On stationary two-dimensional flows around a fast rotating disk

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On stationary two-dimensional flows around a fast rotating disk

Isabelle Gallagher, Mitsuo Higaki, Yasunori Maekawa

To cite this version:

Isabelle Gallagher, Mitsuo Higaki, Yasunori Maekawa. On stationary two-dimensional flows around a fast rotating disk. 2017. �hal-01592269�

On stationary two-dimensional flows around a fast rotating disk

Isabelle Gallagher^∗ Mitsuo Higaki^† Yasunori Maekawa^‡ September 23, 2017

AbstractWe study the two-dimensional stationary Navier-Stokes equations describing flows around a rotating disk. The existence of unique solutions is established for any rotating speed, and qualitative effects of a large rotation are described precisely by exhibiting a boundary layer structure and an axisymmetrization of the flow.

Keywords Navier-Stokes equations·two-dimensional exterior flows·flows around a ro- tating obstacle·boundary layer

Mathematics Subject Classification (2000) 35B35·35Q30·76D05·76D17

1 Introduction

Understanding the structure of flows generated by the motion of a rigid body is an important subject in fluid dynamics. The typical motions of a rigid body (called the obstacle below) are translation and rotation, and it is well known that the flow generated by an obstacle translating with constant speed possesses a wake structure behind it, and that this structure is well described by the Oseen approximation; see Galdi [9] for mathematical results on this topic. Flows around a rotating obstacle have been studied mathematically mainly in three space dimensions, while there are only few mathematical results for two dimensions.

Moreover, in the two dimensional case most known mathematical results are restricted to the case when the Reynolds number is sufficiently small. In general, the motion of the obstacle leads to a drastic change of the decay structure of the flow, and in the two dimensional case it yields a significant localizing effect that enables one to construct corresponding steady state solutions when the motion of the obstacle is slow enough. Although on the one hand a faster motion of the obstacle might give a stronger localizing and stabilizing effect, on the other hand fast motion produces a rapid flow and creates a strong shear near the boundary that can be a source of instability. As a result, rigorous analysis becomes quite difficult for the nonlinear problem in general. Hence it is useful to study the problem under a simple geometrical setting and to understand a typical fluid structure that describes these two competitive mechanisms; localizing and stabilizing effects on the one hand, and the presence of a rapid flow and the boundary layer created by the fast motion of the obstacle on the other hand.

∗IMJ-PRG, Universit´e Paris-Diderot and DMA, Ecole Normale Sup´erieure de Paris, France. E-mail:

gallagher@math.ens.fr

†Department of Mathematics, Kyoto University, Japan. E-mail:mhigaki@math.kyoto-u.ac.jp

‡Department of Mathematics, Kyoto University, Japan. E-mail:maekawa@math.kyoto-u.ac.jp

In this paper we study two dimensional flows around a rotating obstacle assuming that the obstacle is a unit disk centered at the origin, in the case when the rotation speed is sufficiently fast and the Reynolds number is high. Note that in a three dimensional setting, these flows are considered as a model for two dimensional flows around a rotating infinite cylinder with a uniform cross section which is a unit disk.

Let us formulate our problem mathematically. For notational convenience we first state the problem when the obstacle has a general shape. Let us consider the following Navier- Stokes equations for viscous incompressible flows in two dimensions:

(∂_tw−∆w+w· ∇w+∇φ = g , divw = 0, t >0, y∈Ω(t),

w = αy^⊥, t >0, y∈∂Ω(t). (1.1) Herew = w(y, t) = (w₁(y₁, y₂, t), w₂(y₁, y₂, t))^⊤ and φ = φ(y, t) are respectively the unknown velocity field and pressure field, andg(y, t) = (g₁(y, t), g₂(y, t))^⊤is an external force. The vectory^⊥is defined asy^⊥ = (−y₂, y₁)^⊤. HereM^⊤denotes the transpose of a matrixM. The domainΩ(t)is given by

Ω(t) :=

y∈R² |y=O(αt)x , x:= (x₁, x₂)^⊤∈Ω , O(αt) :=

cosαt −sinαt sinαt cosαt

, (1.2)

whereΩis an exterior domain inR² and its complement Ω^c := R²\Ωdescribes the ob- stacle at initial time, while the real numberαrepresents the rotation speed of the obstacle.

The conditionw(t, y) = αy^⊥ on the boundary∂Ω(t)represents the noslip boundary con- dition. Taking into account the rotation of the obstacle, we introduce the following change of variables and unknowns:

y=O(αt)x , u(x, t) =O(αt)^⊤w(y, t), p(x, t) =φ(y, t), f(x, t) =O(αt)^⊤g(y, t). Then (1.1) is equivalent to the equations in the time-independent domainΩ:

(∂_tu−∆u−α(x^⊥· ∇u−u^⊥) +∇p = −u· ∇u+f , divu = 0, t >0, x∈Ω, u = αx^⊥, t >0, x∈∂Ω.

We are interested in stationary solutions to this system when the obstacleΩ^c is a unit disk centered at the origin, and therefore we assume thatf is independent oftand consider the elliptic system







−∆u−α(x^⊥· ∇u−u^⊥) +∇p = −u· ∇u+f , x∈Ω, divu = 0, x∈Ω,

u = αx^⊥, x∈∂Ω,

(NSα)

withΩ := {x ∈ R² | |x| > 1}. We note that in the original coordinates the stationary solution to (NSα) gives a specific time periodic flow with a periodicity ^2π_|α|. Due to the symmetry of the domain there is an explicit stationary solution to (NSα) whenf = 0:

αU, α²∇P

with U(x) := x^⊥

|x|² , P(x) :=− 1

2|x|² · (1.3)

Thus it is natural to consider an expansion around this explicit solution. By using the iden- tityu· ∇u= ¹₂∇|u|²+u^⊥rotuwithrotu:= ∂1u2−∂2u1 androtU = 0forx 6= 0, the equations forv:=u−αUare written as







−∆v−α(x^⊥· ∇v−v^⊥) +∇q+αU^⊥rotv = −v^⊥rotv+f , x∈Ω, divv = 0, x∈Ω,

v = 0, x∈∂Ω.

(NSfα)

The goal of this paper is to show the existence and uniqueness of solutions to (NSfα) for arbitraryα ∈R\ {0}under a suitable condition on the given external forcef in terms of regularity and summability, and moreover, we shall give a detailed qualitative analysis for the fast rotation case|α| ≫1.

Before stating our results let us recall known mathematical results related to this prob- lem. Flows around a rotating obstacle have been studied mainly for the three dimensional case. Borchers [2] proved the existence of global weak solutions for the nonstationary problem, and the existence and uniqueness of local in time regular solutions is shown by Hishida [15] and Geissert, Heck, and Hieber [11]. Global strong solutions for small data are obtained by Galdi and Silvestre [10]. The spectrum of the linear operator related to this problem is studied by Hishida [16], and Farwig and Neustupa [7]. The existence of stationary solutions to the three dimensional problem is proved in [2], Galdi [8], Silvestre [20], and Farwig and Hishida [4]. In particular, [8] constructs stationary flows with a decay of orderO(|x|⁻¹), while [4] discusses the weak L³ framework. The asymptotic profiles of these stationary flows at spatial infinity are studied by Farwig and Hishida [5, 6] and Farwig, Galdi, and Kyed [3], where it is proved that the asymptotic profiles are described by the Landau solutions, which are stationary self-similar solutions to the Navier-Stokes equations inR³ \ {0}. The stability of small stationary solutions has been well studied in the three-dimensional case. Indeed, the global L² stability is proved in [10] and localL³ stability is obtained by Hishida and Shibata [18].

All the results mentioned above are for the three dimensional case. So far there have been only few results in the two dimensional case. Hishida [17] revealed the asymptotic be- havior of the stationary Stokes flow around a rotating obstacle, and showed that the rotation of the obstacle leads to the resolution of the Stokes paradox as in the case of the translation of the obstacle. On the other hand, the linear result in [17] was not sufficient to solve the nonlinear problem due to the nature of the singular perturbation limitα → 0which is spe- cific to the two dimensional case. Recently the linear result of [17] was extended by Higaki, Maekawa, and Nakahara [12], where the existence and uniqueness of solutions to (NSα) decaying at the scale-critical orderO(|x|⁻¹)was also proved when the rotation speedαis sufficiently small and the external forcef is of a divergence formf = divF for someF which is small in a scale critical norm. Moreover, the leading profile at spatial infinity was shown to be C^x⊥

|x|² for some constant C under an additional decay condition on F such asF = O(|x|^−2−r)withr > 0, which is compatible with the work of [17] for the Stokes case. The stability of these stationary solutions, though they are small in a scale-critical norm, is a difficult problem and is still largely open in the two dimensional case. The only known result is by Maekawa [19] for a specific case, which shows the localL² stability of the explicit solution (1.3) in the original frame (1.1) whenα is small enough andΩis the exterior to the unit disk as assumed in this paper. Few results are known in the case of nonsmallα. As a related work in this direction, Hillairet and Wittwer [14] established the existence of stationary solutions to (1.1) for large|α|whenΩ(t) ={y∈R²| |y|>1}as in

this paper, although in [14] the boundary condition isαy^⊥+binstead ofαy^⊥, with some time-independent given databand the external force f is assumed to be zero. In [14] the stationary solution is constructed around (1.3) when|α|is large andbis small. Our problem is in fact essentially different from the one discussed in [14]. Indeed, the stationary solution to (NSα) is a time periodic solution in the original frame (1.1), and therefore, the result in [14] is not applicable to our problem and vice versa. The reader is also referred to [13]

for a model problem of (NS_α) formulated in the whole planeR², where the existence and uniqueness of stationary solutions is proved whenα 6= 0. However, the argument in [13]

relies on the absence of physical boundary, and the key mechanism originating from the boundary is not analyzed.

Let us now return to (NSfα). The novelty of the results in this paper is the followings:

(1) Existence and uniqueness of solutions to (NSfα) for arbitraryα∈R\ {0}.

(2) Relaxed summability condition onf and on the class of solutions, allowing slow spatial decay with respect to scaling.

(3) Qualitative analysis of solutions in the fast rotation case|α| ≫1.

As for (1), the result is new compared with [12] in which the stationary solutions are ob- tained only for nonzero but small |α|, though there is no restriction on the shape of the obstacle in [12]. The reason why we can construct solutions for all nonzeroαis a remark- able coercive estimate for the term−α(x^⊥· ∇v−v^⊥) +αU^⊥rotv in polar coordinates;

see (1.18) below. As for (2), we note that the given dataf and the class of solutions in [12]

are in a scale critical space. A typical behavior forf assumed in [12] is that f = divF with F(x) = O(|x|⁻²), and then the solution v satisfies the estimate |v(x)| ≤ C|x|⁻¹ for|x| ≫1. In this paper the summability condition onf is much weaker than this scaling, see (1.6) below. Moreover, the radial part of the solution constructed in this paper only behaves likeo(1)as|x| → ∞in general, which is considerably slow, while the nonradial part of the solution belongs toL²(Ω)which is just in the scale critical regime. The point (3) is important both physically and mathematically. Understanding the fluid structure around the fast rotating obstacle up to the boundary is one of the main subjects of this paper, and we show the appearance of a boundary layer as well as an axisymmetrization mechanism due to the fast rotation of the obstacle.

Let us state our functional setting. Due to the symmetry of the domain it is natural to introduce the relevant function spaces in terms of polar coordinates. As usual, we set

x₁ = rcosθ , x₂ = rsinθ , r = |x| ≥1, θ ∈[0,2π), e_r = x

|x|, e_θ = x^⊥

|x| = ∂_θe_r, and

v = vre_r + vθe_θ, vr = v·e_r, vθ = v·e_θ.

Next, for eachn∈ Z, we denote byPnthe projection on the Fourier modenwith respect to the angular variableθ:

Pnv := v_r,ne^inθe_r+v_θ,ne^inθe_θ, (1.4)

where

vr,n(r) := 1 2π

Z 2π

0

vr(rcosθ, rsinθ)e^−inθdθ , vθ,n(r) := 1

2π Z _2π

0

vθ(rcosθ, rsinθ)e^−inθdθ . We also set form∈N∪ {0},

Q^mv:=

X∞

|n|=m+1

Pⁿv . (1.5)

For notational convenience we will often writevnforPⁿv. EachPⁿis an orthogonal pro- jection inL²(Ω)², and the spaceL²_σ(Ω) := {f ∈C₀^∞(Ω)²|divf = 0}^L2(Ω)2 is invariant under the action ofPⁿ. Note thatv0 := P⁰v is the radial part ofv, and thus, Q⁰v is the nonradial part of v. We will set PⁿL²(Ω)² := {f ∈ L²(Ω)² | f = Pⁿf}, and similar notation will be used forL²_σ(Ω)andQ0. A vector fieldf inΩis formally identified with the pair(P0f,Q0f). Then, for the class of external forces we introduce the product space

Y :=P0L¹(Ω)²× Q0L²(Ω)². (1.6) For the class of solutions we set

X:=P⁰W₀^1,∞(Ω)²× Q⁰W₀^1,2(Ω)². (1.7) HereW₀^1,r(Ω) :={f ∈W^1,r(Ω)|f = 0on∂Ω}for1< r≤ ∞. Our first result is stated as follows.

Theorem 1.1 There existsγ >0such that the following statements hold.

(i)Let0<|α|<1. Then for any external forcef = (P0f,Q0f)∈Y satisfying

k(P0f)_θkL¹(Ω)≤γ|α|, kQ0fkL²(Ω)≤γ|α|², (1.8) there exists a solutionv∈X∩L^∞(Ω)²∩W_loc^2,1(Ω)²to(NSfα)with a suitable pressureq ∈ W_loc^1,1(Ω), andvsatisfies

kP0vkL^∞(Ω)+∇P0v

L^∞(Ω) ≤Ck(P0f)θkL¹(Ω)+ C

|α|³²kQ0fk²L²(Ω), (1.9) kQ0vkL²(Ω) ≤ C

|α|kQ0fkL²(Ω), (1.10)

X

|n|≥1

kPⁿvkL^∞(Ω) ≤ C

|α|³⁴kQ0fkL²(Ω), (1.11) k∇Q⁰vkL²(Ω) ≤ C

|α|¹²kQ⁰fkL²(Ω). (1.12) This solution is unique in a suitable closed convex set in X (see Subsection 4.2 for the precise description).

(ii)Let|α| ≥1. Then for any external forcef = (P0f,Q0f)∈Y satisfying

k(P0f)_θkL¹(Ω) ≤γ , kQ0fkL²(Ω) ≤γ , (1.13)

there exists a solution v ∈ X ∩L^∞(Ω)² ∩W_loc^2,1(Ω)² to(NSfα) with a suitable pressure q∈W_loc^1,1(Ω), andvsatisfies

kP0vkL^∞(Ω)+∇P0v

L^∞(Ω) ≤Ck(P0f)θkL¹(Ω)+ C

|α|¹²kQ0fk²L²(Ω), (1.14) kQ0vkL²(Ω) ≤ C

|α|¹²kQ0fkL²(Ω), (1.15) X

|n|≥1

kPⁿvkL^∞(Ω) ≤ C

|α|¹⁴kQ0fkL²(Ω), (1.16) k∇Q⁰vkL²(Ω) ≤CkQ⁰fkL²(Ω). (1.17) This solution is unique in a suitable subset ofX.

Note that the summability of f assumed in Theorem 1.1 is much weaker than the scale- critical one. For the radial partP0vwe can show lim_|x|→∞|P0v(x)| = 0 but there is no rate in general under the assumptions of Theorem 1.1. Theorem 1.1 already exhibits the axisymmetrizing effect of the fast rotating obstacle inL² and L^∞, which will be further extended in Theorems 1.2 and 1.3 below. The proof of Theorem 1.1 consists in two ingre- dients: the analysis of the linearized problem (Sα) (defined and studied in Section 3), and the estimate of the interaction between the radial part and the nonradial part in the nonlinear problem (see Section 4). The linear result used in Theorem 1.1 is stated in Proposition 3.1, and the proof is based on an energy method. Although the proof of the linear result is not so difficult, there is a key observation for the term−α(x^⊥· ∇v−v^⊥) +αU^⊥rotv. Indeed, for the linearized problem (Sα) the energy computation forvn=Pⁿvwithn6= 0gives the key identity

αn

kvr,nk²L²(Ω)−(1− 2 n²)vr,n

r ²_L2

(Ω)+kvθ,nk²L²(Ω)−vθ,n

r ²_L2

(Ω)

=−ℑhfn, vniL²(Ω).

(1.18)

HerefndenotesPⁿf and the norm kgkL²(Ω)for the functiong : [1,∞) → Cis defined as(2π)¹²kgkL²((1,∞);rdr). The key point here is that the bracket in (1.18) is nonnegative and provides a bound fork

√|x|²−1

|x| vnk²_L²_(Ω)sinceΩ ={|x|>1}. Then by combining with an interpolation inequality of the form

kgkL²(Ω)≤Ck∂rgk

1 3

L²(Ω)

√r²−1 r g²³

L²(Ω)+C

√r²−1 r g

L²(Ω) (1.19) for any scalar function g ∈ W^1,2((1,∞);rdr)and the dissipation from the Laplacian in the energy computation, we can close the energy estimate for allα6= 0. The proof of (1.19) is given in Appendix B. In solving the nonlinear problem the key observation is that the product of the radial parts in the nonlinear term can always be written in a gradient form and thus regarded as a pressure term, which yields the identity

v^⊥rotv=v^⊥₀rotQ0v+ (Q0v)^⊥rotv₀+ (Q0v)^⊥rotQ0v+∇q˜ (1.20) for a suitableq. Since˜ P0 v^⊥₀rotQ0v+(Q0v)^⊥rotv₀

= 0as long asQ0v∈W₀^1,2(Ω)²the radial part of the velocity in the right-hand side of (1.20) (neglecting∇q) belongs to˜ L¹(Ω)²,

which is the same summability as the spaceY. This is a brief explanation for the reason why we can close the nonlinear estimate and solve (NSfα) inXfor a sourcef ∈Y.

Our second result is focused on the fast rotation case|α| ≫ 1. In this regime there are three fundamental mechanisms in our system:

(I) an axisymmetrization due to the fast rotation of the obstacle,

(II) the presence of a boundary layer for the nonradial part of the flow due to the noslip boundary condition,

(III) the diffusion in high angular frequencies due to the viscosity.

(I) and (II) are potentially in a competitive relation, for the noslip boundary condition and the boundary layer can suppress the effect of the fast rotation to some extent.

(II) and (III) are also competitive. Indeed, it is natural that if the viscosity is strong enough then the boundary layer is diffused and is no longer observable. The important task here is to determine the regime of angular frequencies in which the boundary layer appears, and to estimate the thickness of the boundary layer. We show that the bound- ary layer appears in the regime 1 ≤ |n| ≪ O(|α|¹²), and the thickness of the bound- ary layer is(2|αn|)⁻¹³ for each nin this regime. In constructing the boundary layer the termαU^⊥rotv plays a crucial role as well as the term−α(x^⊥· ∇v−v^⊥). In fact, if we drop the termαU^⊥rotvas a model problem then the thickness of the boundary layer aris- ing from the rotation term−α(x^⊥· ∇v−v^⊥) is|αn|⁻¹², and the leading boundary layer profile is simply described by exponential functions. The termαU^⊥rotvleads to a signif- icant change both in the thickness and in the profile of the boundary layer, and we need to introduce the Airy function to describe the profile of the boundary layer associated with the term−α(x^⊥· ∇v−v^⊥) +αU^⊥rotv.

By performing the boundary layer analysis we can improve the result stated in (ii) of Theorem 1.1 in the regime|α| ≫1, which is briefly described as follows.

Theorem 1.2 There existsγ > 0 such that the following statement holds. For all suffi- ciently large|α| ≥1and for any external forcef = (P⁰f,Q⁰f)∈Y satisfying

k(P0f)θkL¹(Ω)≤γ|α|¹³ , kQ0fkL²(Ω) ≤γ|α|¹³ , (1.21) there exists a solution v ∈ X ∩L^∞(Ω)² ∩W_loc^2,1(Ω)² to(NSfα) with a suitable pressure q∈W_loc^1,1(Ω), andvsatisfies

kP0vkL^∞(Ω)+k∇P0vkL^∞(Ω) ≤Ck(P0f)θkL¹(Ω)+ C

|α|kQ0fk²L²(Ω), (1.22) kQ0vkL²(Ω) ≤ C

|α|²³kQ0fkL²(Ω), (1.23) X

|n|≥1

kPⁿvkL^∞(Ω) ≤ C log|α|¹₂

|α|¹² kQ0fkL²(Ω), (1.24) k∇Q0vkL²(Ω) ≤ C

|α|¹³kQ0fkL²(Ω). (1.25) This solution is unique in a suitable subset ofX.

By fixing the external forcef we can state Theorem 1.2 in a different but more convenient way to understand the qualitative behavior of solutions in the fast rotation limit.

Theorem 1.3 For anyf = (P0f,Q0f) ∈ Y there isα₀ = α₀(kfk^Y) ≥ 1such that the following statements hold. If|α| ≥α₀then there exists a solutionv =v^(α) ∈X∩W_loc^2,1(Ω)² to(NSf_α)with a suitable pressureq^(α) ∈W_loc^1,1(Ω), andv^(α)satisfies

kv^(α)−v^linear₀ kL^∞(Ω)≤ C(log|α|)¹²

|α|¹² · (1.26)

Herev₀^linearis the solution to the linearized problem(Sα)defined in page11withfreplaced byP0f which is in fact independent ofα, andC depends only onkfkY. Moreover, there existsκ >0independent of αandf such that if1 ≤ |n| ≤ κ|α|¹² thenvn^(α) =Pⁿv^(α) is written in the form

v_n^(α) =v_n^(α),slip+v_n^(α),slow+v^(α)_n,BL+ev^(α)_n . (1.27) Here v^(α),slipn satisfies vn,r^(α),slip = rotv^(α),slipn = 0 on ∂Ω, v^(α),slown is irrotational in Ω, andv^(α)_n,BLpossesses a boundary layer structure with the boundary layer thickness|2αn|⁻¹³. Finally the following estimates hold:

kv_n^(α),slipkL²(Ω)+kv^(α),slow_n kL²(Ω)+kv^(α)_n,BLkL²(Ω)≤ C

|αn|²³ , whileevn^(α)is a remainder which satisfies

kev_n^(α)kL²(Ω)≤ C

|αn|· Here the constantCdepends only onkfk^Y.

By going back to (NS_α), Theorems 1.2 and 1.3 show that there exists a unique solu- tionu=u^(α)which satisfies

ku^(α)−αU−v₀^linearkL^∞(Ω)≤ C(log|α|)¹²

|α|¹² , |α| ≫1. (1.28) The expansion (1.28) verifies the axisymmetrizing effect (measured inL^∞) due to the fast rotation. The logarithmic factor(log|α|)¹² is simply due to the regularity off, and iff has more regularity such asP

n6=0kPⁿfk^s_L²_(Ω)<∞for somes <2then the factor(log|α|)¹² in (1.26) and (1.28) can be dropped. Moreover, the power|α|⁻¹² can be also improved by assuming enough regularity off. For example, ifQ0f ∈W₀^1,2(Ω)²in addition, then|α|⁻¹² is replaced by|α|⁻³⁴, though we do not go into the detail on this point. The new ingredient of the proof of Theorems 1.2 and 1.3 is stated in Proposition 3.2 and consists in refined estimates for the linearized problem (S_α). The nonlinear problem is handled exactly in the same manner as in the proof of Theorem 1.1. For (Sα) we observe that in polar coordinates the angular modenof the streamfunction satisfies the ODE inr∈(1,∞)

d² dr² +1

r d dr −n²

r² +iαn 1− 1 r²

d² dr² +1

r d dr −n²

r²

ψn= 0, (1.29)

with the boundary condition ψn(1) = ^dψ_drⁿ(1) = 0when |n| ≥ 1. The thickness of the boundary layer originating from the fast rotation is determined by the balance between _dr^d22

andiαn(1−_r¹²)≈2iαn(r−1)nearr= 1as long as the dissipation−ⁿ_r²² ≈ −n²is moder- ate. This implies that the thickness is|2αn|⁻¹³. Then the regime ofnwhere the dissipation is relatively moderate is estimated from the conditionn²≪ _dr^d22 ≈O(|αn|²³), which leads to|n| ≪O(|α|¹²). From this observation we employ the boundary layer analysis when the angular frequencynsatisfies1 ≤ |n| ≪ O(|α|¹²), while we just apply Proposition 3.1 in the regime |n| ≥ O(|α|¹²)where the boundary layer due to the fast rotation is no longer present. In the regime1 ≤ |n| ≪O(|α|¹²)we first consider (Sα) withf =Pⁿf but under the slip boundary conditionv_r,n = rotv_n = 0on ∂Ω. The estimate of thisslip solution is obtained by the energy method for the vorticity equations thanks to the boundary con- ditionrotvn = 0on ∂Ω. The key point is that the term U^⊥rot in the velocity equation becomesU · ∇in the vorticity equation which is antisymmetric because ofdivU = 0, and thus, it is easy to apply the energy method for the vorticity under the slip boundary condi- tion. Thenoslipsolution is then obtained by correcting the boundary condition. To this end we construct the boundary layer solution called the fast mode. The leading profile of the boundary layer is given by a suitable integral of the Airy function. In order to recover the noslip boundary condition, the fact that

Z _∞

0

Ai(s) ds 6= 0 is crucial and it plays the role of a nondegeneracy condition in our construction of the solution. This construction gives a formula as in (1.27) for the solution to (Sα). Compared with Proposition 3.1, which is based only on an energy computation for the velocity field, the estimate of thenmode v_n is drastically improved for1 ≤ |n| ≪O(|α|¹²)thanks to the boundary layer analysis. On the other hand, in the regime|n| ≥O(|α|¹²), Proposition 3.1 for thenoslipsolution already gives the same decay estimates as in Proposition 3.4 for theslipsolution, as expected.

This paper is organized as follows. In Section 2 we recall some basic facts on operators and vector fields in polar coordinates. In Section 3 the linearized problem (S_α) is studied.

This section is the core of the paper. In Subsection 3.1 we prove the linear estimates which are valid for allα ∈R\ {0}. These are summarized in Proposition 3.1. Subsection 3.2 is devoted to the linear analysis for the case|α| ≫ 1, and the main result of this section is Proposition 3.2. The nonlinear problem is discussed in Section 4. Some basics on the Airy function and the proof of the interpolation inequality (1.19) are given in the appendix.

2 Preliminaries

In this preliminary section we state basic results on some differential operators and vector fields in polar coordinates.

2.1 Operators in polar coordinates

The following formulas will be used frequently:

divv = ∂₁v₁+∂₂v₂ = 1

r∂r(rvr) +1

r∂θvθ, (2.1)

rotv = ∂1v2−∂2v1 = 1

r∂r(rvθ)−1

r∂θvr, (2.2)

|∇v|² = |∂_rv_r|²+|∂_rv_θ|²+ 1

r² |∂_θv_r−v_θ|²+|v_r+∂_θv_θ|²

, (2.3)

and

−∆v =

−∂r

1

r∂r(rvr)

− 1

r²∂_θ²vr+ 2 r²∂θvθ

e_r

+

−∂r

1

r∂r(rvθ)

− 1

r²∂_θ²vθ− 2 r²∂θvr

e_θ,

(2.4)

e_r· ∇v = (∂rvr)e_r + (∂rvθ)e_θ, e_θ· ∇v = ∂_θv_r−v_θ

r e_r + ∂_θv_θ+v_r r e_θ. In particular, we have

x^⊥· ∇v−v^⊥ =|x| e_θ· ∇v

− vre⊥

r +vθe⊥ θ

= (∂θvr−vθ)e_r + (∂θvθ+vr)e_θ− vre^⊥_r +vθe^⊥

θ

=∂θvre_r+∂θvθe_θ. (2.5)

From (2.3) and the definition of Pⁿ in (1.4) it follows that for n ∈ N∪ {0} and forv inW^1,2(Ω)²,

k∇vk²L²(Ω)=X

n∈Z

k∇Pⁿvk²L²(Ω),

|∇Pⁿv|² =|∂rvr,n|²+1 +n² r² |vr,n|² +|∂rvθ,n|²+1 +n²

r² |vθ,n|²−4n

r²ℑ(vθ,nvr,n). In particular, we have

|∇Pⁿv|² ≥ |∂rvr,n|²+(|n| −1)²

r² |vr,n|²+|∂rvθ,n|²+(|n| −1)²

r² |vθ,n|², (2.6) and thus, from the definition ofQmin (1.5),

k∇Qmvk²L²(Ω)≥ k∂_r(Qmv)_rk²L²(Ω)+k∂_r(Qmv)_θk²L²(Ω)+m²k v

|x|k²L²(Ω).

2.2 The Biot-Savart law in polar coordinates

For a given scalar fieldωinΩ, the streamfunction ψis formally defined as the solution to the Poisson equation: −∆ψ=ωinΩ, ψ= 0on∂Ω. Forn∈Zandω ∈L²(Ω)we set

Pⁿω:= 1 2π

Z 2π

0

ω(rcoss, rsins)e^−insds e^inθ, ωn:= Pⁿω

e^−inθ.

(2.7) By using the Laplace operator in polar coordinates, the Poisson equation for the Fourier modenis given by

Hnψn := −ψ^′′_n−1

rψ^′_n+n²

r²ψn=ωn, r >1, ψn(1) = 0. (2.8) Let|n| ≥ 1. Then the solution ψn = ψn[ωn]to the ordinary differential equation (2.8) decaying at spatial infinity is formally given as

ψn[ωn](r) = 1 2|n|

−dn[ωn] r^|n| + 1

r^|n| Z r

1

s^1+|n|ωn(s) ds+r^|n| Z _∞

r

s^1−|n|ωn(s) ds

, dn[ωn] :=

Z _∞

1

s^1−|n|ωn(s) ds .

The Biot-Savart lawVn[ωn]is then written as

Vn[ωn] := Vr,n[ωn]e^inθe_r + Vθ,n[ωn]e^inθe_θ, Vr,n[ωn] := in

r ψn[ωn], Vθ,n[ωn] := − d

drψn[ωn]. (2.9) The velocityVn[ωn]is well defined at least whenr^1−|n|ωn∈L¹((1,∞)), and it is straight- forward to see that

divV_n[ω_n] = 0, rotV_n[ω_n] = ω_ne^inθ in Ω,

e_r·V_n[ω_n] = 0 on∂Ω. (2.10)

The conditionr^1−|n|ωn∈L¹((1,∞))is automatically satisfied whenω ∈L²(Ω)and|n| ≥ 2. When|n|= 1the integral in the definition ofψn[ωn]does not always converge absolutely for generalω ∈ L²(Ω). However, it is well-defined ifω = rotufor someu ∈ W^1,2(Ω)², for one can apply the integration by parts that ensures the convergence of lim

N→∞

Z N

r

ω_ndr even when |n| = 1. As a result, we can check that for any solenoidal vector field v in L²_σ(Ω)∩W^1,2(Ω)², the n mode v_n = Pnv is expressed in terms of its vorticity ω_n by the formula (2.9) when|n| ≥1.

3 Analysis of the linearized system

The linearized system aroundαUfor (NSfα) is







−∆v−α(x^⊥· ∇v−v^⊥) +∇q+αU^⊥rotv = f , x∈Ω, divv = 0, x∈Ω, v = 0, x∈∂Ω.

(Sα)

In this section we study (Sα) forα∈R\ {0}.

3.1 General estimate

In this subsection we establish estimates on solutions to (Sα) that are valid for allα 6= 0.

For convenience we set for scalar functionsg, h: [1,∞)→C, hg, hiL²(Ω):= 2π

Z _∞

1

g(r)h(r)rdr , kgk²L²(Ω):= 2π Z _∞

1 |g(r)|²rdr . Before going into details let us give a remark on the verification of the energy argument.

Let us assume thatf ∈ L²(Ω)² and v ∈ W₀^1,2(Ω)²∩W_loc^2,2(Ω)² is a solution to (Sα) for someq ∈ W_loc^1,2(Ω). We have to be careful when applying the energy argument due to the presence of the termx^⊥· ∇vin the first equation of (Sα), for this term involves a linearly growing coefficient, and therefore it is not clear whether the inner producthx^⊥·∇v, viL²(Ω)

makes sense or not. A similar difficulty appears in taking the inner producth∇q, viL²(Ω), since we are assuming only q ∈ W_loc^1,2(Ω). The most convenient way to overcome this difficulty is to consider the equation forvn := Pⁿv, which is identified with(vr,n, vθ,n).

Note thatvr,n, v_θ,n ∈ W₀^1,2((1,∞);rdr)∩W_loc^2,2([1,∞))ifv ∈ W₀^1,2(Ω)²∩W_loc^2,2(Ω)². Let us denote by ω_n = ω_n(r) then mode of the vorticity of v in the polar coordinates, i.e., ωn(r) = (rotPⁿv)e^−inθ. Similarly, we set qn = qn(r) = (Pⁿq)e^−inθ, where the projectionPⁿfor the scalarqis defined as

Pⁿq:= 1 2π

Z 2π

0

q(rcoss, rsins)e^−insds e^inθ. Then, from (2.4) and (2.5),vr,nandv_θ,nobey the following equations:

−∂r

1

r∂r(rvr,n) +n²

r²vr,n+2in

r² vθ,n−iαnvr,n−αω_n

r +∂rqn=fr,n, (3.1)

−∂_r 1

r∂_r(rv_θ,n) +n²

r²v_θ,n−2in

r² v_r,n−iαnv_θ,n+inq_n=f_θ,n, (3.2) together with the divergence free condition∂r(rvr,n) +invθ,n= 0and the noslip boundary conditionvr,n(1) = vθ,n(1) = 0. Then the key observation is that the factor−iαnis now regarded as a resolvent parameter, and by settingλ:=−iαn, the above system is equivalent to

(λ−∆)vn+∇Pⁿq+αU^⊥rotvn=fn, x∈Ω, (3.3) withdivv_n = 0and v_n|∂Ω = 0, wheref_n = Pnf ∈ PnL²(Ω)². Indeed, system (3.3) in polar coordinates is exactly (3.1) and (3.2). The key point is that there is no term involving a linearly growing coefficient in (3.3), and therefore we can apply the standard regularity theory of the Stokes resolvent system with a resolvent parameterλ. Let us assume thatn6= 0. Thenλ6= 0since we are assuming thatα6= 0. Ifvn∈L²_σ(Ω)∩W₀^1,2(Ω)²∩W_loc^2,2(Ω)² andPnq ∈ W_loc^1,2(Ω)is a solution to (3.3), then (v_n,Pnq)is a weak solution to the Stokes system with source termfn−αU^⊥rotvnwhich clearly belongs toL²(Ω)², and thus, the regularity theory of the Stokes system implies thatvn ∈W^2,2(Ω)² and∇Pⁿq ∈ L²(Ω)². In this way we can recover the summability of ∇²vn,∇Pⁿq ∈ L²(Ω)². Then, by going back to the system (3.1) and (3.2), we also find thatq_n ∈L²((1,∞);rdr)from (3.2), for all the other terms in (3.2) belong toL²([1,∞);rdr). As a summary, for any solutionv∈

L²_σ(Ω)∩W₀^1,2(Ω)² ∩W_loc^2,2(Ω)² and q ∈ W_loc^1,2(Ω)to (Sα), we can rigorously verify the energy computation for the system (3.1)-(3.2) in each n mode (vr,n, vθ,n) with n 6= 0.

The estimate for the0mode is handled in a different way from the energy method, and is discussed in Subsection 3.1.1 below.

Our main result in this subsection is stated as follows. Let us recall that the projectionQ0

is defined asQ0 :=P

n6=0Pn. Proposition 3.1 Letα∈R\ {0}.

(i)For any external forcef₀∈ P0L¹(Ω)²the system(Sα)admits a unique solution(v₀,∇q) withv₀ ∈ P0L^∞(Ω)²∩W₀^1,∞(Ω)²,∇²v₀ ∈ L¹_loc(Ω)^2×2,q ∈ W_loc^1,1(Ω). Moreover,v₀ = v_θ,0e_θand

kv₀kL^∞(Ω)+|x|∇v₀

L^∞(Ω)≤Ckf_θ,0kL¹(Ω). (3.4) Heref_θ,0:=f₀·e_θandCis independent ofα.

(ii)For any external forcef ∈ Q0L²(Ω)²the system(S_α)admits a unique solution(v,∇q) with v ∈ Q0L²_σ(Ω)∩W₀^1,2(Ω)² ∩W_loc^2,2(Ω)² and q ∈ W_loc^1,2(Ω). Moreover, vn = Pⁿv satisfies the following estimates: if1≤ |n|<1 +p

2|α|then kv_nkL²(Ω) ≤ C

|α|¹² 1

|n|+ 1

|α|¹²

kf_nkL²(Ω), (3.5)

p|x|²−1

|x| v_n

L²(Ω) ≤ C

|αn|¹²|α|¹⁴ 1

|n|+ 1

|α|^{12 1}₂

kf_nkL²(Ω), (3.6) kvnkL^∞(Ω) ≤ C

|α|¹⁴ 1

|n|+ 1

|α|¹²

kfnkL²(Ω), (3.7)

k∇vnkL²(Ω) ≤C 1

|n|+ 1

|α|¹²

kfnkL²(Ω), (3.8)

while if|n| ≥1 +p

2|α|then

kvnkL²(Ω)≤ C

|n| 1

|n|+ 1

|α|

kfnkL²(Ω), (3.9)

p|x|²−1

|x| vn

L²(Ω)≤ C

|αn|¹²|n|¹² 1

|n|+ 1

|α| ¹₂

kfnkL²(Ω), (3.10) kv_nkL^∞(Ω)≤ C

|n|³⁴ 1

|n|+ 1

|α| ³₄

kf_nkL²(Ω), (3.11) k∇vnkL²(Ω)≤ C

|n|¹² 1

|n|+ 1

|α| ¹₂

kfnkL²(Ω). (3.12) Finally if|α| ≫1and|n|=O(|α|¹²)then

kvnkL²(Ω) ≤ C

|α|kfnkL²(Ω), (3.13) kvnkL^∞(Ω) ≤ C

|α|³⁴kfnkL²(Ω), (3.14) k∇vnkL²(Ω) ≤ C

|α|¹²kfnkL²(Ω). (3.15) Herefn:=Pⁿf andCis independent ofnandα.

3.1.1 Structure and estimate of the0mode

Firstly we observe that ifv0satisfiesv0 ∈ P⁰L^∞(Ω)²∩W₀^1,∞(Ω)²then the divergence-free condition in polar coordinates (2.1) implies that

d(rv_r,0) dr = 0, and thus, v_r,0 = C

r with some constant C. Then the noslip boundary condition leads toC= 0, and therefore,v_r,0 = 0. So it suffices to consider the angular partv_θ,0. From (2.5) we have

x^⊥· ∇v₀^⊥−v₀^⊥= 0,

and we also note that the termU^⊥rotv₀ in (Sα) withU^⊥ =− x

|x|² is always written in a gradient form, so can be absorbed in a pressure term.

Collecting these remarks, we see that any solution(v₀,∇q)to (Sα) withf₀ ∈ P0L¹(Ω)² satisfying v₀ ∈ P0L^∞(Ω)² ∩W₀^1,∞(Ω)², ∇²v₀ ∈ L¹_loc(Ω)^2×2, q ∈ W_loc^1,1(Ω)must be written asv₀ = v_0,θe_θ, where v_θ,0 = v_θ,0(r) obeys from (3.2) the ordinary differential equation

−d²v_θ,0 dr² − 1

r dv_θ,0

dr + v_θ,0

r² =f_θ,0, r >1, v_θ,0(1) = 0. (3.16) The bounded solution to (3.16) is written as

v_θ,0(r) = 1 2

−1 r

Z _∞

1

f_θ,0ds+1 r

Z r

1

s²f_θ,0ds+r Z _∞

r

f_θ,0ds

. (3.17)

We note that

kf_θ,0kL¹(Ω) = 2π Z _∞

1 |f_θ,0|sds . Thus we see from (3.17) that

kv_θ,0kL^∞(Ω)+krdv_θ,0

dr kL^∞(Ω) ≤Ckf_θ,0kL¹(Ω), which implies (3.4).

3.1.2 A priori estimate of thenmode with|n| ≥1

Letv denote a solution to (Sα) satisfyingv ∈ QL²_σ(Ω)∩W₀^1,2(Ω)² ∩W_loc^2,2(Ω)² forf ∈ Q0L²(Ω)² with some q ∈ W_loc^1,2(Ω). Then, as we have already seen in the beginning of Section 3.1, for each n 6= 0, the n mode vn = Pⁿv belongs in addition to W^2,2(Ω)² and we also have thatPⁿqbelongs to W^1,2(Ω). Hence the energy computation below for (v_r,n, v_θ,n)to the system (3.1)-(3.2) is rigorously verified. With this important remark in mind we multiply both sides of (3.1) byrv¯r,nand of (3.2) byr¯vθ,nand integrate over[1,∞), which results in the following identities:

k∇vnk²L²(Ω) =−αℜhU^⊥rotvn, vniL²(Ω)+ℜhfn, vniL²(Ω), (3.18)