Analytical smoothing effect of solution for the boussinesq equations

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Analytical smoothing effect of solution for the boussinesq equations

F Cheng, C.-J Xu

To cite this version:

F Cheng, C.-J Xu. Analytical smoothing effect of solution for the boussinesq equations. 2017. �hal- 01473882�

ANALYTICAL SMOOTHING EFFECT OF SOLUTION FOR THE BOUSSINESQ EQUATIONS

F. CHENG & C.-J. XU

Abstract. In this paper, we study the analytical smoothing effect of Cauchy problem for the incompressible Boussinesq equations. Precisely, we use the Fourier method to prove that the SobolevH¹-solution to the incompressible Boussinesq equations in periodic domain is analytic for any positive time. So the incompressible Boussinesq equation admet ex- actly same smoothing effect properties of incompressible Navier-Stokes equations.

1. Introduction

In this paper, we consider the following incompressible Boussinesq equa- tions on the torus T^N = [0,2π]^N with N = 2 or 3,



















∂u

∂t −ν∆u+u· ∇u+∇p=θe_N,

∂θ

∂t −κ∆θ+u· ∇θ= 0,

∇ ·u= 0,

u(0, x) =u₀(x), θ(0, x) =θ₀(x),

(1.1)

where u(t, x) = (u₁, . . . , u_N)(t, x),(t, x) ∈ R⁺×T^N, is the velocity vector field, p=p(t, x) is the scalar pressure,θ=θ(t, x) is the scalar temperature in the content of thermal convection and the density in the modeling of geophysical fluids,ν >0 is the viscosity, andκ >0 is the thermal diffusivity, and e_N = (0, . . . ,1) is the unit vector in the x_N-direction. In addition to (1.1), we assume that u, θ, p are periodic for the spatial variable and the average value of u, θ, p on T^N vanishes:

Z

TN

u(t, x)dx= 0, Z

TN

θ(t, x)dx= Z

TN

p(t, x)dx= 0, ∀t≥0. (1.2) In physics, the Boussinesq system (1.1) is commonly used to model large scale atmospheric and oceanic flows, for example, tornadoes, cyclones, and hurricanes. It describes the dynamics of fluid influenced by gravitational force, which plays an very important role in the study of Raleigh-Bernard convection, see [13,17,18,19].

In mathematics, the Boussinesq system is one of the most commonly used simplifed models to understand some key features of the 3-D incompressible Navier Stokes equations and Euler equations. Actually, if we set θ ≡ 0,

2010 Mathematics Subject Classification. 35Q35, 35M30,76B03.

Key words and phrases. Analyticity, smoothing effect of solutions, Boussinesq equation.

1

the Boussinesq system (1.1) reduces to the incompressible Navier Stokes equations.

The well-posedness of the Boussinesq system (1.1) has been studied ex- tensively in recent years. The global well-posedness of weak solutions, or strong solutions in the case of small data for the Boussinesq equations has been considered by many authors. See, e.g. [1,3,8,9,10,11,23]. The global existence and uniqueness of smooth solutions to the 2D Boussinesq system with partial viscosity has also been studied, see [6, 7, 21, 22]. However, similar to three-dimensional Navier Stokes equation, the global existence or finite time blow-up of smooth solutions for the 3-D incompressible Boussi- nesq equations is still an open problem.

In this paper we study the analytical regularity of the strong solution to the incompressible Boussinesq equation (1.1), which is similar to the analytical regularity of Navier Stokes equation. Since Foias and Temam [12]

studied the Gevrey class regularity of Navier Stokes equations in 1989, there are many works on the Gevrey class regularity of solutions for many kinds of equations, see [2,20]. In two dimensions, the authors of [4] studied Gevrey class regularity for the two-dimensional Newton-Boussinesq equations. They considered the vorticity instead of the velocity to eliminate the pressure by transforming the velocity equation into the vorticity equation. In this paper, we use the Leray projection operator to deal with the pressure and the method can be applied to higher dimensions, which improves the previous work.

The paper is organized as follows. In Section2, we will give some notations and state our main results. In Section 3, we first recall some known results and then give some lemmas which are needed to prove the Theorem 2.1. In Section 4, we give the proof of the main Theorem2.1.

2. Notations and Main Theorems

In this section, we will give some notations and function spaces which will be used throughout the following arguments. Throughout the paper, C denotes a generic constant which may vary from line to line.

Denote by C_p^∞(T^N)^N the C^N-valued vector functions space of smooth functions satisfying periodic boundary condition, i.e., for φ∈C^∞(R^N),

φ(x₁,· · ·, x_N) =φ(x₁+ 2nπ, . . . , x_N + 2nπ), ∀n∈Z. Denote

V =

ϕ∈C_p^∞(T^N)^N; ∇ ·ϕ= 0

.

We denote L²(T^N)^N the C^N-valued vector functions space of each vector component in L²(T^N), and it is a Hilbert space for the inner product,

(f, g)_L²_(TN)=

N

X

k=1

Z

T^N

f_k(x) ¯g_k(x)dx,

where f = (f₁, . . . , f_N), g = (g₁, . . . , g_N) ∈ L²(T^N)^N. Let m > 0 be an integer, we denote H^m(T^N)^N be the C^N-valued vector function space of

each vector component in Sobolev space H^m(T^N), and the corresponding norm k · km is

kfk²m= X

|α|≤m N

X

k=1

k∂^αf_kk²L²(TN) .

We denote byL²_σ(T^N) the completion ofV under the normk · k_L²₍T^N)and similarly H_σ^m(T^N) is the closure of V under the normk · km. Using Fourier series expansion, we can identifyL²_σ(T^N) with

L²_σ(T^N) =

u= X

j∈Z^N

ˆ

u_je^ij·x, uˆ_j·j= 0, uˆ_−j = ˆu_j, uˆ₀ = 0, kuk²L²(T^N)= (2π)^N X

j∈ZN

|uˆ_j|² <∞

,

where ˆu₀= 0 meets the condition (1.2). Let r >0 be a real number, we can then identify the Sobolev space H_σ^r(T^N) as

H_σ^r(T^N) =

u= X

j∈ZN

ˆ

u_je^ij·x, uˆ_j·j = 0, uˆ_−j = ˆu_j, uˆ₀ = 0, kuk²r = (2π)^N X

j∈Z^N

|j|^2r|uˆ_j|²<∞

,

Recall that we say a smooth functionf belongs to Gevery classG^s(T^N) for some s >0, if there exists C, τ >0 such that

k∂^αfkL²(T^N)≤C|α|!^s

τ^|α|, ∀α∈N^N.

The parameter τ is called the radius of Gevrey class s. In particular, when s = 1, it is the analytical functions. Let us define the Zygmund operator Λ =√

−∆ with

Λu= X

j∈Z^N

|j|uˆ_je^ij·x, u∈H_σ¹(T^N).

The same definition of Λθfor the scalar function θ∈H¹(T^N).

We define now the function spaceD(Λ^re^τΛ1/s) forr, s, τ >0,u∈ D(Λ^re^τΛ1/s), if u∈L²_σ(T^N) and

kΛ^re^τΛ1/suk²_L²₍TN)= (2π)^N X

j∈Z^N

|j|^2re^2τ|j|1/s|uˆ_j|² <∞.

For the scalar function θ, we also write θ ∈ D(Λ^re^τΛ1/s), which means θ ∈ L²((T^N), ˆθ₀= 0 and

kΛ^re^τΛ1/sθk²_L²₍T^N)= (2π)^N X

m∈ZN

|m|^2re^2τ|m|1/s|θˆ_m|² <∞. It was proved in [16] thatD(Λ^re^τΛ1/s)⊂G^s(T^N) .

With these preparations, we can state our main results.

Theorem 2.1. If the initial data (u₀, θ₀) ∈ H_σ¹(T^N)×H¹(T^N). Then the following results hold:

(1).For two dimensions case, N = 2, the Boussinesq equations (1.1) admit an unique global solution (u, θ) satisfying

u(t, x)∈C [0,+∞[; D(Λe^tΛ)

, θ(t, x)∈C [0,+∞[; D(Λe^tΛ) . (2).For three dimensions case, N = 3, the Boussinesq equations (1.1) admit a unique local solution (u, θ) satisfying

u(t, x)∈C [0, T₁];D(Λe^tΛ)

, θ(t, x)∈C [0, T₁];D(Λe^tΛ) , where T₁ >0 depends on the initial data (u₀, θ₀).

Remark 2.1. .

1. Following the arguments of Foias and Temam [12], one can improve the regularity of time tfor the solution (u, θ) by extending to the complex plane and showing that the solution is actually analytic in time variable. Since the computations are standard, we omit these details.

2. The analytical smoothing effect of Theorem2.1imply the Gevery smooth- ing effect in D(Λ^re^τΛ1/s)⊂G^s(T^N) for any s≥1.

3. Premilinary results

In this Section, we recall some known results on the incompressible Boussi- nesq system (1.1). Let us first recall the definition of weak solutions and strong solutions for the Boussinesq system (1.1).

Definition 3.1 (weak solutions). For any T > 0, we call (u, θ) the weak solution of the Boussinesq system (1.1), if

u∈C [0, T] ;L²_σ(T^N)

∩L² [0, T] ;H_σ¹(T^N) , θ∈C [0, T] ;L²(T^N)

∩L² [0, T] ;H¹(T^N) , and u satisfies the following weak formulation

Z t 0

Z

T^N

∂_tϕ·u+∇ϕ:u⊗u+θe_N ·ϕ+ν∆ϕ·udxds= 0, for all ϕ∈C₀^∞ [0, T]×T^N;R^N

with∇ ·ϕ= 0, andθsatisfies the following weak formulation

Z _t

0

Z

T^N

θ∂_tψ+θ(u· ∇ψ) +κθ∆ψ= 0, for all ψ∈C₀^∞ [0, T]×T^N; R

.

The pressure term p in the velocity equation of (1.1) is the Lagrange multiplier, which is eliminated by taking the L²-inner product with the divergence-free vector field. However, it can be determined by

∆p=−div(u· ∇u) + div(θe_N), with psatisfying periodic boundary conditions and (1.2).

Analogous to the incompressible Navier Stokes equation, the global exis- tence of weak solutions to the Boussinesq equations is standard, see [5, 11, 14,15] and references therein.

Theorem 3.2 (weak solutions). Let (u₀, θ₀) ∈ L²_σ(T^N)×L²(T^N). There exists a weak solution (u, θ) of the Boussinesq equations (1.1) such that,

u∈C [0, T] ;L²_σ(T^N)

∩L² [0, T] ;H_σ¹(T^N) , θ∈C [0, T] ;L²(T^N)

∩L² [0, T] ;H¹(T^N) . Such solutions satisfies, for all t∈[0, T], the energy inequalities

kθ(t)k²L²(TN)+ 2κ Z t

0 k∇θ(s)k²L²(TN)ds≤ kθ₀k²L²(TN), and

ku(t)k²L²(T^N)+ 2ν Z _t

0 k∇u(s)k²L²(T^N)ds≤ ku₀k²L²(T^N)+Ct²kθ₀k²L²(T^N), for all T >0, and some constantC >0.

Remark 3.1. For two dimensions case, N = 2, the weak solution obtained in Theorem3.2is unique. Whether the uniqueness of weak solution for three dimensions case N = 3 is still unknown.

In the following, we study a class of more regular solutions to the Boussi- nesq equations, which is called strong solutions analogous to the Navier- Stokes equations and introduced by Temam in [24].

Definition 3.3 (strong solutions). Let (u₀, θ₀)∈H_σ¹(T^N)×H¹(T^N), Then (u, θ) is a strong solution of the Boussinesq equations (1.1) if

u∈C [0, T] ;H_σ¹(T^N)

∩L² [0, T] ;H_σ²(T^N) , θ∈C [0, T] ;H¹(T^N)

∩L² [0, T] ;H²(T^N) , and (u, θ) satisfies (3.1) and (3.2).

The existence and uniqueness of strong solutions for the incompressible Boussinesq equations (1.1) is analougous to the results of Navier Stokes equations, which we state as follows.

Theorem 3.4. Let(u₀, θ₀)∈H_σ¹(T^N)×H¹(T^N). Then the following results hold.

(1). For two dimensions case N = 2, for any T >0, the Boussinesq equa- tions (1.1) admit an unique strong solution(u, θ) on [0, T]satisfying

u∈C [0, T] ;H_σ¹(T²)

∩L² [0, T] ;H_σ²(T²) , θ∈C [0, T] ;H¹(T²)

∩L² [0, T] ;H²(T²) .

(2). For three dimensions N = 3, there exists T_∗>0 such that the Boussi- nesq equations (1.1)admit a unique strong solution(u, θ)on[0, T_∗]satisfying

u∈C [0, T] ;H_σ¹(T³)

∩L² [0, T] ;H_σ²(T³) , θ∈C [0, T] ;H¹(T³)

∩L² [0, T] ;H²(T³) , where T_∗ >0 depends on the initial data (u₀, θ₀).

Proof. For the self-contenant of paper, we give the `a priori estimate of the solutions, then the construction of approximate solutions follows from the standard Galerkin method which we refer readers to [24].

Let us assume that (u, θ) is the smooth solution of the incompressible Boussinesq equations. We multiply the velocity equation of (1.1) by Λ²u and integrate over T^N. After integrating by parts, we have

1 2

d

dtkΛu(t,·)k²_L²₍T^N)+νkΛ²uk²_L²₍T^N)

= Λ(θe_N),Λu

L²(T^N)− u· ∇u,Λ²u

L²(T^N), (3.1) where ∇p,Λ²u

L²(TN) vanishes becauseu is divergence-free and Λ² =−∆ does not violate this property on the torus. Then we multiply the thermal equation of (1.1) by Λ²θ and integrate over T^N. Integrating by parts, we obtain

1 2

d

dtkΛθ(t,·)k²L²(T^N)+κkΛ²θk²L²(T^N)=− u· ∇θ,Λ²θ

L²(T^N). (3.2) We first recall the following Sobolev inequality (see [24], now usually called the Gagliardo-Nirenberg inequality),

kfkL^∞(T²) ≤Ckfk^1/2_L²₍T²)kΛ²fk^1/2_L²₍T²), ∀f ∈H²(T²), (3.3) kfkL^∞(T³) ≤CkΛfk^1/2_L²₍T³)kΛ²fk^1/2_L²₍T³), ∀f ∈H²(T³). (3.4) Using the Cauchy-Schwartz inequality and the Sobolev inequality, we have

u· ∇u,Λ²u

L²(T²)

≤ ku· ∇ukL²(T²)kΛ²ukL²(T²)

≤ kukL^∞(T²)k∇ukL²(T²)kΛ²ukL²(T²)

≤Ckuk^1/2_L²₍T²)kΛukL²(T²)kΛ²uk^3/2_L²₍T²), (3.5) and

u· ∇u,Λ²u

L²(T³)

≤ ku· ∇ukL²(T³)kΛ²ukL²(T³)

≤ kukL^∞(T³)k∇ukL²(T³)kΛ²ukL²(T³)

≤CkΛuk^3/2_L²₍T³)kΛ²uk^3/2_L²₍T³). (3.6) We have also

u· ∇θ,Λ²θ

L²(T²)

≤ ku· ∇θkL²(T²)kΛ²θkL²(T²)

≤Ckuk^1/2_L²₍T²)kΛ²uk^1/2_L²₍T²)

× k∇θkL²(T²)kΛ²θkL²(T²), (3.7) and

u· ∇θ,Λ²θ

L²(T³)

≤CkΛuk^1/2_L²₍T³)kΛ²uk^1/2_L²₍T³)kΛθkL²kΛ²θkL²(T³). (3.8) Case of N = 2. we add (3.1) and (3.2) and apply the Young’s inequality on (3.5) and (3.7),

1 2

d

dt kΛuk²L²(T²)+kΛθk²L²(T²)

+νkΛ²uk²L²(T²)+κkΛ²θk²L²(T²)

≤Ckuk^1/2_L²₍T²)kΛukL²(T²)kΛ²uk^3/2_L²₍T²)+kΛ(θen)kL²(T²)kΛukL²(T²)

+Ckuk^1/2_L²₍T²)kΛ²uk^1/2_L²₍T²)kΛθkL²(T²)kΛ²θkL²(T²)

≤ ν

2kΛ²uk²L²(T²)+κ

4kΛ²θk²L²(T²)+C_νkuk²L²kΛuk⁴L²(T²)

+C_κ,νkuk²L²(T²)kΛθk⁴L²(T²)+kΛθkL²(T²)kΛukL²(T²). Thus

d

dt kΛuk²_L²₍T²)+kΛθk²_L²₍T²)

+νkΛ²uk²_L²₍T²)+κkΛ²θk²_L²₍T²)

≤C_κ,νkuk²_L²(kΛuk⁴_L²₍T²)+kΛθk⁴_L²₍T²)) +kΛθkL²(T²)kΛukL²(T²). (3.9) Thus if we denote

Y(t) = 1 +kΛu(t,·)k²_L²₍T²)+kΛθ(t,·)k²_L²₍T²), we have

d

dtY(t)≤C_ν,κ^′ (1 +kuk²L²(T²)) 1 +kΛuk²L²(T²)+kΛθk²L²(T²)

Y(t), where C_ν,κ^′ is some large constant depending on ν, κ.

Then the Gronwall’s inequality imply, Y(t)≤Y(0) exp

C_ν,κ^′

Z _t

0

(1 +ku(s,·)k²L²) 1 +kΛu(s)k²L² +kΛθ(s)k²L²

ds

.

Noting that the strong solution is always weak solution, using Theorem 3.2, we have

u∈C [0, T] ;L²_σ

∩L² [0, T] ;H_σ¹

, θ∈C [0, T] ;L²

∩L² [0, T] ;H¹ , and

sup

t∈[0,T]ku(t,·)kL² ≤ ku₀kL², Z _t

0 k∇u(s,·)k²L²ds≤ ku₀k²_L² +Ckθ₀k²_L²t²

2ν ,

Z _t

0 k∇θ(s,·)k²_L²ds≤ kθ₀k²_L² 2κ . Then ,

Y(t)≤Y(0) exp

C_ν,κ^′ (1 +ku₀k²L²)

t+ku₀k²_L²

2ν +Ckθ₀k²_L²t²

2ν +kθ₀k²_L² 2κ

.

Then it gives an upper bound of Y(t), which also implies for anyT >0, u∈L^∞ [0, T] ;H_σ¹(T²)

, θ∈L^∞ [0, T] ;H¹(T²) . If we integrate (3.9) from 0 toT for some fixed T >0, we obtain

u∈L² [0, T] ;H_σ²(T²)

, θ∈L² [0, T] ;H²(T²) . With these facts, one can obtain

∂u

∂t ∈L² [0, T] ;L²_σ(T²) , ∂θ

∂t ∈L² [0, T] ;L²(T²)

Combining these estimates, one can also show the solution (u, θ) is actually satisfying

u∈C [0, T] ;H_σ¹(T²)

, θ ∈C [0, T] ;H¹(T²)

, ∀ T >0.

The continuity oftinH_σ¹(T²) foruand inH¹(T²) forθfollows the arguments of Temam [Chap 3. [24]]. This proves the global strong solution in two- dimensional space.

Case of N = 3. We apply the Young’s inequality on (3.6) and (3.8) to bound the right hand side of (3.1) and (3.2),

1 2

d

dt kΛuk²_L²₍T³)+kΛθk²_L²₍T³)

+νkΛ²uk²_L²₍T³)+κkΛ²θk²_L²₍T³)

≤CkΛuk^3/2_L²₍T³)kΛ²uk^3/2_L²₍T³)+kΛ(θen)kL²(T³)kΛukL²(T³)

+CkΛuk^1/2_L²₍T³)kΛ²uk^1/2_L²₍T³)kΛθkL²(T³)kΛ²θkL²(T³), thus

1 2

d

dt kΛuk²L²(T³)+kΛθk²L²(T³)

+νkΛ²uk²L²(T³)+κkΛ²θk²L²(T³)

≤ ν

2kΛ²uk²_L²₍T³)+κ

4kΛ²θk²_L²₍T³)+C_νkΛuk⁶_L²₍T³)

+C_κ,νkΛuk²_L²₍T³)kΛθk⁴_L²₍T³)+kΛθkL²(T³)kΛukL²(T³). (3.10) If we set Z(t) = 1 +kΛu(t,·)k²_L²₍T³)+kΛθ(t,·)k²_L²₍T³), we obtain

d

dtZ(t)≤C_ν,κ^′ Z(t)³. This implies

Z(t)≤ Z(0) q1−2C_ν,κ^′ Z(0)²t

, 0< t < 1 2C_ν,κ^′ Z(0)². And thus

1 +kΛu(t,·)k²L²(T³)+kΛθ(t,·)k²L²(T³)

≤2(1 +kΛu₀k²L²(T³)+kΛθ₀k²L²(T³)), (3.11) if t≤T₁(kΛu₀kL²(T³),kΛθ₀kL²(T³)) = _8C′ ³

ν,κZ(0)². Then we obtain u∈L^∞ [0, T₁] ;H_σ¹(T³)

, θ∈L^∞ [0, T₁] ;H¹(T³) Integrating (3.10) from 0 toT₁, we can obtain

u∈L² [0, T₁] ;H_σ²(T³)

, θ∈L² [0, T₁] ;H²(T³) . With these facts, one can obtain

∂u

∂t ∈L² [0, T1] ;L²_σ(T³) , ∂θ

∂t ∈L² [0, T1] ;L²(T³) .

Combining these estimates, one can then show the solution (u, θ) is actually satisfying

u∈C [0, T₁] ;H_σ¹(T³)

, θ ∈C [0, T₁] ;H¹(T³) .

This proved the local existence of strong solution for three-dimensional space.

The proof of the existence for both two dimensions and three dimensions can be made rigorous by considering the Galerkin approximation procedure,

which is standard in the book [24].

Remark 3.2. The difference in the Sobolev inequality (3.3) and (3.4) caused by the spatial dimension N makes huge influence on the lifespan of the solution, which is showed in the above proof.

4. Proof of Theorem 2.1

In this Section, we give the proof of Theorem 2.1. In order to prove the main Theorem 2.1, we recall the following Lemmas in [12] concerning the estimates of nonlinear terms.

Lemma 4.1. Let u, v, w be given inD(e^τΛ1/sΛ²) for τ >0, s >0. Then the following estimates hold, for N = 2 or 3,

e^τΛ1/sΛ(u· ∇v), e^τΛ1/sΛw

L²(T^N)

≤Cke^τΛ1/suk^1/21 ke^τΛ1/suk^1/22

× ke^τΛ1/svk1ke^τΛ1/swk2, and

ke^τΛ1/s(u· ∇v)kL²(T^N) ≤Cke^τΛ1/suk^1/21 ke^τΛ1/suk^1/22 ke^τΛ1/svk1, where C >0 is a constant.

Remark 4.1. In Lemma4.1,uisR^N-valued vector function whilev, w can be either R^N-valued vector function orR-valued scalar function.

Then we are able to prove the Theorem 2.1 following the ideal of Foias and Temam [12].

Proof of Theorem 2.1. For the sake of simplicity, we shall only consider the time variable in the real case and we remark that the results can be extended to the complex case following the same arguments of [12]. For this part, we set the radius of Gevrey class τ(t) =t.

We recall that the usual strategy to approximate the Navier Stokes equa- tion is to project the velocity field onto the divergence-free field. Here we also need to write the velocity equation of the Boussinesq system (1.1) into the following form

∂u

∂t −νP∆u+P(u· ∇u) =P(θe_n), (4.1) where P is the Helmholtz-Leray orthogonal projection, i.e., projection onto divergence-free vector fields. In this way, we eliminate the pressure term as Foias and Temam did in [12]. Denote the operator A = −P∆, the eigenvectors {E_j}^∞j=1 of A constitutes the othonormal basis of L²_σ(T^N). We denote the corresponding eigenvalues by {λ_j}^∞j=1 with 0< λ₁ < . . .≤λ_j ≤ λ_j+1 ≤. . ., then AE_j =λ_jE_j. We note that P commutes with −∆ on the torus and A=−∆ when acting on the divergence-free vector field.

The thermal equation of the Boussinesq system (1.1) is the second order parabolic equation and the eigenvectors of −∆ also constitute the orthonor- mal basis of L²(T^N). Let {e_α}^∞α=1 be the orthonormal basis of L²(T^N) and {τ_α}^∞α=1 be the corresponding eigenvalues, which satisfies

0< τ₁ < . . .≤τ_α≤τ_α+1 ≤. . . .

We will then approximate the equation (4.1) and the thermal equation of (1.1) by the Galerkin method. For fixed integer m ∈ N, let P_m be the

projection from L²_σ(T^N) onto the subspace W_m spanned by {E_j, 1 ≤ j ≤ m, j ∈Z} and P_m be the projection from L²(T^N) onto the subspace V_m of L²(T^N) spanned by{e_α, 1≤α≤m, α∈Z}.

We are looking for solution (u_m, θ_m) to the following approximate equa- tions

∂u_m

∂t −νP∆u_m+P_mP(u_m· ∇u_m) =P_mP(θ_me_N), (4.2)

∂θ_m

∂t −κ∆θ_m+P_m(u_m· ∇θ_m) = 0, (4.3) u_m(0) =P_mu₀, θ(0) =P_mθ₀, (4.4) where

u_m(t, x) = X

1≤j≤m

ξ_j,m(t)E_j, θ_α(t, x) = X

1≤α≤m

η_α,m(t)e_α.

The function u_m, θ_m are analytics for x variables, in fact the sequence of E_j’s and e_α’s is the linear combinations of the sequence of functions W_k,ℓ and w_n,

W_k,ℓ=a_k,ℓ e_ℓ− k_ℓk

|k|²

e^ik·x, w_n= 1

(2π)^Ne^in·x, ℓ= 1,· · ·, N, where k = (k₁, . . . , k_N), n = (n₁, . . . , n_N) ∈ Z^N \ {0}, e₁, . . . , e_N is the canonical basis of R^N. a_k,ℓ is the coefficient that make the sequence W_k,ℓ to be orthonormal in L²_σ(T^N).

After takingL²-inner product of (4.2) withE_j and takingL²-inner prod- uct of (4.3) withe_α, for 1≤j ≤m and 1 ≤α ≤m. The Cauchy problem (4.2)-(4.4) is equivalent to the following Cauchy problem for ordinary differ- ential system

d

dtξ_j,m(t) +νλ_jξ_j,m(t) + X

1≤k,ℓ≤m

A_k,ℓ,jξ_k,m(t)ξ_ℓ,m(t)

= X

1≤γ≤m

C_γ,jη_γ,m(t), (4.5) d

dtη_α,m(t) +κτ_αη_α,m(t) + X

1≤j,β≤m

B_j,β,αξ_j,m(t)η_β,m(t) = 0, (4.6) ξ_j,m(0) = (u0,E_j), η_α,m(0) = (θ0,e_α), (4.7) where

A_k,ℓ,j= (E_k· ∇E_ℓ,E_j)_L²₍TN), B_j,β,α = (E_j· ∇e_β,e_α)_L²₍T^N),

C_γ,j = (eγe_N,E_j)_L²₍TN).

The standard theory of ordinary differential equations indicates that the ODE system (4.5)-(4.7) admet a unique local solution ξ_j,m(t), η_α,m(t)

1≤j,α≤m

on some interval [0, T_m].

We prove now the solution of the ODE system (4.5)-(4.7), ξ_j,m(t), ηα,m(t) can be extended to a global in time solution for any fixed m. To show this,

we note that eachE_j ande_αare analytics. Then we can perform integration by parts to obtain

A_k,ℓ,j = (E_k· ∇E_ℓ,E_j)_L²₍T^N)

=−(E_k· ∇E_j,E_ℓ)_L²₍T^N)

=−A_k,j,ℓ, ∀1≤k, ℓ, j ≤m, (4.8) where we used the fact∇ ·E_k= 0. Integrating by parts, one can also obtain B_j,β,α=−B_j,α,β, ∀ 1≤j, β, α≤m. (4.9) So if we multiply (4.5) byξ_j,m(t) and take sum over {1≤j≤m, j ∈Z}, we obtain

1 2

d dt

m

X

j=1

ξ_j,m(t)²+ν

m

X

j=1

λ_jξ_j,m(t)²=

m

X

j=1 m

X

γ=1

C_γ,jη_γ,m(t)ξ_j,m(t), (4.10) where we infer from (4.8) the following fact

X

1≤k,ℓ,j≤m

A_k,ℓ,jξ_k,mξ_ℓ,mξ_j,m=− X

1≤k,ℓ,j≤m

A_k,j,ℓξ_k,mξ_j,mξ_ℓ,m = 0.

If we multiply (4.6) byη_α,m(t) and take sum over {1≤α≤m, α∈Z}, we obtain

1 2

d dt

m

X

α=1

η_α,m(t)²+κ τ_α

m

X

α=1

η²_α,m= 0, (4.11) where we infer from (4.9) the fact

X

1≤j,α,β≤m

B_j,α,βξ_j,mη_α,mη_β,m=− X

1≤j,α,β≤m

B_j,β,αξ_j,mη_β,mη_α,m= 0.

The equation (4.11) implies that, for any m∈N, kθ_m(t,·)k²_L²₍T^N)=

m

X

α=1

η_α,m(t)² ≤

m

X

α=1

η_α,m(0)² ≤ kθ₀k²_L²₍T^N),∀t >0.

With this fact, one can infer from (4.10) that d

dtku_m(t,·)kL²(T^N) ≤ kθ_m(t,·)kL²(T^N), (4.12) by noting thatku_m(t,·)k²_L²₍T^N)=Pm

j=1ξ_j,m(t)². If we integrate (4.12) from 0 to t, we obtain

ku_m(t,·)kL²(T^N)≤ ku_m(0,·)kL²(T^N)+ Z t

0 kθ_m(s,·)kL²(T^N)ds,

≤ ku₀k²_L²₍TN)+tkθ₀kL²(T^N), ∀m∈N, ∀t >0.

The above priori estimates show that the solution (ξ_j,m(t), η_α,m(t)) is bounded only by the initial data, then we can repeat the arguments above to extend the local solution to arbitrary time interval [0, T]. Thus for fixed integer m, the Cauchy problem of ordinary differential equations (4.5)-(4.7) possess a unique solution (ξ_j,m(t), η_α,m(t))∈C¹([0, T]) for allT >0. Equivalently, we have the solution (u_m, θ_m) to the approximate equation (4.2)-(4.4) satisfies

u_m ∈C¹ [0, T] ;H_σ^r(T^N)

, θ_m∈C¹ [0, T] ;H^r(T^N)

, ∀T >0, ∀r >0,

for fixed m∈N.

Once we obtained the approximating solution u_m, θ_m

, we then perform the Gevrey norm estimates on (u_m, θ_m). For fixed integer m, as a finite summation of E_j and e_α, the solution u_m, θ_m

belongs to Gevrey class D(Λ^re^τΛ) for arbitraryr > 0. Let T >0 be fiexed. For 0< t < T, we take L²-inner product of the velocity equation of (4.2) with Λ²e^2tΛu_m, which gives

Λe^tΛ d

dtu_m(t),Λe^tΛu_m(t)

L²(T^N)+νkΛ²e^tΛu_m(t)k²_L²₍T^N)

= Λe^tΛ(θ_me_N), e^tΛΛu_m(t)

L²(T^N) (4.13)

− Λe^tΛ(um· ∇u_m), e^tΛΛum

L²(T^N),

where we used the fact thatu_mis divergence free andPis symmetric. Taking the L²-inner product of the thermal equation of (4.3) with Λ²e^2tΛθ_m, we obtain, similarly,

Λe^tΛ d

dtθ_m(t),Λe^tΛθ_m

L²(T^N)+κkΛ²e^tΛθ_mk²_L²₍T^N)

+ Λe^tΛ(u_m· ∇θ_m),Λe^tΛθ_m

L²(T^N)= 0. (4.14) By Plancherel’s theorem and summing over (4.13) and (4.14), we can write

1 2

d

dt kΛe^tΛu_m(t)k²L²(T^N)+kΛe^tΛθ_m(t)k²L²(T^N)

+νkΛ²e^tΛu_m(t)k²_L²₍TN)+κkΛ²e^tΛθ_mk²_L²₍TN)

= Λ²e^tΛu_m,Λe^tΛu_m

L²(T^N)+ Λ²e^tΛθ_m,Λe^tΛθ_m

L²(T^N)

+ Λe^tΛ(θme_N), e^tΛΛum(t)

L²(T^N) (4.15)

− Λe^tΛ(u_m· ∇u_m), e^tΛΛu_m

L²(T^N)

− Λe^tΛ(u_m· ∇θ_m),Λe^tΛθ_m

L²(TN). By Cauchy-Schwartz inequality, we have

Λ²e^tΛu_m,Λe^tΛu_m

L²(T^N)

≤ ν

8kΛ²e^tΛu_mk²_L²₍TN)+C_νkΛe^tΛu_mk²_L²₍TN), and

Λ²e^tΛθ_m,Λe^tΛθ_m

L²(TN)

≤ κ

8kΛ²e^tΛθ_mk²L²(T^N)+C_κkΛe^tΛθ_mk²L²(T^N). From Lemma 4.1, we have,

Λe^tΛ(u_m· ∇u_m),Λe^tΛu_m

L²(T^N)

≤Cke^tΛu_mk^3/21 ke^tΛu_mk^3/22 , (4.16) and

Λe^tΛ(u_m· ∇θ_m),Λe^tΛθ_m

L²(TN)

≤Cke^tΛu_mk^1/21 ke^tΛu_mk^1/22 ke^tΛθ_mk1ke^tΛθ_mk2. (4.17) Noting that on the torus T^N we have the following Poincar´e inequality,

ke^tΛu_mkL²(T^N)≤Cke^tΛu_mk1, ke^tΛθ_mkL²(T^N)≤Cke^tΛθ_mk1,