n}(r,\theta)|\\ &\leq\frac{C}{|n|}\xi_{n}^{2}\Big{(}\xi_{n}-\frac{\gamma}{2}\Big{)}^{-1}\|F\|_{L^{\infty}_{2(\rho-1)}}\\ &\quad\times\bigg{(}r^{-|n|-1}\int_{1}^{r}s^{|n|-\rho+1}\,{\rm d}s+r^{|n|-1}\int_{r}^{\infty}s^{-|n|-\rho+1}\,{\rm d}s\bigg{)}.\end{split}

It remains to show that vnv_{n} defined in (2.25) is a weak solution of (LPh) for general Fn∈𝒫n​L2​(ρ−1)∞​(Ω)2×2F_{n}\in{\mathcal{P}}_{n}L^{\infty}_{2(\rho-1)}(\Omega)^{2\times 2}. However, the proof by a density argument using the estimates

r−ξn−γ2​∫1rsξn+γ2−1​|H​(s)|​ds\displaystyle r^{-\xi_{n}-\frac{\gamma}{2}}\int_{1}^{r}s^{\xi_{n}+\frac{\gamma}{2}-1}|H(s)|\,{\rm d}s ≤(∫1∞|H​(s)|​s​ds)​r−2,\displaystyle\leq\bigg{(}\int_{1}^{\infty}|H(s)|s\,{\rm d}s\bigg{)}r^{-2},
rξn−γ2​∫r∞s−ξn+γ2−1​|H​(s)|​ds\displaystyle r^{\xi_{n}-\frac{\gamma}{2}}\int_{r}^{\infty}s^{-\xi_{n}+\frac{\gamma}{2}-1}|H(s)|\,{\rm d}s ≤(∫1∞|H​(s)|​s​ds)​r−2\displaystyle\leq\bigg{(}\int_{1}^{\infty}|H(s)|s\,{\rm d}s\bigg{)}r^{-2}

is similar to the one of Proposition 2.1. Thus we omit the details. The proof is complete. ∎

2.3 Vertical Component

Let n∈ℤn\in\mathbb{Z}. Applying 𝒫n{\mathcal{P}}_{n} to v3v_{3} and f3f_{3}

(𝒫n​v3)​(θ,r)=v3,n​(r)​ei​n​θ,(𝒫n​f3)​(θ,r)=f3,n​(r)​ei​n​θ\displaystyle({\mathcal{P}}_{n}v_{3})(\theta,r)=v_{3,n}(r)e^{in\theta},\qquad({\mathcal{P}}_{n}f_{3})(\theta,r)=f_{3,n}(r)e^{in\theta}

and inserting these to (LPv), we see that v3,n​(r)v_{3,n}(r) satisfies

−d2​v3,nd​r2−1+γr​d​v3,nd​r+n2+i​α​nr2​v3,n=f3,n,r>1\displaystyle-\frac{\,{\rm d}^{2}v_{3,n}}{\,{\rm d}r^{2}}-\frac{1+\gamma}{r}\frac{\,{\rm d}v_{3,n}}{\,{\rm d}r}+\frac{n^{2}+i\alpha n}{r^{2}}v_{3,n}=f_{3,n},\quad r>1 (2.26)

and the boundary conditions

v3,n​(1)=0,|v3,n​(r)|→0,r→∞.\displaystyle v_{3,n}(1)=0,\qquad|v_{3,n}(r)|\to 0,\quad r\to\infty. (2.27)

2.3.1 Axisymmetric Part

Proposition 2.3

Let α∈ℝ\alpha\in\mathbb{R}, γ>2\gamma>2 and 2<ρ≤γ2<\rho\leq\gamma.

  1. (1)

    Suppose that f3=f3,0∈𝒫0​L2​ρ−1∞​(Ω)f_{3}=f_{3,0}\in{\mathcal{P}}_{0}L^{\infty}_{2\rho-1}(\Omega). Then there is a unique weak solution v3,0∈𝒫0​W01,2​(Ω)v_{3,0}\in{\mathcal{P}}_{0}W^{1,2}_{0}(\Omega) of (LPv) satisfying

    ‖v3,0‖Lρ−1∞+1γ​‖∇hv3,0‖Lρ∞≤Cγ​(ρ−2)​‖f3,0‖L2​ρ−1∞.\displaystyle\|v_{3,0}\|_{L^{\infty}_{\rho-1}}+\frac{1}{\gamma}\|\nabla_{\rm h}v_{3,0}\|_{L^{\infty}_{\rho}}\leq\frac{C}{\gamma(\rho-2)}\|f_{3,0}\|_{L^{\infty}_{2\rho-1}}. (2.28)
  2. (2)

    Suppose that f3=f3,0f_{3}=f_{3,0} is given by f3,0=divh⁡F0f_{3,0}=\operatorname{div}_{\rm h}F_{0} for some F∈𝒫0​L2​(ρ−1)∞​(Ω)2F\in{\mathcal{P}}_{0}L^{\infty}_{2(\rho-1)}(\Omega)^{2}. Then there is a unique weak solution v3,0∈𝒫0​W01,2​(Ω)v_{3,0}\in{\mathcal{P}}_{0}W^{1,2}_{0}(\Omega) of (LPv) satisfying

    ‖v3,0‖Lρ−1∞+1γ​‖∇hv3,0‖Lρ∞≤Cρ−2​‖F0‖L2​(ρ−1)∞.\displaystyle\|v_{3,0}\|_{L^{\infty}_{\rho-1}}+\frac{1}{\gamma}\|\nabla_{\rm h}v_{3,0}\|_{L^{\infty}_{\rho}}\leq\frac{C}{\rho-2}\|F_{0}\|_{L^{\infty}_{2(\rho-1)}}. (2.29)

The constant CC is independent of α\alpha, γ\gamma and ρ\rho.

Proof.

We may only consider the existence and estimates of solutions.

(1) Putting n=0n=0 in (2.26)–(2.27), we see that v3,0​(r)v_{3,0}(r) solves

−d2​v3,0d​r2−1+γr​d​v3,0d​r=f3,0,r>1,v3,0​(1)=0.\displaystyle-\frac{\,{\rm d}^{2}v_{3,0}}{\,{\rm d}r^{2}}-\frac{1+\gamma}{r}\frac{\,{\rm d}v_{3,0}}{\,{\rm d}r}=f_{3,0},\quad r>1,\qquad v_{3,0}(1)=0. (2.30)

The linearly independent solutions of the homogeneous equation of (2.30) are

r−γand1,\displaystyle r^{-\gamma}\quad{\rm and}\quad 1,

and their Wronskian is γ​r−γ−1\gamma r^{-\gamma-1}. Hence the solution of (2.30) vanishing at infinity is

v3,0(r)=1γ{−(∫1∞s​f3,0​(s)​ds)​r−γ+r−γ∫1rsγ+1f3,0(s)ds+∫r∞sf3,0(s)ds}.\displaystyle\begin{split}v_{3,0}(r)=\frac{1}{\gamma}\bigg{\{}&-\bigg{(}\int_{1}^{\infty}sf_{3,0}(s)\,{\rm d}s\bigg{)}r^{-\gamma}\\ &+r^{-\gamma}\int_{1}^{r}s^{\gamma+1}f_{3,0}(s)\,{\rm d}s+\int_{r}^{\infty}sf_{3,0}(s)\,{\rm d}s\bigg{\}}.\end{split} (2.31)

Thus the estimate (2.28) follows from

r−γ​∫1rsγ−2​ρ+2​ds+∫r∞s−2​ρ+2​ds≤Cρ−2​r−ρ+1.\displaystyle\begin{split}r^{-\gamma}\int_{1}^{r}s^{\gamma-2\rho+2}\,{\rm d}s+\int_{r}^{\infty}s^{-2\rho+2}\,{\rm d}s\leq\frac{C}{\rho-2}r^{-\rho+1}.\end{split} (2.32)

One easily checks that v3,0​(r)v_{3,0}(r) defined in (2.31) is a weak solution of (LPv).

(2) At first we assume that F0∈𝒫0​C0∞​(Ω)2F_{0}\in{\mathcal{P}}_{0}C^{\infty}_{0}(\Omega)^{2}, which gives

f3,0=divh⁡F0=1r​dd​r​(r​Fr,0).\displaystyle f_{3,0}=\operatorname{div}_{\rm h}F_{0}=\frac{1}{r}\frac{\,{\rm d}}{\,{\rm d}r}(rF_{r,0}).

By integration by parts, we rewrite the formula (2.31) as

v3,0​(r)=−r−γ​∫1rsγ​Fr,0​(s)​ds.\begin{split}v_{3,0}(r)=-r^{-\gamma}\int_{1}^{r}s^{\gamma}F_{r,0}(s)\,{\rm d}s.\end{split} (2.33)

Next we let F0∈𝒫0​L2​(ρ−1)∞​(Ω)2F_{0}\in{\mathcal{P}}_{0}L^{\infty}_{2(\rho-1)}(\Omega)^{2}. The estimate (2.29) follows from (2.32). One can verify that v3,0​(r)v_{3,0}(r) defined in (2.33) is a weak solution of (LPv) by a density argument similar to the one in the proof of Proposition 2.1. This completes the proof. ∎

2.3.2 Non-axisymmetric Part

Recall that ξn\xi_{n} is defined in (2.17).

Proposition 2.4

Let n≠0n\neq 0 and let α∈ℝ\alpha\in\mathbb{R}, γ>2\gamma>2 and 2<ρ<32<\rho<3 with ρ≤γ\rho\leq\gamma.

  1. (1)

    Suppose that f3=f3,n∈𝒫n​L2​ρ−1∞​(Ω)f_{3}=f_{3,n}\in{\mathcal{P}}_{n}L^{\infty}_{2\rho-1}(\Omega). Then there is a unique weak solution v3,n∈𝒫n​W01,2​(Ω)v_{3,n}\in{\mathcal{P}}_{n}W^{1,2}_{0}(\Omega) of (LPv) satisfying

    ‖v3,n‖Lρ−1∞+1ξn​‖∇hv3,n‖Lρ∞\displaystyle\|v_{3,n}\|_{L^{\infty}_{\rho-1}}+\frac{1}{\xi_{n}}\|\nabla_{\rm h}v_{3,n}\|_{L^{\infty}_{\rho}} ≤Cξn​(ξn−γ2)−1​‖f3,n‖L2​ρ−1∞.\displaystyle\leq\frac{C}{\xi_{n}}\Big{(}\xi_{n}-\frac{\gamma}{2}\Big{)}^{-1}\|f_{3,n}\|_{L^{\infty}_{2\rho-1}}. (2.34)
  2. (2)

    Suppose that f3=f3,nf_{3}=f_{3,n} is given by f3,n=divh⁡Fnf_{3,n}=\operatorname{div}_{\rm h}F_{n} for some F∈𝒫n​L2​(ρ−1)∞​(Ω)2F\in{\mathcal{P}}_{n}L^{\infty}_{2(\rho-1)}(\Omega)^{2}. Then there is a unique weak solution v3,n∈𝒫n​W01,2​(Ω)v_{3,n}\in{\mathcal{P}}_{n}W^{1,2}_{0}(\Omega) of (LPv) satisfying

    ‖v3,n‖Lρ−1∞+1ξn​‖∇hv3,n‖Lρ∞\displaystyle\|v_{3,n}\|_{L^{\infty}_{\rho-1}}+\frac{1}{\xi_{n}}\|\nabla_{\rm h}v_{3,n}\|_{L^{\infty}_{\rho}} ≤C​(ξn−γ2)−1​‖Fn‖L2​(ρ−1)∞.\displaystyle\leq C\Big{(}\xi_{n}-\frac{\gamma}{2}\Big{)}^{-1}\|F_{n}\|_{L^{\infty}_{2(\rho-1)}}. (2.35)

The constant CC is independent of nn, α\alpha, γ\gamma and ρ\rho.

Proof.

We may only consider the existence and estimates of solutions.

(1) Since v3,n​(r)v_{3,n}(r) satisfies (2.21) with (rot⁡fn)n(\operatorname{rot}f_{n})_{n} replaced by f3,nf_{3,n}, it is given by

v3,n​(r)=12​ζn{−(∫1∞s−ζn+γ2+1f3,n(s)ds)r−ζn−γ2+r−ζn−γ2∫1rsζn+γ2+1f3,n(s)ds+rζn−γ2∫r∞s−ζn+γ2+1f3,n(s)ds}.\displaystyle\begin{split}&v_{3,n}(r)\\ &=\frac{1}{2\zeta_{n}}\bigg{\{}-\bigg{(}\int_{1}^{\infty}s^{-\zeta_{n}+\frac{\gamma}{2}+1}f_{3,n}(s)\,{\rm d}s\bigg{)}r^{-\zeta_{n}-\frac{\gamma}{2}}\\ &\qquad\qquad+r^{-\zeta_{n}-\frac{\gamma}{2}}\int_{1}^{r}s^{\zeta_{n}+\frac{\gamma}{2}+1}f_{3,n}(s)\,{\rm d}s+r^{\zeta_{n}-\frac{\gamma}{2}}\int_{r}^{\infty}s^{-\zeta_{n}+\frac{\gamma}{2}+1}f_{3,n}(s)\,{\rm d}s\bigg{\}}.\end{split} (2.36)

Thus the estimate (2.34) follows from (2.17)–(2.18) and

r−ξn−γ2​∫1rsξn+γ2−2​ρ+2​ds+rξn−γ2​∫r∞s−ξn+γ2−2​ρ+2​ds≤C​(ξn−γ2)−1​r−ρ+1.\displaystyle\begin{split}&r^{-\xi_{n}-\frac{\gamma}{2}}\int_{1}^{r}s^{\xi_{n}+\frac{\gamma}{2}-2\rho+2}\,{\rm d}s+r^{\xi_{n}-\frac{\gamma}{2}}\int_{r}^{\infty}s^{-\xi_{n}+\frac{\gamma}{2}-2\rho+2}\,{\rm d}s\\ &\leq C\Big{(}\xi_{n}-\frac{\gamma}{2}\Big{)}^{-1}r^{-\rho+1}.\end{split} (2.37)

One easily checks that v3,n​(r)​ei​n​θv_{3,n}(r)e^{in\theta} with v3,n​(r)v_{3,n}(r) defined in (2.36) is a weak solution of (LPv).

(2) At first we assume that Fn∈𝒫n​C0∞​(Ω)2F_{n}\in{\mathcal{P}}_{n}C^{\infty}_{0}(\Omega)^{2}, which gives

f3,n=(divh⁡F)n=1r​dd​r​(r​Fr,n)+i​nr​Fθ,n.\displaystyle f_{3,n}=(\operatorname{div}_{\rm h}F)_{n}=\frac{1}{r}\frac{\,{\rm d}}{\,{\rm d}r}(rF_{r,n})+\frac{in}{r}F_{\theta,n}.

By integration by parts, we rewrite the formula (2.36) as

v3,n​(r)=12​ζn[−{∫1∞s−ζn+γ2((ζn−γ2)Fr,n(s)+inFθ,n(s))ds}r−ζn−γ2+r−ζn−γ2​∫1rsζn+γ2​(−(ζn+γ2)​Fr,n​(s)+i​n​Fθ,n​(s))​ds+rζn−γ2∫r∞s−ζn+γ2((ζn−γ2)Fr,n(s)+inFθ,n(s))ds].\begin{split}v_{3,n}(r)&=\frac{1}{2\zeta_{n}}\bigg{[}-\bigg{\{}\int_{1}^{\infty}s^{-\zeta_{n}+\frac{\gamma}{2}}\bigg{(}\Big{(}\zeta_{n}-\frac{\gamma}{2}\Big{)}F_{r,n}(s)+inF_{\theta,n}(s)\bigg{)}\,{\rm d}s\bigg{\}}r^{-\zeta_{n}-\frac{\gamma}{2}}\\ &\qquad\qquad\quad+r^{-\zeta_{n}-\frac{\gamma}{2}}\int_{1}^{r}s^{\zeta_{n}+\frac{\gamma}{2}}\bigg{(}-\Big{(}\zeta_{n}+\frac{\gamma}{2}\Big{)}F_{r,n}(s)+inF_{\theta,n}(s)\bigg{)}\,{\rm d}s\\ &\qquad\qquad\quad+r^{\zeta_{n}-\frac{\gamma}{2}}\int_{r}^{\infty}s^{-\zeta_{n}+\frac{\gamma}{2}}\bigg{(}\Big{(}\zeta_{n}-\frac{\gamma}{2}\Big{)}F_{r,n}(s)+inF_{\theta,n}(s)\bigg{)}\,{\rm d}s\bigg{]}.\end{split} (2.38)

Next we let Fn∈𝒫n​L2​(ρ−1)∞​(Ω)2F_{n}\in{\mathcal{P}}_{n}L^{\infty}_{2(\rho-1)}(\Omega)^{2}. The estimate (2.35) follows from (2.37). One can check that v3,n​(r)​ei​n​θv_{3,n}(r)e^{in\theta} with v3,n​(r)v_{3,n}(r) defined in (2.38) is a weak solution of (LPv) by a density argument similar to the one in the proof of Proposition 2.1. This completes the proof. ∎

3 Proof of Theorem 1.1

We prove Theorem 1.1 in this section. As the proof is similar to [7, Proof of Theorem 1.1], we give only the outline to avoid duplication. For ρ≥0\rho\geq 0, we define the Banach space

l1​(Lρ∞​(Ω))\displaystyle l^{1}\big{(}L^{\infty}_{\rho}(\Omega)\big{)} ={f=∑n∈ℤ𝒫n​f|‖f‖l1​Lρ∞:=∑n∈ℤ‖𝒫n​f‖Lρ∞<∞}.\displaystyle=\bigg{\{}f=\sum_{n\in\mathbb{Z}}\mathcal{P}_{n}f~{}\bigg{|}~{}\|f\|_{l^{1}L^{\infty}_{\rho}}:=\sum_{n\in\mathbb{Z}}\|\mathcal{P}_{n}f\|_{L^{\infty}_{\rho}}<\infty\bigg{\}}.

The following is a corollary to Propositions 2.1–2.4 and the results in [7] for 2D problems.

Corollary 3.1

Let α∈ℝ\alpha\in\mathbb{R}, γ>2\gamma>2 and 2<ρ<32<\rho<3 with ρ≤γ\rho\leq\gamma. Suppose that the external force ff is a distribution on Ω\Omega given by f=g+div⁡Ff=g+\operatorname{div}F where g∈l1​(L2​ρ−1∞​(Ω))3g\in l^{1}\big{(}L^{\infty}_{2\rho-1}(\Omega)\big{)}^{3} and F∈l1​(L2​(ρ−1)∞​(Ω))3×3F\in l^{1}\big{(}L^{\infty}_{2(\rho-1)}(\Omega)\big{)}^{3\times 3}. Then there is a unique weak solution v=(vh,v3)∈(Lσ2​(Ω)×L2​(Ω))∩W01,2​(Ω)3v=(v_{\rm h},v_{3})\in(L^{2}_{\sigma}(\Omega)\times L^{2}(\Omega))\cap W^{1,2}_{0}(\Omega)^{3} of (LPh)–(LPv) satisfying

‖v‖l1​Lρ−1∞+‖∇hv‖l1​Lρ∞≤λ​(‖g‖l1​L2​ρ−1∞+‖F‖l1​L2​(ρ−1)∞),\displaystyle\begin{split}\|v\|_{l^{1}L^{\infty}_{\rho-1}}+\|\nabla_{\rm h}v\|_{l^{1}L^{\infty}_{\rho}}\leq\lambda\big{(}\|g\|_{l^{1}L^{\infty}_{2\rho-1}}+\|F\|_{l^{1}L^{\infty}_{2(\rho-1)}}\big{)},\end{split} (3.1)

where λ=λ​(α,γ,ρ)\lambda=\lambda(\alpha,\gamma,\rho) is given by

λ=C0​γ2​(|α|12+γ)2(ρ−2)2​(3−ρ).\displaystyle\begin{split}\lambda=\frac{C_{0}\gamma^{2}(|\alpha|^{\frac{1}{2}}+\gamma)^{2}}{(\rho-2)^{2}(3-\rho)}.\end{split}

The constant C0C_{0} is independent of α\alpha, γ\gamma and ρ\rho.

Proof.

By [7, Lemma 4.1], for solutions of the 2D problem

{−Δh​wh+(Vh)⊥​rot2​D⁡wh+∇hs=ghin​Ωdivh⁡wh=0in​Ωwh=0on​∂Ωwh​(x)→0as​|x|→∞,\left\{\begin{array}[]{ll}-\Delta_{\rm h}w_{\rm h}+(V_{\rm h})^{\bot}\operatorname{rot}_{{\rm 2D}}w_{\rm h}+\nabla_{\rm h}s=g_{\rm h}&\mbox{in}\ \Omega\\ \operatorname{div}_{\rm h}w_{\rm h}=0&\mbox{in}\ \Omega\\ w_{\rm h}=0&\mbox{on}\ \partial\Omega\\ w_{\rm h}(x)\to 0&\mbox{as}\ |x|\to\infty,\end{array}\right.

there is a constant C0~\tilde{C_{0}} independent of α\alpha, γ\gamma and ρ\rho such that

‖wh‖l1​Lρ−1∞+‖∇hw‖l1​Lρ∞≤C0~​γ​(|α|12+γ)(ρ−2)2​(3−ρ)​‖gh‖l1​L2​ρ−1∞.\displaystyle\begin{split}\|w_{\rm h}\|_{l^{1}L^{\infty}_{\rho-1}}+\|\nabla_{\rm h}w\|_{l^{1}L^{\infty}_{\rho}}\leq\frac{\tilde{C_{0}}\gamma(|\alpha|^{\frac{1}{2}}+\gamma)}{(\rho-2)^{2}(3-\rho)}\|g_{\rm h}\|_{l^{1}L^{\infty}_{2\rho-1}}.\end{split} (3.2)

Hence the desired estimate (3.1) is a consequence of the estimates in Propositions 2.1–2.4 combined with (2.5), (2.17)–(2.18) and (3.2). In addition, the existence and uniqueness of solutions follow from Propositions 2.1–2.4 and [7, Lemma 4.1]. The proof is complete. ∎


Proof of Theorem 1.1: We consider the Banach space

𝒳ρ={w∈W01,2​(Ω)3∩l1​(Lρ−1∞​(Ω))3|wh∈Lσ2​(Ω),∇hw∈l1​(Lρ∞​(Ω))2×3}\displaystyle{\mathcal{X}}_{\rho}=\Big{\{}w\in W^{1,2}_{0}(\Omega)^{3}\cap l^{1}\big{(}L^{\infty}_{\rho-1}(\Omega)\big{)}^{3}~{}\Big{|}~{}w_{\rm h}\in L^{2}_{\sigma}(\Omega),\mkern 9.0mu\nabla_{\rm h}w\in l^{1}\big{(}L^{\infty}_{\rho}(\Omega)\big{)}^{2\times 3}\Big{\}}

equipped with the norm ‖w‖𝒳ρ:=‖w‖l1​Lρ−1∞+‖∇hw‖l1​Lρ∞\|w\|_{{\mathcal{X}}_{\rho}}:=\|w\|_{l^{1}L^{\infty}_{\rho-1}}+\|\nabla_{\rm h}w\|_{l^{1}L^{\infty}_{\rho}}. By Corollary 3.1, for any w∈𝒳ρw\in{\mathcal{X}}_{\rho}, there is a unique weak solution vwv_{w} to (LPh)–(LPv) with the external force

−w⋅∇w+f=g+div⁡(−w⊗w+F)-w\cdot\nabla w+f=g+\operatorname{div}(-w\otimes w+F)

and the solution vwv_{w} satisfies

‖vw‖l1​Lρ−1∞+‖∇hvw‖l1​Lρ∞≤λ​(‖g‖l1​L2​ρ−1∞+‖−w⊗w+F‖l1​L2​(ρ−1)∞)≤λ​(‖w‖𝒳ρ2+‖g‖l1​L2​ρ−1∞+‖F‖l1​L2​(ρ−1)∞).\displaystyle\begin{split}\|v_{w}\|_{l^{1}L^{\infty}_{\rho-1}}+\|\nabla_{\rm h}v_{w}\|_{l^{1}L^{\infty}_{\rho}}&\leq\lambda\big{(}\|g\|_{l^{1}L^{\infty}_{2\rho-1}}+\|-w\otimes w+F\|_{l^{1}L^{\infty}_{2(\rho-1)}}\big{)}\\ &\leq\lambda\big{(}\|w\|_{{\mathcal{X}}_{\rho}}^{2}+\|g\|_{l^{1}L^{\infty}_{2\rho-1}}+\|F\|_{l^{1}L^{\infty}_{2(\rho-1)}}\big{)}.\end{split}

We have used the Young inequality for sequences to estimate −w⊗w-w\otimes w in the last line. Hence the mapping 𝒳ρ∋w↦vw∈𝒳ρ{\mathcal{X}}_{\rho}\ni w\mapsto v_{w}\in{\mathcal{X}}_{\rho} is well-defined, and will be denoted by TT.

It is not hard to check that TT is a contraction on the closed subset

ℬρ​(δ)={w∈𝒳ρ|‖w‖𝒳ρ≤δ},δ>0\displaystyle\mathcal{B}_{\rho}(\delta)=\{w\in{\mathcal{X}}_{\rho}~{}|~{}\|w\|_{{\mathcal{X}}_{\rho}}\leq\delta\},\quad\delta>0

if g,F,δg,F,\delta are small enough depending on λ=λ​(α,γ,ρ)\lambda=\lambda(\alpha,\gamma,\rho). Thus the existence of a weak solution of (NP) in Introduction unique in ℬρ​(δ)\mathcal{B}_{\rho}(\delta) follows from the Banach fixed-point theorem.

Let us set u∈W^1,2​(Ω)3u\in\widehat{W}^{1,2}(\Omega)^{3} by u=V+vu=V+v. Then one can check that uu is a weak solution of (NS) with b=(α​x⟂−γ​x,0)b=(\alpha x^{\perp}-\gamma x,0) by using (1.6). Moreover, uu is unique in the set

{u|u=V+v,v∈ℬρ​(δ)}\displaystyle\{u~{}|~{}u=V+v,\mkern 9.0muv\in\mathcal{B}_{\rho}(\delta)\}

and satisfies the limit (1.3). This completes the proof of Theorem 1.1.  □\Box

Acknowledgements

MH is partially supported by JSPS KAKENHI Grant Number JP 20K14345.

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M. Higaki

Department of Mathematics, Graduate School of Science, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe 657-8501, Japan.

Email: [email protected]

R. Horiuchi

Department of Mathematics, Graduate School of Science, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe 657-8501, Japan.

Email: [email protected]