Three-dimensional stability of Burgers vortices

Three-dimensional stability of Burgers vortices

Thierry Gallay Institut Fourier Universit´e de Grenoble I

BP 74

38402 Saint-Martin-d’H`eres, France Thierry.Gallay@ujf-grenoble.fr

Yasunori Maekawa Faculty of Science

Kobe University 1-1 Rokkodai, Nada-ku

Kobe 657-8501, Japan yasunori@math.kobe-u.ac.jp June 4, 2010

Abstract

Burgers vortices are explicit stationary solutions of the Navier-Stokes equations which are often used to describe the vortex tubes observed in numerical simulations of three- dimensional turbulence. In this model, the velocity field is a two-dimensional perturbation of a linear straining flow with axial symmetry. The only free parameter is the Reynolds number Re = Γ/ν, where Γ is the total circulation of the vortex and ν is the kinematic viscosity. The purpose of this paper is to show that Burgers vortices are asymptotically stable with respect to small three-dimensional perturbations, for all values of the Reynolds number. This general result subsumes earlier studies by various authors, which were either restricted to small Reynolds numbers or to two-dimensional perturbations. Our proof relies on the fact that the linearized operator at Burgers vortex has a simple and very specific dependence upon the axial variable. This allows to reduce the full linearized equations to a vectorial two-dimensional problem, which can be treated using an extension of the techniques developped in earlier works. Although Burgers vortices are found to be stable for all Reynolds numbers, the proof indicates that perturbations may undergo an important transient amplification if Re is large, a phenomenon that was indeed observed in numerical simulations.

1 Introduction

The axisymmetric Burgers vortex is an explicit solution of the three-dimensional Navier-Stokes equations which provides a simple and widely used model for the vortex tubes or filaments that are observed in turbulent flows [1, 27]. Despite obvious limitations, due to oversimplified assumptions, this model describes in a correct way the fundamental mechanisms which are responsible for the persistence of coherent structures in three-dimensional turbulence, namely the balance between vorticity amplification due to stretching and vorticity dissipation due to viscosity. If one believes that vortex tubes play a significant role in the dynamics of turbulent flows, it is an important issue to determine their stability with respect to perturbations in the largest possible class. So far, this problem has been studied only for the axisymmetric Burgers vortex and for a closely related family of asymmetric vortices [24, 18].

As was shown by Leibovich and Holmes [17], one cannot hope to prove energetic stability of Burgers vortex even if the circulation Reynolds number is very small. To tackle the stability problem, it is therefore necessary to have a closer look at the spectrum of the linearized operator.

This is a relatively easy task if we restrict ourselves totwo-dimensionalperturbations. Assuming that the vortex tube is aligned with the vertical axis, this means that the perturbed velocity field lies in the horizontal plane and does not depend on the vertical variable. Under such conditions, Burgers vortex is known to be stable for any value of the Reynolds number. This result was first established by Giga and Kambe [14] for Re≪1 and then in the general case by Gallay and Wayne [9], see also [2, 12, 13]. Moreover, a lot is known about the spectrum of the linearized operator, which turns out to be purely discrete in a neighborhood of the origin in the complex plane. Using perturbative expansions, Robinson and Saffman [24] showed that all linear modes are exponentially damped for small Reynolds numbers. This property was then numerically verified by Prochazka and Pullin [22] for Re≤10⁴, and finally rigorously established in [9].

The situation is much more complicated if we allow for arbitrary three-dimensional pertur- bations. In that case, it was shown by Rossi and Le Diz`es [25] that the linearized operator does not have any eigenfunction with nontrivial dependence in the vertical variable. While this result precludes the existence of unstable eigenvalues, it also implies that stability cannot be deduced from such a simple analysis, and that continuous spectrum necessarily plays an important role.

Unfortunately, the vertical dependence of the perturbed solutions is not easy to determine, as can be seen from the note [3] where a few attempts are made in that direction. The only rigorous result so far is due to Gallay and Wayne [10], who proved that Burgers vortex is asymptoti- cally stable with respect to three-dimensional perturbations in a fairly large class provided that the Reynolds number is sufficiently small. For larger Reynolds numbers, up to Re = 5000, an important numerical work by Schmid and Rossi [26] indicates that all modes are exponentially damped by the linearized evolution, although significant short-time amplification can occur.

In this paper, we prove that the axisymmetric Burgers vortex is asymptotically stable with respect to small three-dimensional perturbations for arbitrary values of the Reynolds number.

As in [10], we assume that the perturbations are nicely localized in the horizontal variables, but we do not impose any decay with respect to the vertical variable. Our approach is based on the fact that the linearized operator has a very simple dependence upon the vertical variable:

the only term involving x₃ is the dilation operatorx₃∂_x3, which originates from the background straining field. This crucial property was already exploited in [25, 3, 26], but we shall show that it allows to reduce the three-dimensional stability problem to a two-dimensional one, which can then be treated using an extension of the techniques developped in [9]. Although the spectrum of the linearized operator remains stable for all Reynolds numbers, the estimates we have on the associated semigroup deteriorate as Re increases, in full agreement with the amplification phenomena observed in [26].

We now formulate our results in a more precise way. We start from the three-dimensional incompressible Navier-Stokes equations:

∂_tV + (V,∇)V = ν∆V −1

ρ∇P , ∇ ·V = 0 , (1.1)

where V = V(x, t) ∈ R³ denotes the velocity field, P = P(x, t) ∈ R is the pressure field, and x= (x₁, x₂, x₃)^⊤∈R³ is the space variable. The parameters in (1.1) are the kinematic viscosity ν >0 and the densityρ >0. To obtain tubular vortices, we assume that the velocity V can be decomposed as follows:

V(x, t) = V^s(x) +U(x, t), (1.2)

whereV^s is an axisymmetric straining flow given by the explicit formula V^s(x) = γ

2





−x1

−x₂ 2x₃



 ≡ γM x , where M =





−¹₂ 0 0 0 −¹₂ 0

0 0 1



 . (1.3)

Hereγ >0 is a parameter which measures the intensity of the strain. Note that∇ ·V^s= 0, and thatV^s is a stationary solution of (1.1) with the associated pressureP^s=−¹₂ρ|V^s|². Our goal is to study the evolution of the perturbed velocity field U(x, t).

To simplify the notations, we shall assume henceforth that γ = ν = ρ = 1. This can be achieved without loss of generality by replacing the variables x,t and the functionsV,P with the dimensionless quantities

˜ x = γ

ν 1/2

x , ˜t = γt , V˜ = V

(γν)^1/2 , P˜ = P ργν .

For further convenience, instead of considering the evolution ofV orU, we prefer working with the vorticity field Ω =∇ ×V =∇ ×U. Taking the curl of (1.1) and using (1.2), (1.3), we obtain for Ω the evolution equation

∂_tΩ + (U,∇)Ω−(Ω,∇)U = LΩ, ∇ ·Ω = 0, (1.4) whereL is the differential operator defined by

LΩ = ∆Ω−(M x,∇)Ω +MΩ. (1.5)

Under mild assumptions that will be specified below, the velocity field U can be recovered from the vorticity Ω via the three-dimensional Biot-Savart law

U(x) = − 1 4π

Z

R3

(x−y)×Ω(y)

|x−y|³ dy =: (K_3D∗Ω)(x) . (1.6) In what follows we shall often encounter the particular situation where the velocity U is two- dimensional and horizontal, namely U(x) = (U₁(x_h), U₂(x_h),0)^⊤ where x_h = (x₁, x₂)^⊤ ∈ R². In that case the vorticity satisfies Ω(x) = (0,0,Ω₃(x_h))^⊤, and the relation (1.6) reduces to the two-dimensional Biot-Savart law

U_h(x_h) = 1 2π

Z

R2

(x_h−y_h)^⊥

|x_h−y_h|² Ω₃(y_h) dy_h =: (K_2D ⋆Ω₃)(x_h), (1.7) whereU_h = (U₁, U₂)^⊤ and x^⊥_h = (−x₂, x₁)^⊤.

We can now introduce the Burgers vortices, which are explicit stationary solutions of (1.4) of the form Ω =αG, where α∈R is a parameter. The vortex profile is given by

G(x) =



 0 0 g(x_h)



 , where g(x_h) = 1

4πe^−|xh|2/4 . (1.8) The associated velocity field U =αU^G can be obtained from the Biot-Savart law (1.7) and has the following form

U^G(x) = u^g(|x_h|²)





−x2

x₁ 0



 , where u^g(r) = 1 2πr

1−e^−r/4

. (1.9)

If Ω =αG, it is easy to verify that α =R

R2Ω3(x_h) dx_h. This means that the parameter α ∈R represents thetotal circulationof the Burgers vortexαG. In the physical literature, the quantity

|α|is often referred to as the (circulation) Reynolds number.

The aim of this paper is to study the asymptotic stability of the Burgers vortices. We thus consider solutions of (1.4) of the form Ω = αG+ω, U = αU^G+u, and obtain the following evolution equation for the perturbation:

∂_tω+ (u,∇)ω−(ω,∇)u = (L−αΛ)ω , ∇ ·ω = 0, (1.10) where Λ is the integro-differential operator defined by

Λω = (U^G,∇)ω−(ω,∇)U^G+ (u,∇)G−(G,∇)u . (1.11) Here and in the sequel, it is always understood thatu=K_3D∗ω.

An important issue is now to fix an appropriate function space for the admissible pertur- bations. Since the Burgers vortex itself is essentially a two-dimensional flow, it is natural to choose a functional setting which allows for perturbations in the same class, but we also want to consider more general ones. Following [10], we thus assume that the perturbations are nicely localized in the horizontal variables, but merely bounded in the vertical direction. As we shall see below, this choice is more or less imposed by the particular form of the linear operator (1.5).

To specify the horizontal decay of the admissible perturbations, we first introduce two- dimensional spaces. Given m ∈[0,∞], we denote by ρ_m : [0,∞) → [1,∞) the weight function defined by

ρ_m(r) =







1 if m= 0,

(1 +_4m^r )^m if 0< m <∞, e^r/4 if m=∞.

(1.12) We introduce the weighted L² space

L²(m) = n

f ∈L²(R²)

kfk²_L2(m) = Z

R2

|f(x_h)|²ρ_m(|x_h|²) dx_h<∞o

, (1.13) which is a Hilbert space with a natural inner product. Using H¨older’s inequality, it is easy to verify thatL²(m)֒→L¹(R²) if m >1. In that case, we also define the closed subspace

L²₀(m) = n

f ∈L²(m)

Z

R2f(x_h) dx_h= 0o

. (1.14)

Next, we define the three-dimensional space X(m) as the set of all φ : R³ → R for which the map x_h 7→ φ(x_h, x₃) belongs to L²(m) for any x₃ ∈ R, and is a bounded and continuous function of x₃. In other words, we set

X(m) = BC(R;L²(m)), X₀(m) = BC(R;L²₀(m)), (1.15) where “BC(R;Y)” denotes the space of all bounded and continuous functions from R into Y. Both X(m) and X0(m) are Banach spaces equipped with the norm

kφk_X(m) = sup

x3∈R

kφ(·, x₃)k_L2(m) . (1.16) Our goal is to study the stability of the Burgers vortex Ω =αGwith respect to perturbations ω∈X(m)³. In fact, we can assume without loss of generality thatω belongs to the subspace

X(m) = X(m)×X(m)×X₀(m) ⊂ X(m)³ , (1.17) which is invariant under the evolution defined by (1.10). Indeed, we have the following result, whose proof is postponed to Section 6.1:

Lemma 1.1 Fix m ∈ (1,∞]. If ω˜ ∈ X(m)³ satisfies ∇ ·ω˜ = 0 in the sense of distributions, then there exists α˜∈R such that

Z

R2

˜

ω₃(x_h, x₃) dx_h = ˜α , for allx₃∈R. (1.18) At a formal level, this is a direct consequence of the divergence-free assumption, since

d dx₃

Z

R2ω˜3(x_h, x3) dx_h = Z

R2∂x3ω˜3(x_h, x3) dx_h = − Z

R2∇_h·ω˜_h(x_h, x3) dx_h = 0 . In view of Lemma 1.1, if Ω =αG+ ˜ω for some ˜ω ∈ X(m)³, we can write Ω = (α+ ˜α)G+ω, where ˜α is given by (1.18) and ω = ˜ω−αG. Then˜ ω ∈X(m) by construction, and we are led back to the stability analysis of the Burgers vortex (α+ ˜α)G with respect to perturbations in X(m).

In what follows we always consider the solutions ω(x, t) of (1.10) as X(m)-valued functions of time, and we often denote byω(·, t) or simply ω(t) the map x7→ω(x, t). A minor drawback of our functional setting is that we cannot expect the solutions of (1.10) to be continuous in time in the strong topology of X(m). This is because the operator L defined in (1.5) contains the dilation operator −x3∂x3, see Section 2.1 below. To restore continuity, it is thus necessary to equip X(m) with a weaker topology. Following [10], we denote byX_loc(m) the space X(m) equipped with the topology defined by the family of seminorms

kφk_Xn(m) = sup

|x3|≤n

kφ(·, x₃)k_L2(m) , n∈N.

In analogy with (1.17), we set X_loc(m) =X_loc(m)×X_loc(m)×X_0,loc(m), whereX_0,loc(m) is of course the spaceX₀(m) equipped with the topology of X_loc(m).

We are now able to formulate our main result:

Theorem 1.2 Fixm∈(2,∞]andα∈R. Then there existδ =δ(α, m)>0andC=C(α, m)≥ 1 such that, for any ω₀ ∈ X(m) with ∇ ·ω₀ = 0 and kω₀kX(m) ≤ δ, Eq. (1.10) has a unique solution ω∈L^∞(R₊;X(m))∩C([0,∞) ;X_loc(m)) with initial data ω0. Moreover,

kω(t)kX(m) ≤ Ckω₀kX(m)e^−t/2 , for allt≥0 . (1.19) Theorem 1.2 shows that the Burgers vortex αG is asymptotically stable with respect to small perturbations in X(m), for any value of the circulation α ∈R. If one prefers to consider perturbations in the larger spaceX(m)³, then our result means that the family{αG}_α∈R of all Burgers vortices is asymptotically stablewith shift, because the perturbations may then modify the circulation of the underlying vortex. The key point in the proof is to show that the linearized operator L−αΛ has a uniform spectral gap for all α∈R. This implies a uniform decay rate in time for the perturbations, as in (1.19). However, it should be emphasized that the constants C and δ in Theorem 1.2 depend on α, in such a way that C(α, m) → ∞ and δ(α, m) → 0 as

|α| → ∞. This is in full agreement with the amplification phenomena numerically observed in [26].

The proof of Theorem 1.2 gives a more detailed information on the solutions of (1.10) than what is summarized in (1.19). First of all, we can prove stability inX(m) for anym >1, but the exponential factore^−t/2in (1.19) should then be replaced bye^−ηt, whereη <(m−1)/2 ifm≤2.

Next, thanks to parabolic smoothing, we can obtain decay estimates not only for ω but also for the spatial derivatives ∂_xjω ≡ ∂ω/∂x_j, j = 1,2,3. For convenience, we shall often use the

multi-index notation ∂_x^β =∂_x^β₁¹∂_x^β₂²∂^β_x3³ forβ = (β₁, β₂, β₃) ∈ N³. Finally, due to the particular structure of the linear operator L−αΛ, it turns out that the horizontal partω_h = (ω1, ω2)^⊤ of the vorticity vector has a faster decay than the vertical componentω₃ ast→ ∞. Thus, a more complete (but less readable) version of our result is as follows:

Theorem 1.3 Fix m ∈ (1,∞], α ∈ R, and take µ ∈ (1,³₂), η ∈ (0,¹₂] such that 2µ < m+ 1 and 2η < m−1. Then there exist δ =δ(α, m) > 0 and C = C(α, m, µ, η) ≥ 1 such that, for all initial data ω0 ∈X(m) with ∇ ·ω0 = 0 and kω0kX(m) ≤δ, Eq. (1.10) has a unique solution ω∈L^∞(R₊;X(m))∩C([0,∞) ;X_loc(m)). Moreover, for all t >0,

k∂_x^βω_h(t)k_X(m)2 ≤ Ckω₀kX(m)

a(t)^|β|/2 e^−µt , (1.20) k∂_x^βω₃(t)k_X(m) ≤ Ckω₀kX(m)

a(t)^|β|/2 e^−ηt , (1.21) where a(t) = 1−e^−t and β ∈N³ is any multi-index of length|β|=β1+β2+β3 ≤1.

The decay rates (1.20), (1.21) are optimal when β= 0, but it turns out that vertical deriva- tives such as ∂x3ω_h(t) or ∂x3ω3(t) have a faster decay as t→ ∞, see Sections 4 and 5 for more details. In any case, we believe that the optimal rates are those provided by the linear stability analysis, as in Proposition 4.1 below.

The rest of this paper is devoted to the proof of Theorems 1.2 and 1.3. Before giving the details, we explain here the important ideas in an informal way. As was already mentioned, the main difficulty is to obtain good estimates on the solutions of the linearized equation

∂tω = (L−αΛ)ω , ∇ ·ω = 0. (1.22) Once this is done, the nonlinear terms in (1.10) can be controlled using rather standard argu- ments, which are recalled in Section 5. To study (1.22), we use the fact that the operatorL−αΛ depends on the vertical variable in a simple and very specific way. Indeed, it is easy to verify that [∂_x3, L] =−∂_x3 and [∂_x3,Λ] = 0, where [A, B] =AB−BAdenotes the commutator of the operators A and B. This key observation, which already plays a crucial role in the previous works [25, 3, 26], implies the following identity:

∂_x^k₃e^t(L−αΛ)ω0 = e^−kte^t(L−αΛ)∂_x^k₃ω0 , (1.23) for all k ∈ N and all t ≥ 0. If we take k ∈ N sufficiently large, depending on |α|, we can use (1.23) to show that∂_x^k₃ω(t) decays exponentially as t→ ∞ ifω(t) is a solution of (1.22). Then, by an interpolation argument, we deduce that all expressions involving at least one vertical derivative play a negligible role in the long-time asymptotics, see Section 4 for more details.

This “smoothing effect” in the vertical direction is due to the stretching properties of the linear flow (1.2).

As a consequence of these remarks, we can restrict our attention to those solutions of (1.22) which are independent of the vertical variable x₃. We call this particular situation thevectorial 2D problem, and we study it in Section 3. Note that the perturbations we allow here are two- dimensional in the sense that ∂_x3u = ∂_x3ω = 0, but that all three components of u or ω are possibly nonzero. This is in contrast with the purely two-dimensional case considered in [9, 10], where in addition u₃ = ω₁ = ω₂ = 0. Extending the techniques developped in [9, 10], it is possible to show that all solutions of (1.22) with ∂_x3ω = 0 converge exponentially to zero as t → ∞, and that the decay rate is uniform in α. This is done using spectral estimates and a

detailed study of the eigenvalue equation (L−αΛ)ω =λω. It is then a rather straightforward task to complete the proof of Theorem 1.2 using the arguments outlined above.

Remark. The vortex tubes observed in numerical simulations are usually not axisymmetric: in general, they rather exhibit an elliptical core region [18, 6]. A simple model for such asymmetric vortices is obtained by replacing the straining flow V^s in (1.3) with the nonsymmetric strain V_λ^s(x) =γM_λx, whereλ∈(0,1) is an asymmetry parameter and

M_λ =





−^1+λ₂ 0 0 0 −^1−λ₂ 0

0 0 1



 . (1.24)

Asymmetric Burgers vortices are then stationary solutions to (1.4), where the operatorLin the right-hand side is defined by (1.5) withM replaced byM_λ. Unlike in the symmetric caseλ= 0, no explicit formula is available and proving the existence of stationary solutions is already a nontrivial task, except perhaps in the perturbative regime where either the asymmetry parameter λ or the circulation number α is very small. In view of these difficulties, asymmetric Burgers vortices were first studied using formal asymptotic expansions and numerical calculations, see e.g. [24, 18, 23]. The mathematical theory is more recent, and includes several existence results which cover now the whole range of parametersλ∈(0,1) andα∈R[10, 11, 19, 20]. In addition, the stability with respect to two-dimensional perturbations is known to hold at least for small values of the asymmetry parameter [11, 19]. However, the only result so far on three-dimensional stability is restricted to the particular case where the circulation numberα is sufficiently small, depending onλ[10].

Using Theorem 1.2 and a simple perturbation argument, it is easy to show that asymmetric Burgers vortices are stable with respect to small three-dimensional pertubations in the space X(m), provided that the asymmetry parameter λis small enough depending on the circulation number α. This follows from the fact the linearized operator at the symmetric Burgers vortex has a uniform spectral gap for all α∈R, and that the asymmetric Burgers vortex isO(λ)-close to the corresponding symmetric vortex in the topology of X(m), uniformly for all α ∈ R [11].

Although this stability result is new and not covered by [10], it is certainly not optimal, and we prefer to postpone the study of the three-dimensional stability of asymmetric Burgers vortices to a future investigation.

2 Preliminaries

In this preliminary section we collect a few basic estimates which will be used throughout the proof of Theorems 1.2 and 1.3. They concern the semigroup generated by the linear operator (1.5), and the Biot-Savart law (1.6) relating the velocity field to the vorticity. Most of the results were already established in [10, Appendix A], and are reproduced here for the reader’s convenience.

As in [10], we introduce the following generalization of the function spaces (1.13) and (1.15).

Givenm∈[0,∞] andp∈[1,∞), we define the weighted L^p space L^p(m) = n

f ∈L^p(R²)

kfk^p_Lp(m)= Z

R2

|f(x_h)|^pρ_m(|x_h|²)^p/2dx_h<∞o , and the corresponding three-dimensional space

X^p(m) = BC(R;L^p(m)), kφk_X^p_(m) = sup

x3∈R

kφ(·, x₃)k_L^p_(m) .

Ifm >2−²_p, we also denote byL^p₀(m) the subspace of allf ∈L^p(m) such thatR

Rfdx_h= 0. In analogy with (1.17), we setX^p(m) = X^p(m)×X^p(m)×X₀^p(m) , whereX₀^p(m) =BC(R;L^p₀(m)).

2.1 The semigroup generated by L

If we decompose the vorticityωinto its horizontal partω_h = (ω₁, ω₂)^⊤and its vertical component ω₃, it is clear from (1.3) and (1.5) that the linear operator L has the following expression:

Lω =

L_hω_h L3ω3

=

(L_h+L₃−³₂)ω_h (L_h+L3)ω3

, (2.1)

whereL_h is the two-dimensional Fokker-Planck operator L_h = ∆_h+ x_h

2 · ∇_h+ 1 =

2

X

j=1

∂_x²_j+

2

X

j=1

x_j

2∂_xj+ 1, (2.2)

and L₃ =∂_x²₃−x₃∂_x3 is a convection-diffusion operator in the vertical variable.

As is shown in [8, appendix A], the operator L_h is the generator of a strongly continuous semigroup inL²(m) given by the explicit formula

(e^tLhf)(x_h) = e^t 4πa(t)

Z

R2

e⁻

|xh−yh|2

4a(t) f(y_he^t/2) dy_h , t >0, (2.3)

wherea(t) = 1−e^−t. Similarly, the operatorL₃generates a semigroup of contractions inBC(R) given by

(e^tL3f)(x3) = 1 p2πa(2t)

Z

R

e⁻

|x3e−t−y3|2

2a(2t) f(y3) dy3 , t >0, (2.4)

see [10, Appendix A]. Note that the semigroup e^tL3 is not strongly continuous in the space BC(R) equipped with the supremum norm. This is mainly due to the dilation factor e^−t in (2.4). However, if we equip BC(R) with the (weaker) topology of uniform convergence on compact sets, then the mapt7→e^tL3f is continuous for anyf ∈BC(R). This observation is the reason for introducing the spaceX_loc(m) in Section 1.

Since the operators L_h and L₃ act on different variables, it is easy to obtain the semigroup generated byL₃=L_h+L₃ by combining the formulas (2.3) and (2.4). We find

(e^tL3φ)(x) = 1 p2πa(2t)

Z

R

e⁻

|x3e−t−y3|2 2a(2t)

e^tLhφ(·, y₃)

(x_h) dy₃ , t >0 . (2.5) In [10, Proposition A.6], it is shown that this expression defines a uniformly bounded semigroup in X(m) for any m > 1, and that the map t 7→ e^tL3 is strongly continous in the topology of X_loc(m). Moreover, the subspace X0(m) is left invariant by e^tL3 for any t ≥ 0. Using these results and the relation (2.1), we conclude that the three-dimensional operator L generates a uniformly bounded semigroup in the space X(m), given by

e^tLω =

e^−3t/2e^tL3ω₁, e^−3t/2e^tL3ω₂, e^tL3ω₃⊤

, t≥0. (2.6)

As is easily verified, if ∇ ·ω= 0, then ∇ ·e^tLω= 0 for all t≥0.

The asymptotic stability of the Burgers vortices relies heavily on the decay properties of the semigroup e^tL as t → ∞. In the proof of Theorems 1.2 and 1.3, we also use the smoothing properties of the operatore^tLfort >0, and in particular the fact thate^tL extends to a bounded operator fromX^p(m) into X²(m) for allp∈[1,2]. All the needed estimated are collected in the following statement.

Proposition 2.1 Let m∈(1,∞], p∈[1,2], and take η∈(0,¹₂] such that2η < m−1. For any β = (β1, β2, β3)∈N³, there exists C >0 such that the following estimates hold:

k∂_x^βe^tLhω_hk_X(m)2 ≤ Ce^−(32+β3)t a(t)^{1p−12+|β|2}

kω_hk_X^p_(m)2 , (2.7)

k∂_x^βe^tL3ω₃k_X(m) ≤ Ce^−(η+β3)t a(t)^{1p−12+|β|2}

kω₃k_X^p_(m) , (2.8)

for anyω ∈X^p(m) and all t >0. Here a(t) = 1−e^−t and |β|=β₁+β₂+β₃.

Proof. We first assume that m∈(1,∞). If p∈[1,2] andβ_h = (β₁, β₂)∈N², it is proved in [8, Appendix A] that

k∂_x^β_h^he^tLhfk_L2(m) ≤ C a(t)^1p−12+

|βh| 2

kfk_L^p_(m) , t >0 , (2.9)

for all f ∈L^p(m). If in addition f ∈L^p₀(m), we have the stronger estimate k∂_x^β_h^he^tLhfk_L2(m) ≤ Ce^−ηt

a(t)^1p−12+

|βh| 2

kfk_L^p_(m) , t >0 , (2.10)

whereη >0 is as in Proposition 2.1. On the other hand, using (2.4), we find by direct calculation k∂_x^β₃³e^tL3fk_L∞(R) ≤ Ce^−β3t

a(t)^β23

kfk_L∞(R) , t >0 . (2.11) Here, as in (1.23), the stabilizing factor e^−β3t comes from the dilation operator −x₃∂_x3 which enters the definition ofL₃. Now, if we start from the representation (2.5) and use the estimates (2.9)–(2.11), we easily obtain (2.7), (2.8) by a direct calculation, see [10, Proposition A.6].

To complete the proof of Proposition 2.1, it remains to show that (2.9), (2.10) still hold when m =∞. If t ∈ (0,1), estimate (2.9) is easily obtained by a direct calculation, based on the representation (2.3). Using this remark and the semigroup property of e^tLh, we conclude that it is sufficient to establish (2.9), (2.10) in the particular case where p = 2 and β_h = 0.

This in turns follows easily from the spectral properties of the generator L_h. Indeed, it is well-known that L_h is a self-adjoint operator in L²(∞) with purely discrete spectrumσ(L_h) = {−^k₂|k= 0,1,2, . . .}. Moreover, the subspaceL²₀(∞) is precisely the orthogonal complement of the eigenspace corresponding to the zero eigenvalue, see for example [9, Lemma 4.7]. It follows that e^tLh is a semigroup of contractions inL²(∞), and that ke^tLhfk_L2(∞) ≤e^−t/2kfk_L2(∞) for all t≥0 if f ∈L²₀(∞). This proves (2.9) and (2.10), withη= 1/2.

2.2 Estimates for the velocity fields

If the velocityuand the vorticityωare related by the Biot-Savart law (1.6), we have|u| ≤J(|ω|), whereJ is the Riesz potential defined by

J(φ)(x) = 1 4π

Z

R3

1

|x−y|² φ(y) dy , x∈R³ . (2.12) Sinceω will typically belong to the Banach spaceX(m), we need estimates on the Riesz potential J(φ) for φ∈X(m). We start with a preliminary result:

Lemma 2.2 Let p₁ ∈[1,2), p₂ ∈[1,2], and assume that φ∈X^p1(0)∩X^p2(0). Ifq₁, q₂ ∈[1,∞]

satisfy

2p₁

2−p₁ < q₁ ≤ ∞ , p₂ < q₂ < 2p₂

2−p₂ , (2.13)

then J(φ) =J₁(φ) +J₂(φ) withJ_i(φ)∈X^qi(0)for i= 1,2, and we have the following estimates kJ₁(φ)k_X^q1(0) ≤ C(p₁, q₁)kφk_X^p1(0) , (2.14) kJ₂(φ)k_X^q2(0) ≤ C(p₂, q₂)kφk_X^p2(0) . (2.15) Proof. We proceed as in [10, Proposition A.9]. We first observe that

J(φ)(x_h, x₃) = Z

|x3−y3|≥1

F(x_h;x₃, y₃) dy₃+ Z

|x3−y3|<1

F(x_h;x₃, y₃) dy₃

= J₁(φ)(x_h, x₃) +J₂(φ)(x_h, x₃) , where

F(x_h;x₃, y₃) = Z

R2

φ(y_h, y₃)

|x_h−y_h|²+ (x₃−y₃)²dy_h , x_h ∈R² , x₃, y₃ ∈R.

For anya∈R, letfa(y_h) = (a²+|y_h|²)⁻¹. Then fa∈L^r(R²) for any r >1 and anya6= 0, and there existsC_r>0 such that

kf_ak_L^r₍R2) ≤ C_r

|a|^2−2r .

Moreover, we have F(·;x3, y3) =φ(·, y3)⋆ fx3−y3 by construction. Thus, if we take 1≤p, q, r≤

∞ such that 1 +¹_q = ¹_p +¹_r, we obtain using Young’s inequality

kF(·;x₃, y₃)k_L^q₍R2) ≤ kφ(·, y₃)k_L^p₍R2)kf_x3−y3k_L^r₍R2) ≤ C_rkφ(·, y₃)k|_L^p₍R2)

|x₃−y₃|^2−2r .

To estimate J₁(φ), we choose p=p₁, q=q₁. In view of (2.13), the corresponding exponent r=r₁ satisfies 2< r₁ ≤ ∞, so that 2−_r²₁ ∈(1,2]. By Minkowski’s inequality, we thus find

kJ₁(φ)(·, x₃)k_L^q1(R2) ≤ Z

|x3−y3|≥1

kF(·;x₃, y₃)k_L^q1(R2)dy₃ ≤ C(r₁) sup

y3∈R

kφ(·, y₃)k_L^p1(R2) . Taking the supremum over x₃ ∈R, we obtain (2.14). Similarly, to boundJ₂(φ), we takep=p₂, q=q2. Then 1< r2 <2, so that 2−_r²₂ ∈(0,1). We thus obtain

kJ₂(φ)(·, x₃)k_L^q2(R2) ≤ Z

|x3−y3|<1

kF(·;x₃, y₃)k_L^q2(R2)dy₃ ≤ C(r₂) sup

y3∈R

kφ(·, y₃)k_L^p2(R2) , and (2.15) follows. Finally, the uniform continuity of J_i(φ)(·, x₃) with respect to x₃ can be

verified exactly as in the proof of [10, Proposition A.9].

As an immediate consequence, we obtain the following useful statements.

Proposition 2.3 Let φ∈X(m) for some m ∈(1,∞]. Then J(φ)∈ X^q(0) for all q ∈(2,∞), and there exists a positive constant C=C(m, q) such that

kJ(φ)k_X^q₍₀₎ ≤ Ckφk_X(m) . (2.16)

Proof. Ifm >1, we recall thatX(m)֒→X^p(0) for allp∈[1,2]. Thus we can apply Lemma 2.2 withp1= 1, p2 = 2,q1 =q2=q ∈(2,∞), and the result follows.

Corollary 2.4 Let φ₁, φ₂ ∈ X(m) for some m ∈ (1,∞]. Then φ₁J(φ₂) ∈ X^p(m) for all p∈(1,2), and there exists a positive constant C =C(m, p) such that

kφ₁J(φ₂)k_X^p_(m) ≤ Ckφ₁k_X(m)kφ₂k_X(m) . (2.17) Proof. We proceed as in [10, Corollary A.10]. Let p ∈ (1,2), and take q ∈ (2,∞) such that

1

q = _p¹− ¹₂. For anyx3 ∈R, we have by H¨older’s inequality kφ1(·, x3)J(φ2)(·, x3)k_L^p_(m) = Z

R2ρm(|x_h|²)^p/2|φ1(x_h, x3)|^p|J(φ2)(x_h, x3)|^pdx_h1/p

≤ Z

R2

ρ_m(|x_h|²)|φ₁(x_h, x₃)|²dx_h1/2Z

R2

|J(φ₂)(x_h, x₃)|^qdx_h1/q

= kφ₁(·, x₃)k_L2(m)kJ(φ₂)(·, x₃)k_L^q₍₀₎ .

Taking the supremum over x₃ ∈ R and using Proposition 2.3, we obtain (2.17). Finally, it is clear that the mapx₃ 7→φ₁(·, x₃)J(φ₂)(·, x₃) is continuous from RintoL^p(m).

We conclude this section with an estimate on the linear operator (1.11) which will be needed in Section 4.

Lemma 2.5 Let p∈[1,2] and 2−²_p < m≤ ∞. For any β∈N³, there existsC >0 such that k∂_x^βΛωkX^p(m) ≤ C X

|β|≤|β|+1˜

k∂_x^β˜ωkX^p(m). (2.18)

Proof. It is sufficient to prove (2.18) for β = 0. The general case easily follows if we use the Leibniz rule to differentiate Λω (we omit the details).

Assume thus that ω belongs to X^p(m), together with its first order derivatives. Since the functionU^Gdefined in (1.9) is smooth and bounded (together with all its derivatives), it is clear that

k(U^G,∇)ωkXp(m)+k(ω,∇)U^GkXp(m) ≤ C X

|β|≤1˜

k∂_x^β˜ωkXp(m) .

We now estimate the term (u,∇)G = (K_3D ∗ω,∇)G, using the fact that |K_3D ∗ω| ≤ J(|ω|).

Since|ω| ∈X¹(0)∩X^p(0) by assumption, we can apply Lemma 2.2 withp₁ = 1,q₁ =∞,p₂=p, and q2 ∈(p,_2−p^2p ). By H¨older’s inequality, we easily find

kJ₁(|ω|)|∇G|k_X^p_(m) ≤ CkJ₁(|ω|)k_X^∞₍₀₎ ≤ Ck|ω|k_X1(0) ≤ CkωkXp(m) , kJ₂(|ω|)|∇G|k_X^p_(m) ≤ CkJ₂(|ω|)k_X^q2(0) ≤ Ck|ω|k_X^p₍₀₎ ≤ CkωkX^p(m) .

We conclude thatk(u,∇)GkX^p(m)=k(K_3D∗ω,∇)GkX^p(m) ≤CkωkX^p(m). In a similar way, com- muting the derivative and the convolution operator, we obtain the estimate k(G,∇)ukX^p(m) ≤ k(G,∇)(K_3D∗ω)kXp(m)≤Ck∇ωkXp(m). This completes the proof.

3 The vectorial 2D problem

In this section we study the linearized equation∂_tω= (L−αΛ)ωin the particular case where the vorticityω does not depend on the vertical variable. As was explained in the introduction, this preliminary step is an essential ingredient in the linear stability proof which will be presented in Section 4. Since the results established here will eventually be applied to the restriction of the 3D vorticity field to a horizontal planex₃ = const., we do not assume in this section that ω is divergence-free.

If ∂_x3ω = 0, then L₃ω = 0, and the expression (2.1) of the linear operator L becomes significantly simpler. On the other hand, we know from (1.11) that

Λω = Λ₁ω−Λ₂ω+ Λ₃ω−Λ₄ω , (3.1) where

Λ₁ω = (U^G,∇)ω = (U_h^G,∇_h)ω , Λ₂ω = (ω,∇)U^G = (ω_h,∇_h)U^G ,

Λ₃ω = (u,∇)G = (u_h,∇_h)G ,

Λ₄ω = (G,∇)u = g∂_x3u . (3.2) Here u = K_3D ∗ω is the velocity field obtained from ω via the three-dimensional Biot-Savart law (1.6). Since∂_x3ω= 0, we have ∂_x3u = 0, hence Λ₄ω = 0 in our case. Moreover, it is easy to verify thatu= (u_h, u₃), whereu_h=K_2D⋆ ω₃. Thus, we see that

(L−αΛ)ω = L_αω , if∂_x3ω = 0, whereL_α is the two-dimensional differential operator defined by

L_αω =

L_α,hω_h L_α,3ω₃

=

(L_h−³₂)ω_h−α(Λ₁−Λ˜₂)ω_h L_hω₃−α(Λ₁+ ˜Λ₃)ω₃

. (3.3)

Here ˜Λ₂ω_h = (ω_h,∇_h)U_h^G and ˜Λ₃ω₃= (K_2D⋆ ω₃,∇_h)g.

For any α ∈ R and any m ∈ (1,∞], the operator L_α defined by (3.3) is the generator of a strongly continuous semigroup in the space L²(m)³. This property can be established by a standard perturbation argument, see Lemma 3.2 below. Our main goal here is to obtain accurate decay estimates for the semigroup e^tLα as t → ∞. As is clear from (3.3), the evolutions for ω_h and ω₃ are completely decoupled, so that we can consider the semigroups e^tLα,h and e^tLα,3 separately. The main contribution of this section is:

Proposition 3.1 Fix m∈(1,∞], α∈R, µ∈(0,³₂), and takeη ∈(0,¹₂] such that1 + 2η < m.

Then there exists C >0 such that

ke^tLα,hω_hk_L²_(m)² ≤ C e^−µtkω_hk_L²_(m)² , t≥0 , (3.4) ke^tLα,3ω₃k_L2(m) ≤ C e^−ηtkω₃k_L2(m) , t≥0 , (3.5) for allω ∈L²(m)²×L²₀(m).

Estimate (3.5) was obtained in [9, Proposition 4.12] for m <∞, and the proof given there extends to the limiting case m=∞ without additional difficulty. We recall that the decay rate e^−ηt is obtained using the fact that ω₃ ∈L²₀(m): If we only assume that ω₃ ∈L²(m) for some m >1, then (3.5) holds with η= 0.

From now on, we focus on the semigroup e^tLα,h, which has not been studied yet. To prove (3.4), we use the same arguments as in [9, Section 4.2]. We first establish a short time estimate:

Lemma 3.2 Fix m∈(1,∞], α∈R, and T >0. There existsC =C(T, m,|α|)>0 such that sup

0≤t≤T

ke^tLα,hω_hk_L2(m)² +a(t)¹²k∇_he^tLα,hω_hk_L2(m)⁴

≤ Ckω_hk_L2(m)² , (3.6) for allω_h∈L²(m)². Here a(t) = 1−e^−t.

Proof. Given ω⁰_h∈L²(m)², the idea is to solve the integral equation ω_h(t) = e^t(Lh−32)ω_h⁰−α

Z t 0

e^{(t−s)(Lh−32)}(Λ1−Λ˜2)ω_h(s) ds , t∈[0, T], (3.7) by a fixed point argument, in the spaceX_T ={ω_h∈C([0, T], L²(m)²| kω_hk_XT <∞}defined by the norm

kω_hkXT = sup

0≤t≤T

kω_h(t)k_L²_(m)² + sup

0≤t≤T

a(t)¹²k∇_hω_h(t)k_L²_(m)⁴ .

From (2.9) we know that ke^t(Lh−32)ω⁰_hkXT ≤C1kω_h⁰k_L²_(m)², for some C1 >0 independent of T. To estimate the integral term in (3.7), we first observe that the velocity field U^G defined by (1.9) satisfies

sup

xh∈R2

(1 +|x_h|)|U^G(x_h)|+ sup

xh∈R2

(1 +|x_h|)²|∇_hU^G(x_h)| < ∞ . (3.8) In view of the definitions (3.2), we thus have

k(1 +|x_h|)Λ1ω_hk_L²_(m)² ≤ Ck∇_hω_hk_L²_(m)⁴ , (3.9) k(1 +|x_h|)²Λ˜₂ω_hk_L2(m)² ≤ Ckω_hk_L2(m)² . (3.10) Using these estimates together with (2.9), we can bound

Z _t

0

e^{(t−s)(Lh−32)}(Λ₁−Λ˜₂)ω_h(s) ds L²(m)²

≤ C Z t

0

e^−32(t−s)

kω_h(s)k_L²_(m)² +k∇_hω_h(s)k_L²_(m)⁴ ds

≤ Ckω_hk_XT Z _t

0

e^−32(t−s)a(s)⁻¹²ds ≤ Ca(T)¹²kω_hk_XT . In a similar way,

∇_h

Z t 0

e^{(t−s)(Lh−32)}(Λ₁−Λ˜₂)ω_h(s) ds

L²(m)⁴ (3.11)

≤ C Z t

0

e^−32(t−s) a(t−s)¹²

kω_h(s)k_L2(m)² +k∇_hω_h(s)k_L2(m)⁴

ds ≤ Ckω_hk_XT .

Summarizing, we have shown that kω_hkXT ≤ C1kω⁰_hk_L²_(m)² +C2|α|a(T)^1/2kω_hkXT, for some positive constants C₁, C₂. If we now take T > 0 small enough so that C₂|α|a(T)^1/2 ≤ 1/2, we see that the right-hand side of (3.7) is a strict contraction in X_T. We deduce that (3.7) has a unique solution, which satisfies kω_hk_XT ≤ 2C₁kω⁰_hk_L2(m)². Since ω_h(t) = e^tLα,hω⁰_h by construction, this proves (3.6) for T sufficiently small, and the general case follows due to the

semigroup property. This concludes the proof.