(1) The Burgers vortex [1] is the steady state solution to (1) of the form VrB(x

ON STABILITY OF BLOW-UP SOLUTIONS OF THE BURGERS VORTEX TYPE FOR THE NAVIER-STOKES EQUATIONS WITH A LINEAR STRAIN

YASUNORI MAEKAWA, HIDEYUKI MIURA, AND CHRISTOPHE PRANGE

AbstractWe study the three-dimensional Navier-Stokes equations in the presence of the axisymmetric linear strain, where the strain rate depends on time in a specific manner. It is known that the system admits solutions which blow up in finite time and whose profiles are in a backward self-similar form of the familiar Burgers vortices. In this paper it is shown that the existing stability theory of the Burgers vortex leads to the stability of these blow-up solutions as well. The secondary blow-up is also observed when the strain rate is relatively weak.

Keywords Navier-Stokes equations·blow-up solutions·Burgers vortex·stability·back- ward self-similarity

Mathematics Subject Classification (2010) 35A01, 35B35, 35B40, 35B44, 35Q30, 35Q35, 76D05, 76E30

1. INTRODUCTION

One of the important mechanisms in three-dimensional turbulent flows is the vorticity amplification due to stretching and vorticity dissipation due to viscosity. As a simple model, the vorticity amplification induced by linear straining flows has been widely studied. A famous example is the Burgers vortex [1], which describes a vortical structure localized in a tubelike domain due to the background axisymmetric linear strain (see also [26,12,21,22]

for the study when the linear strain is not necessarily axisymmetric). Let us consider the three-dimensional Navier-Stokes equations for viscous incompressible flows

∂tV −∆V +∇P+V · ∇V = 0 ∇ ·V = 0, t >0, x∈R³. (1)

The Burgers vortex [1] is the steady state solution to (1) of the form V^rB(x) = γ

2





−x₁

−x₂ 2x₃



 + α 1−e^−γ|x0|

2

4

2π|x⁰|²





−x₂ x1

0



, x⁰ = (x1, x2), P^rB(x) =−1

2|V^rB(x)|²− α²γ 16π²

ˆ ∞ γ|x⁰|²

1

r(1−e^−r4)e^−r4 dr, (2)

whereγ >0andα∈Rare given constants which respectively represent the strain rate and the circulation at infinity. The corresponding vorticity fieldΩ^rB=∇ ×V^rBis given by

Ω^rB(x) =



 0 0 αγg(γ¹²x⁰)



, g(X⁰) = 1 4πe^−|X0|

2 4 . (3)

Date: April 8, 2019.

1

To fix the notation, let

(4) U^G(x⁰) = 1−e^−|x0|

2 4

2π|x⁰|²





−x₂ x₁

0



 and G(x⁰) =



 0 0 g(x⁰)



.

The stability of the Burgers vortex is studied in detail by now, and remarkably, it is stable for any γ > 0and allα ∈ R, locally with respect to three-dimensional perturbations and globally with respect to two-dimensional perturbations. Indeed, the local two-dimensional stability for small circulation is proved by Giga-Kambe [14] and global stability for arbitrary size of circulation is proved by Gallay-Wayne [10]. The three-dimensional local stability for small circulation is shown by Gallay-Wayne [11], and the restriction on the size of circulation is removed by Gallay-Maekawa [7] also for the three-dimensional perturbations.

The reader is referred to the survey of Gallay-Maekawa [8] about the existence and the stability problem related to the Burgers vortex.

Although the Burgers vortex presents in a simple way a nontrivial swirling (whenα6= 0) flow exhibiting the balance between the vorticity stretching and dissipation, it is a nonde- caying (or even growing) solution to the Navier-Stokes equations (1), and it is well known that we do not have uniqueness for such flows in general. A typical example is given by the so-called parasitic solutions. Let ρ : [0,∞) → Rwithρ(0) =ρ₀ be a bounded differen- tiable function. Then

V(x, t) =Cρ(t), P(x, t) =−ρ⁰(t)C·x, C ∈R³

is a solution to (1) with initial dataCρ₀. If one considers the initial data in the form of the linear strainρ₀(−x₁,−x₂,2x₃)^>one finds that

(5) V^lin(x, t) =ρ(t)





−x₁

−x₂ 2x₃



, P(x, t) =−¹₂|V^lin(x, t)|²−¹₂∂_tV^lin(x, t)·x is a solution to (1), with a pressure growing quadratically. In these examples, the function ρ(t) is in principle taken arbitrary, and in particular, one may take it in a singular way so that it blows up at a finite time T∗ > 0. In view of the linear strainV^lin above, it is then natural to look for a solution of the Burgers vortex type but with a time-dependent linear strain which blows up in a finite time, though the functionρ(t)for the strain rate cannot be arbitrary any longer and should be chosen suitably in this case. Significant contributions in this direction have been made by Moffatt [19] and Ohkitani-Okamoto [29]. Indeed, Moffatt [19] provided a family of blowing-up solutions as follows: forµ >1andα∈R,

V^sB(x, t) = µ 2(T^∗−t)





−x₁

−x₂ 2x₃



 + α q

β_µ(t)U^G q

β_µ(t)x⁰

, β_µ(t) = µ−1 T^∗−t, P^sB(x, t) = −¹₂∂tV^lin(x, t)·x−1

2|V^sB(x, t)|²−α²β_µ(t) 16π²

ˆ ∞ βµ(t)|x⁰|²

1

r(1−e^−r4)e^−r4 dr, (6)

whereV^linis (5) withρ(t) = _2(T^µ∗−t) and the formula of the pressure is obtained by using the general identity u· ∇u = ¹₂∇|u|²−u×ω withω = ∇ ×u, by also noting the fact thatU^G×(∇ ×U^G) = 0and∇ ×V^lin = 0. Indeed, by noticing that the corresponding vorticity field is

Ω^sB(x, t) =αβµ(t)G q

βµ(t)x⁰ (7)

withGdefined in (4), we have

∂tV^sB +V^sB · ∇V^sB−∆V^sB

=∂_tV^lin+∂_t α

q

β_µ(t)U^G q

β_µ(t)x⁰ +1

2∇|V^sB|²−V^sB ×Ω^sB+∇ ×Ω^sB

=∂_tV^lin+1

2∇|V^sB|²−α q

β_µ(t)U^G q

β_µ(t)x⁰

×Ω^sB +∂t

α

q

βµ(t)U^G q

βµ(t)x⁰

−V^lin×Ω^sB +∇ ×Ω^sB.

Then the first three terms in the right-hand side are written as the potential form and hence define the pressure as stated above. On the other hand, we see from∂_tp

β_µ(t) =

1

2(µ−1)β_µ(t)³²,

∂_t α

q

β_µ(t)U^G q

β_µ(t)x⁰

=αβ_µ(t)³² µ−1

1 2U^G

q

β_µ(t)x⁰ +1

2 q

β_µ(t)x⁰· ∇⁰U^G q

β_µ(t)x⁰ and

−V^lin×Ω^sB +∇ ×Ω^sB = αβµ(t)² 2(µ−1)g(

q

βµ(t)x⁰)



 x₂

−x₁ 0





= αβµ(t)³² µ−1 ∆U^G

q

βµ(t)x⁰ .

Thus the conclusion holds from the identity∆U^G(ξ⁰) +¹₂ξ⁰· ∇⁰U^G(ξ⁰) +¹₂U^G(ξ⁰) = 0.

The flow (6) blows up in a backward self-similar form and has a very similar vortical structure to the Burgers vortex. In this paper we callV^sB the singular Burgers vortex. Let us notice that, due to the presence of the linear strain, the singular Burgers vortex is out of the theory of Caffarelli-Kohn-Nirenberg [2] forε-regularity for Navier-Stokes equations, and of Neˇcas-Ruˇziˇcka- ˇSver´ak [28] and Tsai [38] for the non existence of backward self- similar solutions.

What is interesting in (6) is the restrictionµ >1, that is, the strain rate has to be strong enough to define the singular Burgers vortex as arealflow. For the specific choice of the circulation numberα, Ohkitani-Okamoto [29] provided an alternative interpretation for this time-dependent strain rate in the singular Burgers vortex. Indeed, whenα=α_µ= _µ−1^4πµ one finds that the identity _2(T^µ_∗−t) = ^k∇×VsB₂^(·,t)kL∞ holds, that is, the strain rate behaves as if it depends on the unknown variable, i.e., theL^∞norm of the vorticity field. This gives an interesting perspective in view of the Taylor expansion aboutx of the velocity around the origin. The analysis in this direction has been developed further by Nakamura-Okamoto- Yagisita [27] and Okamoto [30].

From now on we focus our attention on the singular Burgers vortexV^sB. Without loss of generality, we normalize the blow-up time as1, i.e.,T^∗ = 1. The aim of this paper is to study the asymptotic stability of the explicit blowing-up solutions (6)-(7). To simplify the notations we set

U_µ(x⁰, t) = q

β_µ(t)U^G q

β_µ(t)x⁰ , Gµ(x⁰, t) =βµ(t)G

q

βµ(t)x⁰ .

Let us go back to (1) and recall that the vorticity fieldΩ =∇ ×V satisfies the equations

∂tΩ−∆Ω +V · ∇Ω−Ω· ∇V = 0, ∇ ·Ω = 0, (8)

which are formally equivalent to (1). We note that the divergence free condition∇·Ω = 0is preserved under the evolution equation in (8), and thus, if the initial vorticity is divergence free then the second equation in (8) is automatically satisfied. Hence we will always drop the divergence free condition for the vorticity field from now on. To study the stability of (6)-(7), we consider the solution(Ω, V)to (8) of the form(Ω, V) = (Ω^sB+ω, V^sB+u).

Then we have from (8) the evolution equations for the perturbation vorticityω, which reads

∂_tω−∆ω+ µ 1−t

M x· ∇ω−M ω

+αΛ_µ(t)ω =−u· ∇ω+ω· ∇u ω|_t=0 =ω₀,

(9) and

Λµ(t)ω=Uµ(t)· ∇ω+u· ∇G_µ(t)−ω· ∇U_µ(t)−Gµ(t)· ∇u.

(10)

Here the matrixM is given by

M =





−¹₂ 0 0 0 −¹₂ 0

0 0 1



,

and the perturbation velocityuis formally recovered from the vorticityωby the Biot-Savart law

u(x, t) =− 1 4π

ˆ

R³

(x−y)×ω(y, t)

|x−y|³ dy= K_3D∗ω(t) (x), (11)

by assuming a suitable spatial decay onω. The goal of this paper is to study the behavior of ω up toT^∗ = 1 for a suitable class of the initial data ω0. In particular, to show the asymptotic stability of the blowing-up solution (Ω^sB, V^sB), we aim at establishing (1− t)¹²ku(t)k_L^∞ →0ast↑1.

As in the work of Lundgren [18] for the strained Navier-Stokes equations and of Giga- Kohn [15] for the nonlinear heat equation, it is convenient to introduce the self-similar variables:

τ =−(µ−1) log(1−t), ξ = q

βµ(t)x, βµ(t) = µ−1 1−t , and

u(x, t) = q

βµ(t)V(ξ, τ), ω(x, t) =βµ(t)W(ξ, τ).

(12)

Then the system forW is written as

∂_τW −(L_µ−αΛ)W =−V · ∇W+W · ∇V τ >0, ξ∈R³, W|_τ=0=W₀,

(13) where

LµW = ∆W+ξ⁰

2 · ∇⁰W − 2µ+ 1

2(µ−1)ξ3∂3W+ 1

µ−1(µM−I)W (14)

and

ΛW =U^G· ∇W+V · ∇G− W · ∇U^G−G· ∇V.

(15)

HereV = K_3D ∗ W, andU^G andGare defined in (4). Hence, by using the self-similar variables(ξ, τ)the study of the behavior ofωaround the blow-up timeT^∗ = 1is translated into the large time behavior ofW, and the problem shares common features in essence with

the stability problem of the Burgers vortex for which a detailed analysis was done. Indeed, if we set the differential operatorLas

Lw= ∆w+ ξ⁰

2 · ∇⁰w−ξ3∂3w+M w, (16)

which is formally obtained by taking the limit µ → ∞ in (14), then the linear operator L−αΛ is exactly the linearized operator around the Burgers vortex with circulationα.

It is studied in [7] for all circulation numberα ∈ R, and the uniform spectral gap of−¹₂ is achieved for all α in a suitable functional setting. Since the only difference between L_µ and L is the stretching term related to ξ₃∂₃, it is natural to expect that the operator Lµ−αΛalso has a uniform spectral gap under the similar functional framework as in [7].

This implies that the original blowing-up solution (6)-(7) is asymptotically stable under the small perturbations as long asµ > 1. Our main theorem is the following stability result stated in Theorem1below, which is in the spirit of stability results for blowing-up solutions of the nonlinear heat equation [25] or of dispersive equations [24]. As far as the authors know, this is the first result of this type for the Navier-Stokes equations, though the key stability mechanism is brought by the singular linear strain and hence the spatial growth of the solution plays an important role. The function spaces used in the theorem are defined in the paragraphNotationsbelow. It is shown thatLµ−αΛgenerates a semigroupe^{τ(Lµ−αΛ)} in the function spaceX(m)defined inNotationsand then the mild solutionWto (13) (with the initial dataW₀) is defined as the solution to the integral equation

W(τ) =e^{τ(Lµ−αΛ)}W₀+ ˆ _τ

0

e^{(τ−s)(Lµ−αΛ)}

− V · ∇W+W · ∇V ds, V =K_3D∗ W.

Then we callωthe mild solution to (9) whenωis given by the transformation (12) forW.

Theorem 1(stability of the singular Burgers vortex). Letα∈R,m∈(2,∞],µ∈(1,∞).

Letω0,2d∈L²₀(m). Then there existsε=ε(α, m, µ, ω0,2d)∈(0,1)such that the following statement holds. For all divergence-freeω₀ ∈X(m)satisfying

ω₀ = (0,0, ω_0,2d(x⁰)) +ω_0,3d(x⁰, x₃), kω_0,3dk_X(m)≤ε,

there exists a unique mild solution ω ∈ L^∞_loc([0,1);X(m)) to (9) such that t¹²∂x^βω ∈ L^∞_loc([0,1);X(m))with|β| ≤1. This solution satisfies

lim sup

t→1

(1−t)^12−µ−12 ku(·, t)k_L∞(R³)<∞, whereu=K_3d∗ω. Moreover, ifα6= 0, letting

λi =−p µ−1

ˆ

R²

xiω_0,2d(x⁰)dx⁰, i={1,2}, we have the following asymptotic estimate foru:

(17) lim sup

t→1

(1−t)^12−µ−12

u(·, t)−(1−t)^µ−12 X

i=1,2

λ_i

rµ−1

1−t(∂_iU^G)

rµ−1 1−t·

L^∞(R³)

≤Cε.

HereCdepends onα,m,µ, andω_0,2d.

Remark 1. (1) Let us notice thatV = V^sB +u is a solution to the Navier-Stokes system (1). A way to rephrase the asymptotic expansion (17) is as follows:

V(x, t) =V^sB(x, t) +p

µ−1(1−t)^µ2−1 X

i=1,2

λi(∂iU^G)

rµ−1 1−tx +OL^∞ (1−t)^µ2−1

+oL^∞ (1−t)^µ2−1 .

Hence in the vicinity ofV^sB, we have constructed a whole family of blowing-up solutions.

Moreover, when α, λi 6= 0, if the strain is sufficiently weak so thatµ ∈ (1,2), then we identify a leading part (in ε) of the secondary blow-up profile in the sense that the term

√µ−1(1−t)^µ2−1P

i=1,2λi(∂iU^G) qµ−1

1−tx

blows up as well in theL^∞norm. In fact, as seen in the proof (and the last statement of Theorem2and its remark below) the secondary blow up profile is still a linear combination of √

µ−1(1−t)^µ2−1(∂_iU^G) qµ−1

1−tx but in a weaker topology. Precisely, we can show that there existd_i ∈R,i= 1,2, such that

limt→1(1−t)^12−µ−12 u(·, t)

−(1−t)^µ−12 X

i=1,2

(λ_i+d_i)

rµ−1

1−t(∂_iU^G)

rµ−1 1−t ·

L^∞_x3([−N,N];L^∞_x0)= 0, (18)

for anyN > 0. Hered_i satisfies|d_i| ≤Ckω_0,3dk_X(m) 1withCdepending only onα, m,µ, andω_0,2d.

(2) In Theorem 1the initial dataω0,2d ∈ L²₀(m) is taken arbitrary, while we do not have a quantitative information between kω_0,2dk_L2(m) and the small constant for the three- dimensional perturbation. On the other hand, for small initial data in X(m) the con- dition in Theorem 1 can be stated in a more quantitative way as follows; there exists = (α, m, µ) > 0 such that, if the (divergence-free) initial data ω₀ ∈ X(m) satisfies kω₀k_X(m) ≤ , then the stability estimate such as (17) or (18) for the solution is verified with a constantC depending only on α,m, andµ(hereC is taken independently ofω₀, sincekω₀k_X(m)≤andis small enough).

(3) It should be emphasized that the uniqueness of the solution in Theorem 1is claimed for the equation (9), and not for the original Navier-Stokes equation (1). Indeed, as already explained, we do not have the uniqueness of solutions to (1) for nondecaying initial data.

Roughly speaking, the uniqueness holds for the class of solutions having the formV = V^sB+uwith decaying (in the horizontal direction)u.

Outline of the paper. In Section2we handle the stability analysis of the linear equation

∂_τw−(L_µ−αΛ)w= 0.

The goal of the analysis is to extend the arguments of Gallay-Maekawa [7] to the case of the operatorLµ−αΛ. Section3is devoted to the proof of nonlinear stability. More precisely, we prove that the zero solution of the nonlinear equation (13) is stable under arbitrarily large two-dimensional perturbations, and small genuinely three-dimensional perturbations, see Theorem 2. Such a result is in the spirit of [31], though the technique we use is dif- ferent. Instead of continuing the solution via a blow-up criteria as in [31], we construct a mild solution iteratively until the source becomes small enough for global in time solutions to exist. We conclude this by investigating the existence of a secondary blow-up profile.

Theorem1immediately follows from the results of Section3by scaling back to the original variables(x, t).

Notations. As is usual, we always decomposex= (x⁰, x₃)∈R³orξ = (ξ⁰, ξ₃)∈R³into horizontal componentx⁰ = (x1, x2) ∈ R² orξ⁰ = (ξ1, ξ2) ∈ R², and vertical component x₃ orξ₃. Similary, we write∇⁰ = (∂_x1, ∂_x2)and∆⁰ = ∂_x²₁ +∂_x²₂. Throughout the paper, we work in the following functional setting. Form∈[0,∞]set

ρ_m(r) =







1, m= 0,

(1 +_4m^r )^m, 0< m <∞, e^r4, m=∞.

Forp∈[1,∞), we define the weighted spaces L^p(m) =

n

w∈L^p(R²)| kwk²_Lp(m)= ˆ

R²

|w(x⁰)|^pρm(|x⁰|²)^p2 dx⁰ o

, L^p₀(m) =n

w∈L^p(m)| ˆ

R²

wdx⁰ = 0o

for m >2−²_p. Moreover, we use the following product spaces: forp∈[1,∞)andm >2−²_p

X^p(m) =BC(R;L^p(m)), X₀^p(m) =BC(R;L^p₀(m)), with kφk_Xp(m)= sup

x3∈R

kφ(·, x₃)k_Lp(m), X^p(m) =X^p(m)×X^p(m)×X₀^p(m),

where “BC(R;Y)” denotes the space of all bounded and continuous functions fromRto a Banach spaceY. We denote byX_loc^p (m)the subspace ofX^p(m)endowed with the topology given by the seminorms(k · k_X^p

n(m))n∈Ndefined by kφk_X^p

n(m)= sup

|x₃|≤n

kφ(·, x₃)k_Lp(m), ∀φ∈X^p(m), n∈N.

We then set in analogy with aboveX^p_loc(m) = X_loc^p (m)×X_loc^p (m)×X_loc,0^p (m), where X_loc,0(m) isX₀^p(m) equipped with the topology ofX_loc^p (m). When p = 2andm > 1, we use the abbreviationsX(m),X0(m)andX(m)and analogously for the “loc” versions.

These spaces are used in [7].

2. ANALYSIS OF THE LINEARIZED OPERATOR

This section is centered on the analysis of the semigroupe^{τ(Lµ−αΛ)}for the linear evolu- tion. The main result is the following.

Proposition 2(linear stability). Letα ∈R,m∈(1,∞],µ∈(1,∞)andp ∈[1,2]. Then for allη ∈(0,¹₂]withη < ^m−1₂ ,κ ∈(0, η+ _2(µ−1)^2µ+1 ), andβ ∈N³, there exists a constant C(α, m, µ, κ, η, β)<∞such that

k∂_ξ^β(e^{τ(Lµ−αΛ)}w0)⁰k_X(m)2 ≤ Ce^−(κ+

2µ+1 2(µ−1)β3)τ

a(τ)^p1−12+

|β|

2

kw₀k_Xp(m)

(19)

k∂_ξ^β(e^{τ(Lµ−αΛ)}w₀)₃k_X(m)≤ Ce^−(η+

2µ+1 2(µ−1)β3)τ

a(τ)^1p−12+

|β|

2

kw₀k_Xp(m)

(20)

for all divergence-freew₀ ∈X^p(m)andτ ∈(0,∞), wherea(τ) = 1−e^−τ. Moreover, ˆ

R²

(e^{τ(Lµ−αΛ)}w0)3(ξ⁰, ξ3)dξ⁰ = 0,

and∇ ·w₀ = 0implies∇ ·e^{τ(Lµ−αΛ)}w₀ = 0for allτ ∈(0,∞). Ifη < ¹₂ thenκis taken asκ=η+_2(µ−1)^2µ+1 .

Corresponding estimates for the Burgers votex are obatined and used in [7]. The gain in the decay forξ3derivatives in (19) and (20) is due to the commutation property stated in (40) below, which is already used in [3,35,36,7] . This property is an effect of the stretching in the vertical direction due to the structure of the linear strain, which is a key stabilizing effect of the Burgers vortex. Another remark concerns the transient growth. There is a factora(τ)^−(1p−12+

|β|

2 )

related to the parabolic-type smoothing effect ofe^{τ(Lµ−αΛ)}, which is large in short time. This factor does not depend onα. The constantC in (19) and (20), though, gets large whenα→ ∞,µ→1⁺. We note that the estimate ofe^{τ(Lµ−αΛ)}for local time is not difficult to show:

k∂_ξ^βe^{τ(Lµ−αΛ)}w0k_X(m)≤ C a(τ)^1p−12+

|β|

2

kw₀k_Xp(m), 0< τ ≤1, p∈[1,2].

(21)

HereCdepends only onα,µandm; see, e.g., the argument of [7, Proposition 4.2]. Thus, by recalling the semigroup property, we may focus on the estimates (19)-(20) but only for τ ≥1andp= 2.

The argument to achieve the linear stability is rather parallel to the one of Gallay-Maekawa in [7]. The idea is to decompose the full operatorL_µ−αΛinto a dominant two-dimensional (but vectorial) part and a three-dimensional part whose contribution to the solution decays fast and is negligible in the longtime. Hence we first focus on the operator L_µ. Second, we address the vectorial 2d problem, i.e. on the action ofLµ−αΛon fieldsw = w(x⁰) independent ofx3. Finally, we analyze the full operatorLµ−αΛusing the stretching in the vertical direction which makes the three-dimensional part of the solution decay fast.

2.1. Analysis of Lµ. Let us start from the analysis of Lµ. We first rewrite the operator similarly to [7] so as to make the comparison easier. We first expand (14) as

Lµ= ∆⁰+ξ⁰

2 · ∇⁰+∂²₃− 2µ+ 1

2(µ−1)ξ3∂3+







−_2(µ−1)^µ+2 0 0

0 −_2(µ−1)^µ+2 0

0 0 1





. Hence, we can write the action ofLµin the following form

(22) Lµw=

L_µ,hw⁰ L_µ,3w₃

=

L_hw⁰+L₃w⁰− 1 +_2(µ−1)^µ+2 w⁰ L_hw3+L₃w3

, where

L_h = ∆⁰+ξ⁰

2 · ∇⁰+ 1, L₃ =∂₃²− 2µ+ 1

2(µ−1)ξ3∂3.

According to [11, Appendix A], the operatorL_h is the generator of a strongly continuous semigroup inL²(m)given by the explicit formula

(23) (e^τLhf)(ξ⁰) = e^τ 4πa(τ)

ˆ

R²

e⁻

|ξ0−η0|2

4a(τ) f(η⁰e^τ2)dη⁰, a(τ) = 1−e^−τ,

for all τ ∈ (0,∞). Moreover, it is well known that−L_h is self-adjoint in L²(∞) and satisfies the following lower bounds (cf. [10, Lemma 4.7]):

−L_h ≥0 inL²(∞), −L_h ≥ 1

2 inL²₀(∞), −L_h ≥1 inL²₁(∞), (24)

whereL²₁(∞) ={f ∈L²₀(∞)| ´

R²ξjf dξ = 0, j = 1,2}. As forL₃, it is the generator of a semigroup of contractions inBC(R)with the following explicit formula, which is derived

from [11, Appendix A]) for the semigroup generated by the differential operator of the form

∂₃²−bξ3∂3−b.

(25) (e^τL3f)(ξ₃) = 2χ 4π(e^2χτ−1)

¹₂ ˆ

R

exp

−2χ|ξ₃−η₃|² 4(e^2χτ −1)

f(η₃e^−χτ)dη₃, for allτ ∈(0,∞), where

(26) χ=χ_µ= 2µ+ 1

2(µ−1). Notice thatχ >1. We can rewrite formula (25):

(27) (e^τL3f)(ξ₃) =

χ 2πa(2χτ)

¹₂ˆ

R

e⁻

χ|e−χτ ξ3−η3|2

2a(2χτ) f(η₃)dη₃, a(τ) = 1−e^−τ, for allτ ∈ (0,∞). From the previous formula, we immediately obtain the fast temporal decay of derivatives ofe^τL3f. Indeed, for allk∈N, there exists a constantC(k)<∞such that for allτ ∈(0,∞),

(28)

∂₃^ke^τL3f

L^∞(R)≤Cχ^k2e^−χkτ a(2χτ)^k2

kfk_L^∞_(R), whereχis defined in (26). Moreover, we also have from (27),

τ→∞lim

e^τL3f − χ 2π

¹₂ ˆ

R

e^−χ|η3|

2

2 f(η₃)dη₃

L^∞([−e^(χ−δ)τ,e^(χ−δ))τ]) = 0, (29)

for anyδ >0.

The fast decay (28), due to the strong stretching in the vertical direction, plays a key role in the fact that the linear evolution becomes independent ofx₃at the main order. Hence, the leading dynamics in the longtime is driven by the2d vectorial problem, which we analyze in the next subsection.

2.2. Localization in the horizontal direction: the2d vectorial problem. In this subsec- tion we analyze the2d vectorial problem, which corresponds with the action ofe^{τ(Lµ−αΛ)} on the functions of the formw = w(ξ⁰) ∈ R³, i.e.,∂3w = 0. We define forα ∈ Rand µ∈(1,∞),

(30) Lµ,αw=

Lµ,α,hw⁰ Lµ,α,3w3

= (L_h−1−_2(µ−1)^µ+2 )w⁰−α(Λ1−Λ˜2)w⁰ L_hw3−α(Λ1+ ˜Λ3)w3

! . Here

Λ₁w=U^G· ∇w= (U^G)⁰· ∇⁰w, Λ˜2w⁰ =w⁰· ∇⁰(U^G)⁰,

Λ˜₃w₃ = (K_2d∗w₃)· ∇⁰g.

The operatorsΛ₁ andΛ˜₃ come from the vorticity transport, whileΛ˜₂ originates from the vortex stretching term. One can check that Lµ−αΛw = Lµ,αw when ∂3w = 0; see identity (42) and inequality (43). Notice that the horizontalw⁰and the vertical components w₃ are completely decoupled. This makes the2d vectorial problem more tractable than the full original one. The operatorLµ,α,3 and the associated semigroup inL²₀(m)are already analyzed in [10, Section 4]. The main result of this section is stated as follows.

Proposition 3 (2d vectorial problem). Let α ∈ R, m ∈ (1,∞], µ ∈ (1,∞). Then for all κ ∈ (0,1 + _2(µ−1)^µ+2 ), for allη ∈ (0,¹₂]such that η < ^m−1₂ , there exists a constant C(α, m, µ, κ, η)<∞such that

ke^τLµ,α,hw_0,hk_L2(m)² ≤Ce^−κτkw_0,hk_L2(m)², ke^τLµ,α,3w_0,3k_L2(m)≤Ce^−ητkw_0,3k_L2(m),

for all τ ∈ [0,∞) and for all w₀ ∈ L²(m)² ×L²₀(m). Moreover, if m > 2 then for η ∈(¹₂,1]withη < ^m−1₂ ,

e^τLµ,α,3w_0,3−e^−τ2 X

i=1,2

θ_i∂_ig

L²(m)≤Ce^−ητkw_0,3k_L2(m). (31)

Hereθi =−´

R²ξiw0,3(ξ⁰)dξ⁰andg(ξ⁰) = _4π¹ e^−|ξ0|

2 4 .

Note that the result fore^τLµ,α,3w_0,3in Proposition3is due to [10, Section 4], in particular [10, Proposition 4.12]. The key observation there is that for suitably largemthe spectrum of Lµ,α,3 in L²(m) near the imaginary axis consists of the isolated eigenvalues whose eigenfunctions actually belong toL²(∞), and thus, the analysis of the large time behavior ofe^τLµ,α,3 inL²(m)is essentially reduced to the analysis inL²(∞), in whichL_h is self- adjoint and moreoverΛ1+ ˜Λ3 is skew-symmetric; see [10, Lemma 4.8]. Then, the lower bounds in (24) enable us to conclude the expansion (31) inL²₀(m)for large enoughm, by also using the fact that the eigenspace of the eigenvalue−¹₂ ofLµ,α,3inL²(∞)is spanned by∂ig,i= 1,2. On the other hand, the estimate ofe^τLµ,α,h is obtained in the same manner as in [7, Proposition 3.1], as sketched below for reader’s convenience. The following lemma is the key for the study ofe^τLµ,α,h. Letress(A;X)be the radius of the essential spectrum of a bounded linear operatorAonX; see [4, IV-1.20].

Lemma 4. Letm∈(1,∞]. Then the following statements hold.

(i)r_ess(e^τLµ,α,h;L²(m)) = (

e⁻⁽¹²⁺

µ+2 2(µ−1)+^m₂)τ

, m6=∞

0, m=∞ .

(ii) Ifλ ∈ CwithReλ ≥ −¹₂ − _2(µ−1)^µ+2 − ^m₂ is an eigenvalue ofLµ,α,h inL²(m)² then Reλ≤ −1−_2(µ−1)^µ+2 .

The proof of Lemma 4 (i) is identical to the proof of [7, Proposition 3.3]. Indeed, we see that the operator ∆_α(τ) = e^τLµ,α,h −e^τ(Lh−1−

µ+2 2(µ−1))

is compact inL²(m)² for any m ∈ (1,∞]. Hence, Weyl’s theorem implies that both semigroups have the same essential spectrum and hence have same essential radii: for all m ∈ (1,∞], we have ress(e^τLµ,α,h;L²(m))² = ress(e^τ(Lh−1−

µ+2 2(µ−1))

;L²(m)²). Then the fact ress(e^τ(Lh−1−

µ+2 2(µ−1))

;L²(m)) = (

e⁻⁽¹²⁺

µ+2 2(µ−1)+^m₂)τ

, m6=∞

0, m=∞ ,

which was proved in [11, Appendix A], yields the statement (i) of Lemma 4. Next, the proof of Lemma4(ii) is sketched below. By a standard argument [7, Proposition 3.4 and Section 6.2], every eigenfunction inL²(m)² associated to an eigenvalueλwithRe(λ) >

−¹₂ − _2(µ−1)^µ+2 − ^m₂ belongs toL²(∞)², i.e. has Gaussian decay. It is therefore enough to study the discrete spectrum of Lµ,α,h inL²(∞)², for which the same argument as in [7, Proposition 3.5] is applied as follows. The eigenfunctionw⁰ ∈ L²(∞)² associated to the

eigenvalueλsatisfies, by its definition, (32) λw⁰ =L_hw⁰−

1 + µ+ 2 2(µ−1)

w⁰−α(U^G)⁰· ∇⁰w⁰+αw⁰· ∇⁰(U^G)⁰.

By the direct computation we also have the equations that are respectively satisfied byξ⁰·w⁰ and∇⁰·w⁰:

λξ⁰·w⁰ =L_h(ξ⁰·w⁰)−2∇⁰·w⁰− 1

2ξ⁰·w⁰−

1 + µ+ 2 2(µ−1)

ξ⁰·w⁰−αU^G· ∇⁰(ξ⁰·w⁰), (33)

λ∇⁰·w⁰ =L_h(∇⁰·w⁰) +1

2∇⁰·w⁰−

1 + µ+ 2 2(µ−1)

∇⁰·w⁰−αU^G· ∇⁰(∇⁰·w⁰).

(34)

The upper bound of Reλ is obtained from these identities (32), (33), and (34). Indeed, testing the equation (32) againstw⁰(complex conjugate ofw⁰), we obtain

(35) Reλkw⁰k² =hL_hw⁰, w⁰i −

1 + µ+ 2 2(µ−1)

kw⁰k² + 2αRe

ˆ

R²

e^|ξ0|

2

4 (ξ⁰·w⁰)(ξ^0⊥·w⁰)∂r(u^g)(|ξ⁰|²)dξ⁰

, where we used the skew-symmetry ofw⁰ 7→U^G· ∇⁰w⁰inL²(∞)²equipped with the scalar producth·,·idefined by

hw⁰¹, w⁰²i= ˆ

R²

e^|ξ0|

2

4 w⁰¹w⁰²dξ⁰. Similarly, we have from (33) and (34),

(36) Reλkξ⁰·w⁰k² =hL_h(ξ⁰·w⁰), ξ⁰·w⁰i − 3

2+ µ+ 2 2(µ−1)

kξ⁰·w⁰k²

−2Reh∇⁰·w⁰, ξ⁰·w⁰i, and

(37) Reλk∇⁰·w⁰k²=hL_h(∇⁰·w⁰),∇⁰·w⁰i − 1

2+ µ+ 2 2(µ−1)

k∇⁰·w⁰k².

Now suppose that∇⁰·w⁰is not identically zero. Then (37) and (24) with the fact∇⁰·w⁰ ∈ L²₀(∞)(i.e.,−L_h ≥ ¹₂ inL²₀(∞)) implyReλ ≤ −1− _2(µ−1)^µ+2 . Next suppose that∇⁰·w⁰ is identically zero but thatξ⁰ ·w⁰ is not identically zero. In this case (36) and (24) (i.e.,

−L_h≥0inL²(∞)) implyReλ≤ −³₂ −_2(µ−1)^µ+2 . Finally, suppose thatξ⁰·w⁰is identically zero. Then (35) and (24) give the boundReλ≤ −1−_2(µ−1)^µ+2 . Therefore, we conclude

(38) Reλ≤ −1− µ+ 2

2(µ−1). The statement of Lemma4(ii) is proved.

The estimate ofe^τLµ,α,hstated in Proposition3follows from Lemma4and the standard theory ofC₀-semigroup [4, Corollary IV-2.11], and we conclude that the growth bound of e^τLµ,α,h inL²(m)²,m ∈ (1,∞], is estimated from above by−1− _2(µ−1)^µ+2 . Thus, for any κ∈(0,1 + _2(µ−1)^µ+2 )we haveke^τLµ,α,hw0,hk_L2(m)² ≤ Ce^−κτkw_0,hk_L2(m)², which proves Proposition3.

We stress that, in Proposition3,wis a function of the horizontal variableξ⁰ ∈R² only.

Notice that we do not yet have the restrictionκappearing in Proposition2. This additional restriction comes from the analysis of the full three-dimensional problem.

2.3. The regularizing effect of vertical stretching: full3d linear stability problem. In this subsection we complete the proof of Proposition 2. As is mentioned in the beginning of Section2, we may focus on the caseτ ≥1andp= 2. Our first goal is to show

ke^{(τ+1)(Lµ−αΛ)}w0k_X(m)≤Ce^−ητkw₀k_X(m), (39)

which in particular proves (20) with β = 0 and p = 2 for the vertical component of e^{τ(Lµ−αΛ)}w₀. To this end we see that, due to the stretching effect in the vertical direc- tion, the longtime dynamics of the semigroup e^{τ(Lµ−αΛ)} is dominated by the2d vectorial problem analyzed in Proposition 3. The following result shows a simple but important stabilizing effect brought by the linear strain to realize this idea.

Lemma 5. We have the following commutation property: for everyµ >1andα∈R, [∂3, Lµ−αΛ] = [∂3,L₃] =− 2µ+ 1

2(µ−1)∂3.

As a consequence, we have for allµ >1,α∈R,k∈N, for allτ ∈(0,∞), (40) ∂₃^ke^{τ(Lµ−αΛ)}=e⁻

2µ+1 2(µ−1)kτ

e^{τ(Lµ−αΛ)}∂₃^k.

This property ofL₃ is due to the stretching in the vertical direction and plays a crucial role in reducing the longtime dynamics to the2d vectorial problem studied above. Indeed, since it is not difficult to show the naive boundke^{τ(Lµ−αΛ)}k_X(m)→X(m) ≤ C1e^C0τ for all τ >0, whereC₀ andC₁may depend onα,µ, andm, (40) gives the bound

k∂₃^ke^{(τ+1)(Lµ−αΛ)}w₀k_X(m)≤C₁e^C0τ−

2µ+1 2(µ−1)kτ

k∂₃^ke^(Lµ−αΛ)w₀k_X(m)

≤C_1,ke^C0τ−

2µ+1 2(µ−1)kτ

kw₀k_X(m). (here(21)is used) Hence, ifk₀is large enough depending onC₀, we have

k∂₃^k0e^{(τ+1)(Lµ−αΛ)}w₀k_X(m)≤Ce⁻

2µ+1

2(µ−1)τkw₀k_X(m), τ >0.

(41)

To obtain the decay estimate (39) fore^{(τ+1)(Lµ−αΛ)}, rather than for∂₃^k0e^{(τ+1)(Lµ−αΛ)}, we decompose the operatorL_µ−αΛinto two-dimensional part and three-dimensional part as follows:

(42) (L_µ−αΛ)w=Lµ,αw+L₃w−α





0 0 Λ₃w−Λ˜₃w₃



+αΛ₄w, whereLµ,αis defined in (30) and

Λ₃w−Λ˜₃w₃ = (K_3d∗w)⁰−K_2d∗w₃

· ∇⁰g, Λ₄w=g∂₃(K_3d∗w).

The proof of the linear stability for the full three-dimensional problem relies on the de- composition (42) and on the following estimates for the three-dimensional part: for all m∈(1,∞]andσ ∈(0,1), there existsC(m, σ)such that

kΛ₃w−Λ˜₃w₃k_X(m)≤C k∂₃wk_X(m)+kwk^σ

X(m)k∂₃wk^1−σ

X(m)

, kΛ₄wk_X(m)≤Ck∂₃wk_X(m).

(43)

These estimates are exactly given in [7, Proposition 4.5], so we omit the proof of (43). The three-dimensional part is then treated as a perturbation of the2d vectorial problem. The rest of the analysis leading to Proposition2is rigorously identical to [7, Section 4]; namely, it suffices to solve the integral equation

˜

w(τ) :=e^{(τ+1)(Lµ−αΛ)}w₀ =e^{τ(Lµ,α+L3)}

e^(Lµ−αΛ)w₀

−α ˆ _τ

0

e^{(τ−s)(Lµ,α+L3)} (



0 0 Λ₃w˜−Λ˜₃w˜₃



−Λ₄w˜ )

ds , (44)

with the a priori knowledge of the exponential decay of∂₃^k0e^{(τ+1)(Lµ−αΛ)}as in (41). Note that the semigroupe^{τ(Lµ,α+L3)} is factorized ase^τL3 ⊗e^τLµ,α, and hence, the estimate of e^{τ(Lµ,α+L3)} is a consequence of (28) and Proposition 3. Then, by applying also (43) we have for (44),

kw(τ˜ )k_X(m)

≤Ce^−ητkw˜0k_X(m)+C ˆ _τ

0

e^{−η(τ−s)} k∂₃wk˜ _X(m)+kwk˜ ^σ_X(m)k∂₃wk˜ ^1−σ_X(m) (s)ds.

(45)

Here w˜₀ = e^(Lµ−αΛ)w₀ and C depends only on α, µ, m, and σ ∈ (0,1). Then the interpolation inequalityk∂₃wk˜

X(m)≤Ckwk˜ ¹⁻

1 k0

X(m)k∂^k₃⁰wk˜

1 k0

X(m)yields for any∈(0,1), kw(τ˜ )k_X(m)≤Ce^−ητkw˜0k_X(m)+C

ˆ _τ

0

e^{−η(τ−s)} kwk˜ _X(m)+Ck∂₃^k0wk˜ _X(m) (s)ds.

Note that the term´_τ

0 e^{−η(τ−s)}k∂₃^k0w(s)k˜ _X(m)dsis bounded from above byCe^−ητkw₀k_X(m), in virtue of (41) and η ∈ (0,¹₂]. Then, by taking small enough, one can show that kw(τ˜ )k_X(m) ≤ Ce^−η2τkw˜0k_X(m) +Ce^−ητkw₀k_X(m). This estimate combined with the local (in time) estimate implies, in the end,

ke^{τ(Lµ−αΛ)}w0k_X(m)≤Ce^−η2τkw₀k_X(m), τ >0.

(46)

The decay rate is then improved as follows. From (40) and (46) we have k∂₃e^{(τ+1)(Lµ−αΛ)}w₀k_X(m) ≤Ce⁻

η

2τ−_2(µ−1)^2µ+1 τk∂₃w˜₀k_X(m)≤Ce⁻

η

2τ−_2(µ−1)^2µ+1 τkw₀k_X(m). (47)

Then, (45) withσclose to0and (47) imply (39) and hence (20) forβ= 0. Forβ 6= 0, (20) follows from (21), (39), and the identity

∂_ξ^βe^{τ(Lµ−αΛ)}w0 =e⁻

2µ+1 2(µ−1)β3(τ−¹

2)

∂_ξ^β0⁰e^{12(Lµ−αΛ)}e^{(τ−1)(Lµ−αΛ)}∂^β₃³e^{12(Lµ−αΛ)}w0. Next, to show (19), it suffices to consider the case β = 0. Fix a given number κ ∈ (0, η+ _2(µ−1)^2µ+1 ). As for the horizontal component ofe^{(τ+1)(Lµ−αΛ)}w0, denoted byw˜⁰(τ), we have again from (44) and Proposition3,

kw˜⁰(τ)k_X(m)2 ≤Ce^−κ0τkw˜₀⁰k_X(m)2 +C ˆ _τ

0

e^{−κ0(τ−s)}k∂₃wk˜ _X(m)(s)ds.

(48)

Here κ⁰ ∈ (0,1 + _2(µ−1)^µ+2 ) is taken so thatκ⁰ > κ, which is possible since η ∈ (0,¹₂] and κ ∈ (0, η + _2(µ−1)^2µ+1 ). Note that (40) and (39) imply k∂₃e^{(τ+1)(Lµ−αΛ)}w₀k_X(m) ≤ Ce^−ητ−

2µ+1 2(µ−1)τ

kw₀k_X(m). Thus (48) gives the bound kw˜⁰(τ)k_X(m)2 ≤ Ce^−κτkw₀k_X(m),