MI Preprints|九州大学 マス・フォア・インダストリ研究所 math 5ab0a2cd4f483

MI Preprint Series

Kyushu University

Strongly stratified limit for the 3D

inviscid Boussinesq equations

Ryo Takada

MI 2018-1

( Received March 20, 2018 )

Institute of Mathematics for Industry

Graduate School of Mathematics

BOUSSINESQ EQUATIONS

RYO TAKADA

Abstract. We consider the initial value problem of the 3D inviscid Boussinesq equations for stably stratified fluids. We prove the long time existence of classical solutions for large initial data when the buoyancy frequency is sufficiently high. Furthermore, we consider the singular limit of the strong stratification, and show that the long time classical solution converges to that of 2D incompressible Euler equations in some space-time Strichartz norms.

1. Introduction

Let us consider the initial value problem for the 3D inviscid Boussinesq equations, describing the motion of perfect incompressible fluids in R3_:

        

∂tv+ (v· ∇)v =−∇q+ηe3 t >0, x∈R3,

∂tη+ (v· ∇)η = 0 t >0, x∈R3,

∇ ·v = 0 t>0, x_∈R3_,

v(0, x) =v0(x), η(0, x) =η0(x) x∈R3.

(1.1)

The unknown functions v = (v1(t, x), v2(t, x), v3(t, x))T, η =η(t, x) and q = q(t, x) represent the velocity field, the temperature and the scalar pressure of the fluids, respectively, while v0 = (v0,1(x), v0,2(x), v0,3(x))T is the given initial velocity field satisfying the compatibility condition _{∇ ·}v0 = 0 and η0 =η0(x) is the given initial temperature. The vertical unit vector is denoted by e3 = (0,0,1)T.

In this manuscript, we prove the long time existence of classical solutions to (1.1) around the explicit stratified solution (vs, ηs, qs) = (0, N2x3, N2x23/2) when the constant temperature gradient N = √dηs/dx3 > 0 is sufficiently large. More precisely, we shall show that for given initial disturbanceϕ = (v0,(η0−N2x3)/N)T ∈

Hs+4_(R3_{) with s > 3 and for given finite time T, there exists a positive parameter}

Nϕ,T such that the 3D inviscid stratified Boussinesq system (1.3) admits a unique

classical solution on the time interval [0, T] provided N > Nϕ,T. Furthermore, we

consider the singular limit of the strong stratification as N _{→ ∞}, and show that the long time classical solution vN _{to (1.3) strongly converges to that of the 2D}

incompressible Euler equations in the space-time norm Lq_{(0, T;W}1,∞_(R3_{)) with the} convergence rate O(N−1q) for 4₆q <∞.

Before stating our result, we first review the local existence results on the inviscid Boussinesq equations. In the Sobolev spaces Hs_{-framework, it is known that for}

initial data (v0, η0) ∈ Hs(R3) with ∇ ·v0 = 0 and s > 5/2 there exists a T0 =

T0(s,∥(v0, η0)∥Hs)>0 such that (1.1) possesses a unique classical solution (v, η) in

the class C([0, T0];Hs(R3)). See [6–8, 12, 27] for the local existence theory of (1.1) 2010Mathematics Subject Classification. 76B70, 76B03.

Key words and phrases. the 3D inviscid Boussinesq equations, stable stratification, long time existence, strongly stratified limit.

in function spaces embedded in C1 _{class such as the H¨older spaces and the Besov} spaces, and the blow-up criteria of local solutions including the 2D cases.

Next, let us consider the solution of (1.1) around a stratified solution. It is known that the system (1.1) has an elementary explicit stationary solution (vs, ηs, qs) of the

form

vs ≡0, ηs(x3) = N2x3, qs(x3) =

N2 2 x

2

3 (1.2)

satisfying the hydrostatic balance dqs

dx3

= ηs, where N > 0 is called the buoyancy

or the Brunt-V¨ais¨al¨a frequency and represents the strength of stable stratification. Let us set

θ(t, x) =η(t, x)₋ηs(x3), qη(t, x) = q(t, x)−qs(x3),

where ηs and qs are given by (1.2). We consider the time evolution of the

perturba-tions about a mean state in hydrostatic balance, and then (v, θ, qη) should satisfy

        

∂tv+ (v· ∇)v =−∇qη +θe3,

∂tθ+ (v· ∇)θ=−N2v3, ∇ ·v = 0,

v(0, x) = v0(x), θ(0, x) =θ0(x) = η0(x)−N2x3,

(1.3)

whereθ0 denotes the initial thermal disturbance. The system (1.3) exhibits a disper-sive nature due to the presence of the stable stratification (θe3,−N2v3)T. This phe-nomenon is closely related to the dispersive estimates for the propagatore±iN t|Dh|/|D|

defined by the Fourier integral

e±iN t|Dh|D||_{f(x) :=} 1

(2π)3

∫

R3

eix·ξ±iN t|ξh|ξ||_fb_{(ξ)dξ, (t, x)∈}R1+3_.

Here, ξh = (ξ1, ξ2) ∈ R2 so that |ξh| =

√ ξ2

1 +ξ22 and fb denotes the Fourier transform of f. The sharp dispersive estimate for e±iN t|Dh|/|D| was established

in [26]. Widmayer [30] proved the local well-posedness of (1.3) in Hs_(R3_{) with}

s > 3 for all N > 0. Furthermore, it is shown in [30] that for initial data (v0, θ0)T ∈ Hs+3(R3)∩W5,1(R3) with s > 3, the local solution (vN, θN) to (1.3) on [0, T0] can be decomposed into two parts as

(vN, θN/N) = (wN,0,0) + (uN, ρN), wN = (w₁N, w₂N), uN = (uN₁ , uN₂ , uN₃ ),

and there holds for every 0< t6T0

∥(uN, ρN)(t)_∥W1,∞₍R3₎ →0, ∥wN(t)−w(t)∥_L2₍R3₎→0

as N _{→ ∞}, where w= (w1(t, x), w2(t, x)) solves the 2D incompressible Euler equa-tions (see (1.7) below). For the related singular limit problems to the rotating Navier-Stokes equations and the viscous rotating Boussinesq equations, we refer to [1–3, 9–11]. See also [5, 13–15] for compressible stratified flows.

To state our result more precisely, we rewrite the sytem (1.3) in the Craya-Herring cyclic basis. Let us combine the velocity field with the rescaled thermal disturbance into the new unknown function

u:=

( v, θ

N )T

=

(

v1, v2, v3,

θ N

)T

Put

J :=

   

0 0 0 0

0 0 0 0

0 0 0 ₋1

0 0 1 0

  

, ∇e := (∇,0)T.

Then, the perturbed system (1.3) can be written as

{

∂tu+N Ju+ (u·∇e)u+∇eqη = 0, ∇ ·e u= 0,

u(0, x) = ϕ(x), (1.4)

where ϕ := (u0, θ0/N)T. Next, let P be the Helmholtz projection of the velocity v onto the divergence-free vector fields which is defined by

P_:=

(

(δjk +RjRk)₁6j,k63 0

0 1

) .

Here_{Rj}₁6j63 denote the Riesz transforms on R

3_{. Applying the Helmholtz}

projec-tion P _{to (1.4) gives the following evolution equation:}

{

∂tu+NPJPu+P(u·∇e)u= 0, ∇ ·e u= 0,

u(0, x) =ϕ(x). (1.5)

Here, we have used the facts that P_∇e_{qη = 0 and}P_{u=u since ∇ ·}e _{u= 0.}

In this paper, we address the long time existence of classical solutions to (1.5) when the buoyancy frequency N is sufficiently high, and then we show that the long time classical solution vN _{to (1.5) converges to that of the 2D incompressible Euler}

equations in the space-time Strichartz norm Lq(0, T;W1,∞(R3_{)) for 46q <∞.}

The main result of this paper reads as follows:

Theorem 1.1. Let s _∈ N _{satisfy s} > 3, and let 4 6 q < _∞. Then, for every

ϕ = (ϕ1, ϕ2, ϕ3, ϕ4)T ∈ Hs+4(R3) satisfying ∇ ·e ϕ = 0 and for every 0 < T < ∞,

there exists a positive constant Nϕ,T depending on s, q, T and ∥ϕ∥Hs+4 such that if

N >Nϕ,T then (1.5) possesses a unique classical solution uN in the class

uN _∈C([0, T];Hs+4(R3_))∩C1_{([0, T];H}s+3₍R3_)).

Furthermore, there exists a positive constant C =C(s, q, T,_∥ϕ_∥Hs+4) such that

∥uN ₋u0_∥Lq_(0,T;W1,∞₎ 6CN−

1

q (1.6)

for all N >Nϕ,T, where u0 = (w,0,0)T andw= (w1(t, x), w2(t, x))T is the classical

solution of the two dimensional Euler equations

        

∂tw+Ph(w· ∇h)w= 0 t >0, x∈R3,

∇h·w= 0 t >0, x∈R3,

w(0, x) =P_{hϕh(x) x∈}R3_,

w_∈C([0, T];Hs+4_(R3_))∩C1_{([0, T];H}s+3_(R3_)).

(1.7)

Here, ϕh = (ϕ1, ϕ2)T, ∇h = (∂1, ∂2)T and Ph = (δjk+∂j∂k(−∆h)−1)₁6j,k62 denotes

the two dimensional Helmholtz projection.

arguments based on the Strichartz estimate for the linear propagator with the blow-up criteria of the Beale-Kato-Majda type. However, the situation is different for the 3D inviscid stratified Boussinesq equations. Indeed, the linear solution of (1.5) is given explicitly by

e−tNP_JP

ϕ=eiN t|Dh|D||P

+ϕ+e−iN t

|_Dh| |D| P

−ϕ+P0ϕ (1.8)

(see Proposition 2.1 in Section 2 for details), which has the stationary mode P0ϕ. Thus, the continuation arguments in [24,29] cannot be applied directly. To overcome this, we adapt the ideas in [9] for the viscous rotating stratified fluids and the arguments in [19, 28] to extend the local solutions of the 3D Euler equations, and employ the stability method for the limit system. In the proof of Theorem 1.1, we first show the global regularity of the limit system (1.7) and give the global a priori Hs+3_(R3_{)-estimate for the solution u}0 _{= (w,0,0)}T _{to the limit system with}

u0_{(0) =P}

0ϕ. Next, we introduce the modified linear dispersive equations

  

∂tu±∓iN|

Dh|

|D_|u

±_+P±(u0_·∇e_)u0 _{= 0, ∇ ·}e _u± _{= 0,}

u±_{(0, x) = P±ϕ(x)}

(1.9)

(see (4.2) in Section 4), and establish the space-time Strichartz estimates for the solutions u± _{in L}q_{(0, T;W}1,∞_(R3_{)) with the decay rate N}−1_q

for 46q <_∞. Then, the difference vN _=uN _−u0_−u+_−u− _{of u}0_{, u}± _{and the local solution u}N _{to (1.5)}

with uN_{(0) =ϕ satisfies}

∂tvN +NPJPvN +P(uN ·∇e)vN +

∑

j=0,±

P_(vN _·∇e_)uj₊ ∑

j,k=0,±

(j,k)̸=(0,0)

P_(uj _·∇e_)uk_{= 0}

with vN_{(0) = 0 on some local time interval. We shall show that the H}s_-norm

of vN _{can be taken arbitrarily small provided that the buoyancy frequency N is}

large enough depending only on the given data s, q, T and _∥ϕ_∥Hs+4. Then, the local

solutionuN _{has a uniformH}s_{-bound, and can be continued to the given time interval}

[0, T]. Furthermore, the estimate (1.6) of the singular limit immediately follows from the Hs_{-bound for v}N _{and the space-time estimates foru}±_.

This paper is organized as follows. In Section 2, we derive the explicit formula (1.8) of linear solutions e−tNPJP

ϕ, and establish the space-time Strichartz estimates for the linear propagator e±iN t|Dh|/|D|_{. In Section 3, we show the global regularity}

of the limit system (1.7). In Section 4, we introduce the modified linear dispersive systems (1.9) and show the space-time decay estimates for u±_{. In Section 5, we}

present the proof of Theorem 1.1.

Throughout this paper, we denote by C the constants which may differ at each occurrence. In particular, C = C(_·,_{· · ·} ,_·) will denote the constant which depends only on the quantities appearing in parentheses.

2. Linear Estimates

In this section, we derive the explicit representation for the time evolution semi-group generated by the linear operator₋NP_JP_{, and establish the homogeneous and}

Linear solutions. We follow the argument in [26, Section 2]. Let us consider the linear equation of (1.5):

{

∂tu+NPJPu= 0, ∇ ·e u= 0,

u(0, x) =ϕ(x). (2.1)

Applying the Fourier transform to (2.1), we have

{

∂tub+N P(ξ)JP(ξ)ub= 0, (ξ,0)T ·bu= 0,

b

u(0, ξ) =ϕb(ξ). (2.2)

Here, P(ξ) is the multiplier matrix of the projectionP_{defined by}Pc_{u(ξ) =P(ξ)ub(ξ),}

which is given explicitly by

P(ξ) :=

 

( δjk −

ξjξk

|ξ_|2

)

16j,k63 0

0 1

 .

Set S(ξ) :=₋P(ξ)JP(ξ). Then, direct calculation yields

S(ξ) = 1

|ξ_|2

   

0 0 0 ₋ξ1ξ3

0 0 0 ₋ξ2ξ3

0 0 0 ξ21 +ξ22

ξ1ξ3 ξ2ξ3 −(ξ12+ξ22) 0

   ,

and then

det_{λI ₋S(ξ)_}=λ2 (

λ2+ ξ 2 1 +ξ22

|ξ_|2

) .

Thus, the eigenvalues of S(ξ) are {_±i|ξh|

|ξ|, 0, 0 }

, where ξh = (ξ1, ξ2) and |ξh| =

√ ξ2

1 +ξ22. Moreover, the corresponding eigenvectors are given by

a±(ξ) = √ 1

2_|ξh||ξ|

   

±iξ1ξ3 ±iξ2ξ3 ∓i_|ξh|2

|ξh||ξ|

  

, a0(ξ) = 1

|ξh|

   

−ξ2

ξ1 0 0   

, b0(ξ) = 1

|ξ_|     ξ1 ξ2 ξ3 0    . (2.3) We see that _{a+(ξ), a−(ξ), a0(ξ), b0(ξ)} is an orthonormal basis in C4 and satisfies

S(ξ)a±(ξ) = _±i|ξh|

|ξ_| a±(ξ), S(ξ)a0(ξ) =S(ξ)b0(ξ) = 0.

Hence the solution to (2.2) can be written as

b

u(t, ξ) =eN tS(ξ)ϕb(ξ) = ∑

σ∈{±,0}

eσiN t|ξh|ξ||_⟨ϕb_{(ξ), a}

σ(ξ)⟩C4a_σ(ξ)

Here, we remark that _⟨ϕb(ξ), b0(ξ)⟩C4 = 0 by the divergence-free condition ∇ ·e ϕ = 0.

Therefore, the solution to (2.1) is explicitly given in terms of the evolution semigroup, and we obtain the following proposition.

Proposition 2.1. For every N >0 and for every ϕ_∈L2_(R3_{) with ∇ ·}e _{ϕ = 0, there}

exists a unique solution u to (2.1) which is given explicitly by

u(t, x) =e−tNP_JP

ϕ(x)

where

Pjϕ:=F−1[⟨ϕb(ξ), aj(ξ)⟩C4a_j(ξ)] (j =±,0), (2.4)

e±iN tp(D)_{f(x) :=} 1 (2π)3

∫

R3

eix·ξ±iN tp(ξ)_fb_{(ξ)dξ (2.5)}

and

p(ξ) := |ξh|

|ξ_| = √

ξ2 1 +ξ22

|ξ_| , ξ∈R

3

\ {0_}. (2.6)

Strichartz estimates. In this subsection, we shall prove the homogeneous and inhomogeneous space-time Strichartz estimates for the linear propagatore±iN t|Dh|/|D|

defined by (2.5)–(2.6). Since the phase p(ξ) = _|ξh|/|ξ| is homogeneous of degree 0,

by the Littlewood-Paley decomposition and scaling, the matter is reduced to the frequency localized case. Also, the sign_±does not have any role. Hence we consider the operators

UN(t)f(x) :=

∫

R3

eix·ξ+iN tp(ξ)_ψ(ξ)2_fb_(ξ)dξ,

VN(t)f(x) :=

∫

R3

eix·ξ+iN tp(ξ)ψ(ξ)fb(ξ)dξ, (t, x)_∈R1+3_,

where ψ is a real-valued function in S₍R3_{) satisfying suppψ ⊂ {2}−2 6_|ξ_|622_} and ψ(ξ) = 1 on _{2−1 _{6|ξ|62}. The sharp dispersive estimates for U}

N(t) and

VN(t) are obtained in [26].

Lemma 2.2 ([26, Theorem 1.1]). There exists a positive constant C = C(ψ) > 0

such that

∥UN(t)f∥L∞ 6C(1 +N|t|)−

1 2∥f∥_L1

for all t _∈R _{and f ∈L}1₍R3_{). The same is true for VN(t). Also, the decay rate 1/2} cannot be improved to a larger one.

Now we investigate the boundedness of UN(t). We use the notation for the

space-time norm

∥f_∥LqtLrx :=∥f∥Lq(R;Lr(R3)).

The following results are the homogeneous and inhomogeneous space-time estimates for the linear operator UN(t).

Lemma 2.3. Let the exponents q,q, r,˜ r˜satisfy

2

q +

1

r 6

1 2,

2 ˜

q +

1 ˜

r 6

1

2, 46q,q˜6∞, 26r,˜r6∞. (2.7)

Then, there exist positive constants C1 = C1(ψ, q, r) and C2 =C2(ψ, q,q, r,˜ r˜) such

that

∥UN(t)f∥LqtLrx 6C1N

−1

q∥f∥

L2, (2.8)

∫ t

−∞

UN(t−s)F(s)ds

LqtLrx

6C2N−

1

q−

1 ˜

q∥F∥

Lqt˜′Lr˜

′

x (2.9)

for f _∈L2_(R3_{) and F ∈L}˜q′

Proof. We remark that the L1_-L∞ _{decay rate of U}

N(t) is −1/2 and the admissible

range (2.7) does not include the endpoint q = 2. Hence the proof is based on the standard T T∗ _{argument and the interpolation (See for examples, [16, 22, 23]).}

For the homogeneous estimate (2.8), it suffices to show its adjoint estimate

∫ R

UN(−s)F(s)ds

L2

6CN−1q∥F∥

Lqt′Lr

′

x, (2.10)

and also (2.10) follows from the estimate

∫

R

∫

R|⟨

UN(−s)F(s), UN(−t)G(t)⟩_L2|dsdt6CN

−2

q∥F∥

Lqt′Lr

′

x∥G∥Lq

′

t Lr

′

x. (2.11)

Now we shall show (2.11). By Lemma 2.2 and the L2_{-boundedness of U}

N(t) with

∥UN(t)f∥L2 6C∥f∥_L2, we have for 26r6∞

∥UN(t)f∥Lr 6C(1 +N|t|)−

1 2(1−

2

r)∥f∥

Lr′ (2.12)

for all t _∈R_{. Then, it follows from (2.12) and the Hausdorff-Young inequality that}

|⟨UN(−s)F(s), UN(−t)G(t)⟩_L2|=

∫ R3

UN(t−s)F(s)· F−1[ψ2]∗G(t)dx

6_∥UN(t−s)F(s)∥Lr∥F−1[ψ2]∗G(t)∥_Lr′

6 C

(1 +N_|t₋s_|)12−1r∥

F(s)_∥Lr′∥G(t)∥_Lr′. (2.13)

For (q, r) = (_∞,2), we have by (2.13)

∫

R

∫

R|⟨

UN(−s)F(s), UN(−t)G(t)⟩_L2|dsdt6C∥F∥L1

tL2x∥G∥L1tL2x. (2.14)

In the case 2_q + 1_r = ₂1 with (q, r) _̸= (_∞,2), it follows from (2.13) and the Hardy-Littlewood-Sobolev inequality that

∫

R

∫

R|⟨

UN(−s)F(s), UN(−t)G(t)⟩_L2|dsdt

6CN−(1₂−1_r)

∫

R

∫

R

1

|t₋s_|12− 1

r∥

F(s)_∥Lr′∥G(t)∥_Lr′ds dt

6CN−2q

∫ R 1

|t₋s_|2q∥

F(s)_∥Lr′ds

Lqt

∥G_∥Lq′ t Lr

′

x

6CN−2q∥F∥

Lqt′Lr

′

x∥G∥L q′ t Lr

′

In the case 2

q +

1

r <

1

2, we have by (2.13) and the Hausdorff-Young inequality

∫

R

∫

R

|⟨UN(−s)F(s), UN(−t)G(t)⟩L2|dsdt

6C ∫ R ∫ R 1

(1 +N_|t₋s_|)12− 1

r∥

F(s)_∥Lr′∥G(t)∥_Lr′ ds dt

6C ∫ R 1

(1 +N_|t₋s_|)12− 1

r∥

F(s)_∥Lr′ ds

Lq_t

∥G_∥Lq′ t Lr

′ x 6C 1 (1 +N_|t_|)12−

1 r L q 2 t

∥F_∥Lq′

t Lrx′∥G∥L q′ t Lrx′

=CN−2q∥F∥

Lqt′Lr

′

x∥G∥L q′ t Lr

′

x. (2.16)

Hence we obtain the homogeneous estimate (2.8) by (2.14)–(2.16). Note that it also holds

∫

R

∫

R

|⟨VN(−s)F(s), VN(−t)G(t)⟩L2|dsdt6CN− 2

q∥F∥

Lqt′Lr

′

x∥G∥L q′ t Lr

′

x. (2.17)

by the exactly same procedure as above.

Next, we shall prove the inhomogeneous estimate (2.9). By duality, it suffices to show that ∫ R ∫ t −∞⟨

VN(−s)F(s), VN(−t)G(t)⟩_L2ds dt

6CN−

1

q−

1 ˜

q∥F∥

Lqt˜′L˜r

′

x∥G∥L q′ t Lr

′

x.

Firstly, it easily follows from (2.17) that

∫ R ∫ t −∞⟨

VN(−s)F(s), VN(−t)G(t)⟩_L2ds dt

6 ∫ R ∫ R|⟨

VN(−s)F(s), VN(−t)G(t)⟩L2|dsdt

6CN−2q∥F∥

Lq_t′Lr′ x∥G∥L

q′ t Lr

′

x.

Hence we have for the case (q, r) = (˜q,r˜)

∫ t −∞

UN(t−s)F(s)ds

LqtLrx

6C2N−

2

q∥F∥

Lqt′Lr

′

x. (2.18)

Also, the estimate (2.10) for VN(t) gives that

∫ R ∫ t −∞⟨

VN(−s)F(s), VN(−t)G(t)⟩_L2ds dt

6 ∫ R ∫ t −∞

VN(−s)F(s)ds

L2∥

VN(−t)G(t)∥L2dt

6CN−1q˜∥F∥ Lqt˜′L˜r

′

x∥G∥L

1

tL2x,

which yields for 2_q˜+1_r˜ 6 1

2 ∫ t −∞

UN(t−s)F(s)ds

L∞

t L2x

6C2N−

1 ˜

q∥F∥

Lqt˜′Lr˜

′

Therefore, interpolating (2.18) and (2.19), and using the duality argument, we have for _q2 +1_r = 1₂, _q2_˜+1_r˜ = 1₂

∫ t −∞

UN(t−s)F(s)ds

LqtLrx

6C2N−

1

q−

1 ˜

q∥F∥

Lqt˜′L˜r

′

x. (2.20)

Next, we consider the case 26r 6_∞and 2 ˜ q + 1 ˜ r 6 1

2. Since there holds

∥VN(−t)G(t)∥L2 =∥e−iN tp(ξ)ψ(ξ)Gb(t)∥_L2

ξ

=_∥F−1[ψ]_∗G(t)_∥L2 6_∥F−1[ψ]_∥

L( 12 +1r)−1∥G(t)∥Lr′,

we have by (2.10) for VN(t) that

∫ R ∫ t −∞⟨

VN(−s)F(s), VN(−t)G(t)⟩_L2ds dt

6 ∫ R ∫ t −∞

VN(−s)F(s)ds

L2

∥VN(−t)G(t)∥L2dt

6CN−1q˜∥F∥ Lqt˜′L˜r

′

x∥G∥L

1

tLr

′

x,

which yields for 26r 6_∞and 2 ˜ q + 1 ˜ r 6 1 2 ∫ t −∞

UN(t−s)F(s)ds

L∞

t Lrx

6C2N−

1 ˜

q∥F∥

Lqt˜′Lr˜

′

x. (2.21)

Then, since every (q, r) satisfying 2

q +

1

r 6

1

2 is an interpolation between (∞, r) and (q0, r) with _q02 +1_r = 1₂, it follows from (2.20), (2.21) and the interpolation argument that the inhomogeneous estimates (2.9) hold true for 2_q +1_r 6 1

2 and 2 ˜ q + 1 ˜ r = 1 2. By duality, we then have for _q2 + 1_r = 1₂ and _q2_˜+ 1_˜r 6 1₂

∫ t −∞

UN(t−s)F(s)ds

Lq_tLr x

6C2N−

1

q−

1 ˜

q∥F∥

Lqt˜′L˜r

′

x. (2.22)

Again, since every (q, r) satisfying 2_q+1_r 6 1

2 is an interpolation between (∞, r) and (q0, r) with _q02 +1_r = 1₂, it follows from (2.21), (2.22) and the interpolation argument that the inhomogeneous estimates (2.9) hold for every (q, r) and (˜q,˜r) satisfying

(2.7). This completes the proof of Lemma 2.3.

From (2.8), (2.9), the Littlewood-Paley theory and scaling, we can show the space-time Strichartz estimates for the original propagator e±iN t|Dh|/|D| as a corollary of

Lemma 2.3. Let φ0 be a function in S(R3) satisfying 06φ0(ξ)61 for all ξ∈R3, suppφ0 ⊂

{

ξ_∈R3 ₂−1 6_|ξ_|62}

and _∑

j∈Z

φj(ξ) = 1 for every ξ∈R3\ {0},

where φj(ξ) := φ0(2−jξ). We set ∆jf := F−1[φj(ξ)fb(ξ)] for j ∈ Z. Then, for

˙

Bs

r,σ(R3) as

∥f_∥B˙s r,σ :=

{2sj_∥∆ jf∥_Lr

}

j∈Z

ℓσ₍Z). Also, we define the following space-time norm for 16q6_∞:

∥F_∥Lfq

tB˙sr,σ :=

{

2sj_∥∆jF∥_Lq tLrx

}

j∈Z

ℓσ_(Z)

.

Lemma 2.4. Let the exponents q,q, r,˜ r˜satisfy

2 q + 1 r 6 1 2, 2 ˜ q + 1 ˜ r 6 1

2, 46q,q˜6∞, 26r,˜r6∞.

Then, there exist positive constants C1 =C1(q, r) and C2 =C2(q,q, r,˜ ˜r) such that

∥e±iN tp(D)f_∥Lfq tB˙0r,σ

6C1N−

1

q∥f∥ ˙

B3( 12−1r)

2,σ , (2.23) ∫ t −∞

e±iN(t−s)p(D)F(s)ds

_f

Lq_tB˙0

r,σ

6C2N−

1

q−

1 ˜

q∥F∥

g Lqt˜′B˙

3(1−1_r−1_˜r) ˜

r′,σ

(2.24)

for all 16σ 6_∞, f _∈B˙3(12− 1

r)

2,σ (R3) and F ∈Lfq˜

′

(R_{; ˙B}3(1− 1 r− 1 ˜ r) ˜

r′_,σ (R3)).

Proof. Since ψ(ξ) = 1 on the support of φ0, we see that

UN(t)∆0f(x) =

∫

R3

eix·ξ+iN tp(ξ)ψ(ξ)2φ0(ξ)fb(ξ)dξ

= (2π)3∆0eiN tp(D)f(x).

Hence we have by (2.8) and (2.9) in Lemma 2.3

∥∆0eiN tp(D)f∥LqtLrx 6CN

−1

q∥∆

0f∥L2, (2.25)

∫ t −∞

∆0eiN(t−s)p(D)F(s)ds

LqtLrx

6CN−1q−

1 ˜

q∥∆ 0F∥_L˜q′

t Lr˜

′

x. (2.26)

Note that p(ξ) = _|ξh|/|ξ| is homogeneous of degree 0. Hence scaling ξ 7→ 2jξ gives

that for j _∈Z

∆jeiN tp(D)f(x) = ∆0eiN tp(D)

[

f(2−j_·)](2jx),

∆jf(x) = ∆0

[

f(2−j_·)](2jx).

Therefore, we obtain by (2.25) and (2.26)

∥∆jeiN tp(D)f∥Lq_tLr

x 6CN

−1

q(2j)3(12−1r)∥∆_jf∥

L2,

∫ t −∞

∆jeiN(t−s)p(D)F(s)ds

LqtLrx

6C2N−

1

q−

1 ˜

q(2j)3(1−1r−

1 ˜

r)∥∆_jF∥

Lqt˜′Lr˜

′

x.

3. The Limit System

In this section ,we shall show the global regularity of the limit system (1.7), and give the global a priori Hs+3_(R3_{)-estimate for the solution to (1.7). We remark that} the projection P0 onto the stationary mode of the linear solution to (2.1) defined in (2.3) and (2.4) is also written as

d P0ϕ(ξ) =

( (

δjk− _|ξ_ξj_hξ_|k2

)

16j,k62 0

0 0

) b ϕ(ξ).

Hence we see that P0 corresponds to the two dimensional Helmholtz projection

P_{h =}(_{δjk+∂j∂k(−∆h)}−1)

16j,k62, P0 =

(

P_{h 0}

0 0

)

. (3.1)

Now, let us consider the limit system of (1.5):

    

∂tw+Ph(w· ∇h)w= 0 t >0, x∈R3,

∇h·w= 0 t>0, x∈R3,

w(0, x) =P_{hϕh(x) x∈}R3_,

(3.2)

where w= (w1(t, x), w2(t, x))T, ϕh = (ϕ1(x), ϕ2(x))T and ∇h = (∂1, ∂2)T. The global regularity result for (3.2) reads as follows:

Theorem 3.1. Let s _∈ N _{satisfy s > 3. Then, for every ϕh ∈ H}s+3₍R3_{) and for} every 0< T <_∞, there exists a unique classical solution w to (3.2) in the class

w_∈C([0, T];Hs+3(R3_))∩C1_{([0, T];H}s+2₍R3_)).

Moreover, there exists a positive constant CL =CL(s, T,∥ϕh∥Hs+3) such that

sup 06t6T∥

w(t)_∥Hs+3 6C_L(s, T,∥ϕ_h∥_Hs+3). (3.3)

Proof. We first remark that for fixedx3 ∈Rthe system (3.2) for w=w(·, x3) is the two dimensional incompressible Euler equations. Hence it is well-known by [17, 20] that for the initial data ϕh(·, x3) ∈ Hs(R2) there exists a unique classical global solution w(_·, x3) to (3.2) satisfying

w(_·, x3)∈C([0,∞);Hs(R2))∩C1([0,∞);Hs−1(R2)). Let us first derive the a priori estimate for the norm

∥w(t)_∥L∞

x₃H_xhs :=∥w(t)∥L∞(Rx₃;Hs(R2)).

By the standard energy method and the Gronwall inequality (see [18, 21]), we have

∥w(t,_·, x3)∥Hs₍R2₎6∥ϕ_h(·, x₃)∥_Hs₍R2₎exp

{ C

∫ t

0 ∥∇

hw(τ,·, x3)∥L∞₍R2₎dτ

}

. (3.4)

By the Biot-Savart law, we have a representation of w in terms of the vorticity

ω =∂1w2−∂2w1 as

w=₋(₋∆h)−1∇⊥hω, ∇⊥h = (−∂2, ∂1)T, which yields

∇hw=

( Rh

1Rh2 (Rh2)2 −(Rh

1)2 −Rh1Rh2

Here, Rh

j = −∂j(−∆h)− 1

2 (j = 1,2) denotes the two dimensional Riesz transform.

Then, since the Riesz transform is bounded in BMO(R2_{) and there holdsL}∞₍R2_)֒→

BMO(R2_{), it follows from the logarithmic Sobolev inequality by [25, Theorem 1] that}

∥∇hw(τ,·, x3)∥L∞₍R2₎

6C{1 +_∥ω(τ,_·, x3)∥BMO(R2₎

(

1 + log+_∥∇hw(τ,·, x3)∥Hs−1₍R2₎

)}

6C{1 +_∥ω(τ,_·, x3)∥L∞₍R2₎log

(

∥w(τ,_·, x3)∥Hs₍R2₎+e

)}

. (3.5)

Let Xh(t) be the trajectory flow defined by the solution of the ordinary differential

equation _

 

dXh

dt (t) =w(t, Xh(t), x3), Xh(0) =xh ∈R2.

Then, since the vorticity w satisfies ∂tω+w· ∇hω= 0, we have

∂

∂t{ω(t, Xh(t), x3)}= 0,

which yields ω(t, Xh(t), x3) = ω(0, xh, x3) for allt >0. Hence we have ∥ω(t,_·, x3)∥L∞₍R2₎ =∥ω(0,·, x₃)∥_L∞₍R2₎

6C_∥ϕh(·, x3)∥Hs₍R2₎. (3.6)

Then, it follows from (3.4), (3.5) and (3.6) that

∥w(t,_·, x3)∥Hs₍R2₎+e 6(_∥ϕh(·, x3)∥Hs₍R2₎+e)

×exp

{

Ct+C_∥ϕh(·, x3)∥Hs₍R2₎

∫ t

0

log(_∥w(τ,_·, x3)∥Hs₍R2₎+e)dτ

}

. (3.7)

Defining z(t) = log(_∥w(t,_·, x3)∥Hs₍R2₎+e), we have from (3.7)

z(t)6z(0) +Ct+C_∥ϕh(·, x3)∥Hs₍R2₎

∫ t

0

z(τ)dτ.

The Gronwall inequality gives

z(t)6(z(0) +Ct) exp{Ct_∥ϕh(·, x3)∥Hs_(R2₎},

which implies that

∥w(t,_·, x3)∥Hs₍R2₎+e

6(_∥ϕh(·, x3)∥Hs_(R2₎+e)exp{Ct∥ϕh(·,x3)∥Hs(R2)}eCtexp{Ct∥ϕh(·,x3)∥Hs(R2)}.

Therefore, we obtain for all t >0

∥w(t)_∥L∞

x₃H_xhs 6

(

∥ϕh∥L∞

x₃H_xhs +e

)exp{Ct∥ϕh∥L∞_x

3Hsxh

}

eCtexp

{

Ct∥ϕh∥L∞x₃Hs_xh

}

. (3.8)

Now, we shall show that w belongs to C([0, T];Hs+3_(R3_))∩C1_{([0, T];H}s+2_(R3₎₎ and satisfies the estimate (3.3). It follows from the local well-posedness theory of the Euler equations by [4, 18, 21] that there exists a local time T0 > C/∥ϕh∥Hs+3

such that the unique solution w to (3.2) belongs to the classC([0, T0];Hs+3(R3))∩

C1_{([0, T}

By the standard energy method, we have

∥w(t)_∥Hs+3 6∥ϕ_h∥_Hs+3+C

∫ t

0 ∥∇

w(τ)_∥L∞∥w(τ)∥_Hs+3dτ 6_∥ϕh∥Hs+3+C

∫ t

0 ∥∇

hw(τ)∥L∞∥w(τ)∥_Hs+3dτ

+C ∫ t

0 ∥

∂3w(τ)∥L∞∥w(τ)∥_Hs+3dτ. (3.9)

Let us set the right hand side of (3.8) by

A(t,_∥ϕh∥L∞

x₃H_xhs ) :=

(

∥ϕh∥L∞

x₃H_xhs +e

)exp{Ct∥ϕh∥L∞x₃Hs_xh

}

eCtexp

{

Ct∥ϕh∥L∞x₃Hs_xh

}

.

Since s>3, the Sobolev embedding Hs_(R2_)֒→C1_(R2_{) and (3.8) give that} ∥∇hw(t)∥L∞ 6∥w(t)∥_L∞

x₃H_xhs

6A(t,_∥ϕh∥L∞

x₃H_xhs ). (3.10)

Next, we shall derive the estimate for _∥∂3w(t)∥L∞. By the Sobolev embedding

H2_(R2_)֒→L∞_(R2_{), we have}

∥∂3w(t)∥L∞ 6C∥∂₃w(t)∥_L∞

x₃H_xh2 . (3.11)

Hence we consider theH2_(R2_{)-estimate for∂}

3w(t,·, x3). By (3.2), ∂3wshould satisfy

∂t(∂3w) +Ph(∂3w· ∇h)w+Ph(w· ∇h)∂3w= 0. (3.12) Taking the L2_(R2_{)-inner product of (3.12) with ∂}

3w, we have by the divergence-free condition _∇h·w= 0 that

1 2

d

dt∥∂3w(t,·, x3)∥

2

L2₍R2₎+⟨(∂3w· ∇h)w, ∂3w⟩_L2₍R2₎ = 0. (3.13)

Since it holds

⟨(∂3w· ∇h)w, ∂3w⟩_L2₍R2₎

6_∥∇hw_∥L∞₍R2₎∥∂₃w∥2_L2₍R2₎,

we have the L2_(R2_{)-estimate by (3.13)} 1

2

d

dt∥∂3w(t,·, x3)∥

2

L2₍R2₎6∥∇hw(t,·, x3)∥L∞₍R2₎∥∂₃w(t,·, x₃)∥2_L2₍R2₎. (3.14)

For the ˙H1_(R2_{)-estimate for ∂}

3w, it follows from (3.12) that

∂t(∂l∂3w) +Ph(∂l∂3w· ∇h)w+Ph(∂3w· ∇h)∂lw

+P_{h(∂lw· ∇h)∂3w+}P_{h(w· ∇h)∂l∂3w= 0 (3.15)}

for l = 1,2. Taking the L2_(R2_{)-inner product of (3.15) with ∂}

l∂3wgives 1

2

d

dt∥∂l∂3w(t,·, x3)∥

2

L2 +⟨(∂l∂3w· ∇h)w, ∂l∂3w⟩_L2₍R2₎

Here, we have used the fact that _⟨(w_{· ∇}h)∂l∂3w, ∂l∂3w⟩_L2₍R2₎ = 0 by ∇h·w = 0.

By the H¨older inequality, we have

⟨(∂l∂3w· ∇h)w, ∂l∂3w⟩_L2₍R2₎

6_∥∇hw_∥L∞₍R2₎∥∇_h∂₃w∥2_L2₍R2₎,

⟨(∂3w· ∇h)∂lw, ∂l∂3w⟩_L2₍R2₎

6_∥∇2_hw_∥L∞₍R2₎∥∂₃w∥_L2₍R2₎∥∇_h∂₃w∥_L2₍R2₎,

⟨(∂lw· ∇h)∂3w, ∂l∂3w⟩_L2₍R2₎

6_∥∇hw_∥L∞₍R2₎∥∇_h∂₃w∥2_L2₍R2₎.

Substituting these estimates into (3.16) gives

1 2

d

dt∥∇h∂3w(t,·, x3)∥

2

L2₍R2₎

6C_∥∇hw(t,·, x3)∥L∞₍R2₎∥∇_h∂₃w(t,·, x₃)∥2_L2₍R2₎

+C_∥∇2hw(t,·, x3)∥L∞₍R2₎∥∂₃w(t,·, x₃)∥_L2₍R2₎∥∇_h∂₃w(t,·, x₃)∥_L2₍R2₎. (3.17)

For the ˙H2_(R2_{)-estimate for ∂}

3w, we have by (3.15)

∂t(∂k∂l∂3w) +Ph(∂k∂l∂3w· ∇h)w+Ph(∂l∂3w· ∇h)∂kw

+P_{h(∂k∂3w· ∇h)∂lw+}P_{h(∂3w· ∇h)∂k∂lw}

+P_{h(∂k∂lw· ∇h)∂3w+}P_{h(∂lw· ∇h)∂k∂3w}

+P_{h(∂kw· ∇h)∂l∂3w+}P_{h(w· ∇h)∂k∂l∂3w= 0 (3.18)}

for k, l= 1,2. Then, there holds by the H¨older inequality that

⟨(∂k∂l∂3w· ∇h)w, ∂k∂l∂3w⟩L2₍R2₎

6_∥∇hw_∥L∞₍R2₎∥∇2_h∂₃w∥2_L2₍R2₎,

⟨(∂l∂3w· ∇h)∂kw, ∂k∂l∂3w⟩L2₍R2₎

6_∥∇2_hw_∥L∞₍R2₎∥∇_h∂₃w∥_L2₍R2₎∥∇2_h∂₃w∥_L2₍R2₎,

⟨(∂k∂3w· ∇h)∂lw, ∂k∂l∂3w⟩L2₍R2₎

6_∥∇2_hw_∥L∞₍R2₎∥∇_h∂₃w∥_L2₍R2₎∥∇2_h∂₃w∥_L2₍R2₎,

⟨(∂3w· ∇h)∂k∂lw, ∂k∂l∂3w⟩L2₍R2₎

6_∥∇3_hw_∥L∞₍R2₎∥∂₃w∥_L2₍R2₎∥∇2_h∂₃w∥_L2₍R2₎,

⟨(∂k∂lw· ∇h)∂3w, ∂k∂l∂3w⟩L2₍R2₎

6_∥∇2_hw_∥L∞₍R2₎∥∇_h∂₃w∥_L2₍R2₎∥∇2_h∂₃w∥_L2₍R2₎,

⟨(∂lw· ∇h)∂k∂3w, ∂k∂l∂3w⟩L2₍R2₎

6_∥∇hw_∥L∞₍R2₎∥∇2_h∂₃w∥2_L2₍R2₎,

⟨(∂kw· ∇h)∂l∂3w, ∂k∂l∂3w⟩L2₍R2₎

6_∥∇hw_∥L∞₍R2₎∥∇2_h∂₃w∥2_L2₍R2₎.

Taking theL2_(R2_{)-inner product of (3.18) with ∂}

k∂l∂3w and substituting the above estimates, we have

1 2

d

dt∥∇

2

h∂3w(t,·, x3)∥2L2₍R2₎

6C_∥∇hw(t,·, x3)∥L∞₍R2₎∥∇2_h∂₃w(t,·, x₃)∥2_L2₍R2₎

+C_∥∇2_hw(t,_·, x3)∥L∞₍R2₎∥∇_h∂₃w(t,·, x₃)∥_L2₍R2₎∥∇2_h∂₃w(t,·, x₃)∥_L2₍R2₎

Therefore, we obtain from (3.8), (3.14), (3.17), (3.19) and Hs−1_(R2_)֒→L∞_(R2_{) for}

s >3 that 1 2

d

dt∥∂3w(t,·, x3)∥

2

H2₍R2₎

6_∥∇hw(t,_·, x3)∥L∞₍R2₎∥∂₃w(t,·, x₃)∥2_L2₍R2₎

+C_∥∇hw(t,·, x3)∥L∞₍R2₎∥∇_h∂₃w(t,·, x₃)∥2_L2₍R2₎

+C_∥∇2_hw(t,_·, x3)∥L∞₍R2₎∥∂₃w(t,·, x₃)∥_L2₍R2₎∥∇_h∂₃w(t,·, x₃)∥_L2₍R2₎

+C_∥∇hw(t,·, x3)∥L∞₍R2₎∥∇2_h∂₃w(t,·, x₃)∥2_L2₍R2₎

+C_∥∇2_hw(t,_·, x3)∥L∞₍R2₎∥∇_h∂₃w(t,·, x₃)∥_L2₍R2₎∥∇2_h∂₃w(t,·, x₃)∥_L2₍R2₎

+C_∥∇3_hw(t,_·, x3)∥L∞₍R2₎∥∂₃w(t,·, x₃)∥_L2₍R2₎∥∇2_h∂₃w(t,·, x₃)∥_L2₍R2₎ 6C_∥∂3w(t,·, x3)∥2H2₍R2₎

3

∑

l=1

∥∇lhw(t,·, x3)∥L∞₍R2₎ 6C_∥w(t,_·, x3)∥Hs+2₍R2₎∥∂₃w(t,·, x₃)∥2_H2₍R2₎ 6CA(t,_∥ϕh∥L∞

x₃H s+2

xh )∥∂3w(t,·, x3)∥ 2

H2₍R2₎,

which yields by the Gronwall inequality that

∥∂3w(t,·, x3)∥H2₍R2₎ 6∥∂₃ϕ_h(·, x₃)∥_H2₍R2₎exp

{

CtA(t,_∥ϕh∥L∞

x₃Hxhs+2)

}

. (3.20)

Hence we obtain from (3.11) and (3.20)

∥∂3w(t)∥L∞ 6C∥∂₃ϕ_h∥_L∞

x₃H_xh2 exp

{

CtA(t,_∥ϕh∥_L∞

x₃Hxhs+2)

}

. (3.21)

Therefore, we have by (3.9), (3.10) and (3.21)

∥w(t)_∥Hs+3 6∥ϕ_h∥_Hs+3 +CA(T,∥ϕ_h∥_L∞

x₃Hs_xh)

∫ t

0 ∥

w(τ)_∥Hs+3dτ

+C_∥∂3ϕh∥L∞

x₃H_xh2 exp

{

CT A(T,_∥ϕh∥L∞

x₃Hxhs+2)

} ∫ t

0 ∥

w(τ)_∥Hs+3dτ

(3.22) for all 06t6T. Here, let us set

B(t,_∥ϕh∥Hs) :=(∥ϕ_h∥_Hs +e)

exp{Ct∥ϕh∥Hs}

eCtexp{Ct∥ϕh∥Hs}_.

Then, it follows from the continuous embedding H1_(R)֒→L∞_{(R) that} A(t,_∥ϕh∥L∞

x₃H_xhs )6CB(t,∥ϕh∥Hs+1). (3.23)

Hence we have by (3.22) and (3.23)

∥w(t)_∥Hs+3 6∥ϕ_h∥_Hs+3 +CB(T,∥ϕ_h∥_Hs+1)

∫ t

0 ∥

w(τ)_∥Hs+3dτ

+C_∥ϕh∥H4exp{CT B(T,∥ϕ_h∥_Hs+3)}

∫ t

0 ∥

w(τ)_∥Hs+3dτ.

Therefore, we obtain by the Gronwall inequality that sup

06t6T∥

w(t)_∥Hs+3

This gives the global a priori estimate for _∥w(t)_∥Hs+3, and we complete the proof of

Theorem 3.1.

4. Modified Linear Dispersive Solutions

In this section, we adapt the idea in [9] and introduce the modified linear dispersive equations (1.9) (and (4.2) below). Making use of Lemma 2.4, we shall establish the global space-time estimates for the solutions u± _{to those systems.}

Let s _∈ N _{satisfy s} > 3, and let 0 < T < _∞. Then, for the initial data

ϕ = (ϕh, ϕ3, ϕ4)T ∈Hs+4(R3) with∇·e ϕ= 0, letw = (w1, w2)∈C([0, T];Hs+4(R3))∩

C1([0, T];Hs+3(R3_{)) be the classical solution to (3.2) with w(0, x) =} P_hϕh(x)

con-structed in Theorem 3.1 satisfying the Hs+4_-estimate sup

06t6T∥

w(t)_∥Hs+4 6C_L(s, T,∥ϕ_h∥_Hs+4). (4.1)

Now, we put u0 _{= (w,0,0)}T_{, and consider the solution to the following linear}

systems with the external forces P±(u0_·∇e_)u0_:

    

∂tu±∓iN p(D)u±+P±(u0 ·∇e)u0 = 0 t >0, x∈R3,

e

∇ ·u±_{= 0 t>0, x∈R}3_,

u±_{(0, x) =P±ϕ(x) x∈R}3_,

(4.2)

wherep(D) = _|Dh|/|D|is the Fourier multiplier, and the projections P± are defined

in (2.3) and (2.4). By the Duhamel principle, the solutions to (4.2) are given by

u±(t) = e±iN tp(D)P±ϕ₋ ∫ t

0

e±iN(t−τ)p(D)P±(u0(τ)_·∇e)u0(τ)dτ. (4.3)

Lemma 4.1. Let s _∈ N _{satisfy s} > 3, and let 0 < T < _∞. Then, for every

ϕ _∈ Hs+4_(R3_{) satisfying ∇ ·e ϕ = 0, there exists a unique classical solution u}± _to

(4.2) in the class

u± _∈C([0, T];Hs+3(R3_))∩C1_{([0, T];H}s+2₍R3_)).

Moreover, there exists a positive constant C =C(s, T,_∥ϕ_∥Hs+4) such that

sup 06t6T ∥

u±(t)_∥Hs+3 6∥ϕ∥_Hs+3+C(s, T,∥ϕ∥_Hs+4). (4.4)

Also, for46q <_∞there exist positive constantsCq =C(q)andC =C(s, q, T,∥ϕ∥Hs+4) such that

∥∇lu±_∥Lq_(0,T;L∞₎6C_qN−

1

q (∥ϕ∥

H2+l+C(s, q,∥ϕ∥_Hs+4)) (4.5) for l = 0,1,2, . . . , s+ 1.

Proof. Let us first show the Hs+3_{-estimate (4.4). Taking the H}s+3 _{inner product of} (4.2) with u±_{, and considering the real part, we have}

1 2

d dt∥u

±_(t)

∥2Hs+3+⟨(u0(t)·∇e)u0(t), u±(t)⟩_Hs+3 = 0. (4.6)

It follows from the Hs+4_{-estimates (4.1) for w(t) that}

⟨(u0(t)_·∇e)u0(t), u±(t)_⟩Hs+3

6_∥(w(t)_{· ∇}h)w(t)∥Hs+3∥u±(t)∥Hs+3 6C_∥w(t)_∥2_Hs+4∥u±(t)∥_Hs+3

Substituting (4.7) into (4.6), we have

1 2

d dt∥u

±_(t)

∥2Hs+3 6C(s, T,∥ϕh∥Hs+4)∥u±(t)∥_Hs+3,

which implies that

∥u±(t)_∥Hs+3 6∥P_±ϕ∥_Hs+3 +tC(s, T,∥ϕ_h∥_Hs+4)

for all 06t6T. This yields the desired estimate (4.4).

Next, we shall prove the space-time estimate (4.5). For the homogeneous term in (4.3), by the continuous embedding ˙B0

∞,1(R3)֒→L∞(R3), the Minkowski inequality and (2.23) in Lemma 2.4, we have for l= 0,1,2, . . . , s+ 1

_∇l_e±iN tp(D)_P±ϕ

Lq_(0,T;L∞₎ 6C

_∇l_e±iN tp(D)_P±ϕ

Lq_{(0,T; ˙B}0

∞,1) 6C∑

j∈Z

∆j∇le±iN tp(D)P±ϕ

Lq_(0,T;L∞₎

=C_∥∇le±iN tp(D)P±ϕ_∥Lfq_{(0,T; ˙B}0

∞,1) 6CN−1q∥∇lP±ϕ∥

˙

B32 2,1

6CN−1q∥ϕ∥

H2+l. (4.8)

For the inhomogeneous term in (4.3), similarly to (4.8), it follows from (2.24) in Lemma 2.4 with (˜q,r˜) = (_∞,2) that

∇l ∫ t 0

e±iN(t−τ)p(D)P±(u0(τ)_·∇e)u0(τ)dτ

Lq_(0,T;L∞₎

6C ∫ t 0 ∇

l_e±iN(t−τ)p(D)_P

±(u0(τ)·∇e)u0(τ)dτ

_f

Lq_{(0,T; ˙B}0

∞,1) 6CN−1q

∇lP±(u0_·∇e)u0

f L1_{(0,T; ˙B}32

2,1)

. (4.9)

Here, we have by the Hs+4_{-estimates (4.1) for w(t)}

∇lP±(u0_·∇e)u0

f L1_{(0,T; ˙B}32

2,1)

=∑

j∈Z

232j

∫ T

0

∆j∇lP±(u0(t)·∇e)u0(t)

L2 dt

=

∫ T

0

∇l_P

±(u0(t)·∇e)u0(t) _˙

B32 2,1

dt 6C ∫ T 0

(u0(t)_·∇e)u0(t)

H2+l dt

6C ∫ T

0 ∥

w(t)_∥2_H3+l dt

6C ∫ T

0 ∥

w(t)_∥2_Hs+4 dt6C(s, T,∥ϕh∥Hs+4). (4.10)

5. Proof of Theorem 1.1

We are now ready to present the proof of Theorem 1.1.

Proof of Theorem 1.1. Let s _∈N _{with s} > 3, and let ϕ = (ϕh, ϕ3, ϕ4)T ∈ Hs+4(R3) satisfying _{∇ ·}e ϕ = 0. Since P_JP _{is skew-symmetric and then ⟨}P_JP_{u, u⟩}

Hs = 0, it

follows from the standard local well-posedness theory for the 3D Euler equations in

Hs_(R3_{) by [18, 21, 24] that there exists a local timeT}

0 =T0(s,∥ϕ∥Hs)>0 such that

(1.5) possesses a unique classical solution uN _{for all N >0 in the class}

uN _∈C([0, T0];Hs(R3))∩C1([0, T0];Hs−1(R3)). (5.1) In particular, there exist positive constants C0 =C0(s) and C1 =C1(s) such that

T0 >

C0 ∥ϕ_∥Hs

, sup

06t6T0∥

uN(t)_∥Hs 6C₁∥ϕ∥_Hs. (5.2)

Let 0 < T < _∞. We shall first show that the local solution uN _{in the class}

(5.1) can be extended to the arbitrary finite time interval [0, T] provided that the buoyancy frequency N is sufficiently high.

Let w = (w1, w2) ∈ C([0, T];Hs+4(R3))∩ C1([0, T];Hs+3(R3)) be the classical solution to the limit system (3.2) with w(0, x) = P_{hϕh(x) constructed in Theorem}

3.1. We put u0 _{= (w,0,0)}T_{. Then, by (3.1), we see thatu}0 _{is the classical solution} to the system

{

∂tu0+P0(u0 ·∇e)u0 = 0, ∇ ·e u0 = 0,

u0_{(0, x) = P} 0ϕ.

Also, let u± _{∈ C([0, T];H}s+3_(R3_))∩C1_{([0, T];H}s+2_(R3_{)) be the classical solutions} to the linear systems (4.2) constructed in Lemma 4.1 satisfying (4.4) and (4.5).

Now we set

vN :=uN ₋u+₋u−₋u0.

Then, since there hold ϕ =P_{ϕ = P+ϕ+P−ϕ+P0ϕ and Pju}j _=uj _{for j ∈ {0,±},}

the perturbation vN _{should solve}

        

∂tvN +NPJPvN +P(uN ·∇e)vN +

∑

j=0,±

P_(vN _·∇e_)uj₊ ∑

j,k=0,±

(j,k)̸=(0,0)

P_(uj _·∇e_)uk_{= 0,}

e

∇ ·vN _{= 0,}

vN_{(0, x) = 0}

(5.3) on the local time interval [0, T0]. Let us derive the Hs-estimate for vN(t). Taking the Hs _{inner product of (5.3) with v}N _gives

1 2

d dt∥v

N_(t)

∥2Hs +⟨(uN(t)·∇e)vN(t), vN(t)⟩_Hs

+ ∑

j=0,±

⟨(vN(t)_·∇e)uj(t), vN(t)_⟩Hs +

∑

j,k=0,±

(j,k)̸=(0,0)

⟨(uj(t)_·∇e)uk(t), vN(t)_⟩Hs = 0.

(5.4)

Since it holds _∫

R3

for α_∈(N_{∪ {0})}3 _{with |α|}6s by the divergence-free condition, we have

⟨(uN _·∇e)vN, vN_⟩Hs

= ∑

|α|6s

∫

R3

∂α(uN _·∇e)vN _·∂αvNdx = ∑

|α|6s

∑

0<β6α

Cα,β

∫

R3

(∂βuN _·∇e)∂α−βvN _·∂αvNdx 6 ∑

|α|6s

∑

0<β6α

Cα,β

(∂β_uN _·∇e)∂α−β_vN

L2

∂α_vN

L2

6C_∥uN_∥Hs∥vN∥2_Hs. (5.5)

Here, we have used the estimates (see [18, Lemma in page 302])

(∂βuN _·∇e)∂α−βvN

L2 6

{

C_∥uN_∥

H3∥vN∥_H|α| 0< β 6α, |β|= 1,2,

C_∥uN_∥

H|β|∥vN∥_H|α|−|β|+3 0< β 6α, |β|>3.

For the third term in the left hand side of (5.4), since s>3 andHs_(R3_{) is a Banach} algebra, we see that

∑

j=0,±

⟨(vN _·∇e)uj, vN_⟩Hs

6 ∑

j=0,±

(vN _·∇e)uj

Hs

vN_Hs

6C ∑

j=0,±

∥uj_∥Hs+1∥vN∥2_Hs. (5.6)

For the fourth term in the left hand side of (5.4), the Schwartz inequality gives

∑

j,k=0,±

(j,k)̸=(0,0)

⟨(uj_·∇e)uk_{, v}N_⟩ Hs 6 ∑

j,k=0,±

(j,k)̸=(0,0)

∥(uj _·∇e)uk_∥

Hs∥vN∥_Hs. (5.7)

Let us derive the estimates for _∥(uj _·∇e_)uk_∥

Hs. It follows from the the Leibniz rule

that

∥(uj_·∇e)uk_∥2_Hs =

∑

|α|6s

∫

R3

∂α(uj _·∇e)uk_·∂α(uj _·∇e)ukdx

= ∑

|α|6s

∑

β6α

∑

γ6α

Cα,β,γ

∫

R3

(∂βuj _·∇e)∂α−βuk_·(∂γuj _·∇e)∂α−γukdx.

(5.8)

For (j, k) = (_±,_±), we have by the H¨older inequality

∑

|α|6s

∑

β6α

∑

γ6α

Cα,β,γ

∫

R3

(∂βu±_·∇e)∂α−βu±_·(∂γu±_·∇e)∂α−γu±dx

6 ∑

|α|6s

∑

β6α

∑

γ6α

Cα,β,γ∥∂βu±∥L∞∥∂γu±∥_L∞∥∇∂α−βu±∥_L2∥∇∂α−γu±∥_L2 6C_∥u±_∥2_Hs+1

( _s

∑

l=0

∥∇lu±_∥L∞

)2

Similarly to (5.9), we see that for (j, k) = (_±,_∓)

∑

|α|6s

∑

β6α

∑

γ6α

Cα,β,γ

∫

R3

(∂βu±_·∇e)∂α−βu∓_·(∂γu±_·∇e)∂α−γu∓dx

6C_∥u∓_∥2_Hs+1

( _s

∑

l=0

∥∇lu±_∥L∞

)2

. (5.10)

For (j, k) = (_±,0), it follows from the H¨older inequality that

∑

|α|6s

∑

β6α

∑

γ6α

Cα,β,γ

∫

R3

(∂βu±_·∇e)∂α−βu0_·(∂γu±_·∇e)∂α−γu0dx

6 ∑

|α|6s

∑

β6α

∑

γ6α

Cα,β,γ∥∂βu±∥L∞∥∂γu±∥_L∞∥∇∂α−βu0∥_L2∥∇∂α−γu0∥_L2 6C_∥u0_∥2_Hs+1

( _s

∑

l=0

∥∇lu±_∥L∞

)2

. (5.11)

Similarly to (5.11), we have for (j, k) = (0,_±)

∑

|α|6s

∑

β6α

∑

γ6α

Cα,β,γ

∫

R3

(∂βu0_·∇e)∂α−βu±_·(∂γu0_·∇e)∂α−γu±dx

6C_∥u0_∥2_Hs

(_s+1 ∑

l=0

∥∇lu±_∥L∞

)2

. (5.12)

Combining (5.7)–(5.12), we obtain

∑

j,k=0,±

(j,k)̸=(0,0)

⟨(uj_·∇e)uk, vN_⟩Hs

6C ∑

j=0,±

∥uj_∥Hs+1 s+1

∑

l=0

(

∥∇lu+_∥L∞ +∥∇lu−∥_L∞)∥vN∥_Hs.

(5.13)

Substituting (5.5), (5.6) and (5.13) into (5.4), we have

1 2

d dt∥v

N_(t)

∥2Hs 6C∥uN∥_Hs∥vN∥2

Hs+C

∑

j=0,±

∥uj_∥Hs+1∥vN∥2_Hs

+C ∑

j=0,±

∥uj_∥Hs+1 s+1

∑

l=0

(

∥∇lu+_∥L∞ +∥∇lu−∥_L∞)∥vN∥_Hs,

which yields

d dt∥v

N_(t)

∥Hs 6C

(

∥uN_∥Hs+

∑

j=0,±

∥uj_∥Hs+1

)

∥vN_∥Hs

+C ∑

j=0,±

∥uj_∥Hs+1 s+1

∑

l=0

(

Here, it follows from the uniform Hs+3 _{estimates (3.3), (4.4) and (5.2) that there} exists a positive constant C =C(s, T,_∥ϕ_∥Hs+4) such that

∥uN(t)_∥Hs +

∑

j=0,±

∥uj(t)_∥Hs+1 6 sup

06t6T0∥

uN(t)_∥Hs+

∑

j=0,±

sup 06t6T∥

uj(t)_∥Hs+3

6C(s, T,_∥ϕ_∥Hs+4). (5.15)

for 06t6T0. Then, by (5.14), (5.15) and vN(0) = 0, we have

∥vN(t)_∥Hs 6C(s, T,∥ϕ∥_Hs+4) s+1

∑

l=0

∫ t

0

(

∥∇lu+(τ)_∥L∞ +∥∇lu−(τ)∥_L∞)dτ

+C(s, T,_∥ϕ_∥Hs+4)

∫ t

0 ∥

vN(τ)_∥Hsdτ. (5.16)

Here, it follows from the H¨older inequality and the space-time estimates (4.5) in Lemma 4.1 that for 46q <_∞

s+1

∑

l=0

∫ t

0 ∥∇

l_u±_(τ)

∥L∞dτ 6T1−

1

q

s+1

∑

l=0

∥∇lu±_∥Lq_(0,T;L∞₎

6C(s, q, T,_∥ϕ_∥Hs+4)N− 1

q (5.17)

for 06t6T0 < T. Hence we have by (5.16) and (5.17)

∥vN(t)_∥Hs 6C(s, q, T,∥ϕ∥_Hs+4)N− 1

q +C(s, T,∥ϕ∥

Hs+4)

∫ t

0 ∥

vN(τ)_∥Hsdτ. (5.18)

The Gronwall inequality yields

sup 06t6T0∥

vN(t)_∥Hs 6C(s, q, T,∥ϕ∥_Hs+4)N− 1

qeC(s,T,∥ϕ∥Hs+4)T. (5.19)

Therefore, there exists a positive constant N0 =N0(s, q, T,∥ϕ∥Hs+4) > 0 such that

there holds

sup 06t6T0∥

vN(t)_∥Hs 61 (5.20)

for all N > N0. Then, since vN = uN −u0 −u+ −u−, it follows from (3.3), (4.4) and (5.20) that there exists a positive constant C∗ =C∗(s, T,_∥ϕ_∥Hs+4) such that

∥uN(T0)∥Hs 6∥vN(T₀)∥_Hs +

∑

j=0,±

∥uj(T0)∥Hs

6 sup 06t6T0∥

vN(t)_∥Hs+

∑

j=0,±

sup 06t6T∥

uj(t)_∥Hs+3

61 +C∗(s, T,_∥ϕ_∥Hs+4). (5.21)

Note that the constant C∗(s, T,∥ϕ∥Hs+4) is independent of the local time T₀.

There-fore, the local solution uN _{can be extended to [T}

0, T1], where

T1−T0 >

C0

1 +C∗(s, T,∥ϕ∥Hs+4)

, (5.22)

and there holds

sup

T06t6T1∥

uN(t)_∥Hs 6C₁(1 +C_∗(s, T,∥ϕ∥_Hs+4)). (5.23)

argument for the initial data _∥v(T0)∥Hs and theHs estimates foruN as in (5.2) and

(5.23). Then, similarly to (5.18), we have

∥vN(t)_∥Hs 6∥vN(T₀)∥_Hs+ ˜C(s, q, T,∥ϕ∥_Hs+4)N− 1

q

+ ˜C(s, T,_∥ϕ_∥Hs+4)

∫ t

T0∥

vN(τ)_∥Hsdτ (5.24)

forT0 6t6T1. Therefore, it follows from (5.24), (5.19) and the Gronwall inequality that

sup

T06t6T1∥

vN(t)_∥Hs 6C˜(s, q, T,∥ϕ∥_Hs+4)N− 1

qeC˜(s,T,∥ϕ∥Hs+4)T

for N >N0. Hence one can takeN1 =N1(s, q, T,∥ϕ∥Hs+4)>N₀ so that there holds

sup

T06t6T1∥

vN(t)_∥Hs 61 (5.25)

for all N >N1. Then, we have by (3.3), (4.4) and (5.25)

∥uN(T1)∥Hs 6∥vN(T₁)∥_Hs +

∑

j=0,±

∥uj(T1)∥Hs

6 sup

T06t6T1∥

vN(t)_∥Hs +

∑

j=0,±

sup 06t6T∥

uj(t)_∥Hs+3

61 +C∗(s, T,_∥ϕ_∥Hs+4) (5.26)

for all N >N1. Note that the above bound (5.26) is exactly same as (5.21). Hence the local solution uN _{can be uniquely extended to the solution of (1.5) on the time}

interval [T1, T1+ (T1−T0)] (defined in (5.22)) for N >N1 and satisfies

sup

T16t62T1−T0∥

uN(t)_∥Hs 6C₁(1 +C∗(s, T,∥ϕ∥_Hs+4)). (5.27)

Also note that the bound (5.27) is exactly same as (5.23). SinceT is arbitraryfinite

time, we repeat a finite number of the same procedures in the above, and continue the local solutionuN _{to the given time interval [0, T] in the classC([0, T];H}s_(R3_))∩

C1_{([0, T];H}s−1_(R3_{)) for N > N}

ϕ,T, where Nϕ,T = N(s, q, T,∥ϕ∥Hs+4) is some large

positive constant.

Next, we shall show that the solutionuN belongs to the classC([0, T];Hs+4(R3_))∩

C1_{([0, T];H}s+3_(R3_{)). Since the initial data ϕ is in H}s+4_(R3_{) and PJP is} skew-symmetric, it follows from the standard local existence theory for the 3D Euler equations in Hs_(R3_{) by [18, 21, 24] that u}N _{belongs to}

uN _∈C([0, TL];Hs+4(R3))∩C1([0, TL];Hs+3(R3))

with some local time TL>Cs/∥ϕ∥Hs+4 for all N > 0. Hence it suffices to show the

global a priori estimate for _∥uN_(t)∥

Hs+4 on [0, T] when N >N_ϕ,T.

By the above procedure on the extension of solutions, we see that the long time solution uN _{on [0, T] satisfies the uniform H}s _{estimate as}

sup 06t6T∥

uN(t)_∥Hs 6C(s, q, T,∥ϕ∥_Hs+4) (5.28)

with some positive constant C(s, q, T,_∥ϕ_∥Hs+4) for N > N_ϕ,T. Therefore, the