# On the Existence of Long-Time Classical Solutions for the 2D Inviscid Boussinesq Equations

Authors: Jiafa Xu; Lishan Liu
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we consider the Cauchy problem for the 2D inviscid Boussinesq equations withN being the buoyancy frequency. It is proved that for general initial data u0∈Hs with s>3, the life span of the classical solutions satisfies T>ClnN3/4.


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## Body

## 1. Introduction

The 2D inviscid Boussinesq equations read as(1)∂tη+u⋅∇η=0,x,t∈ℝ2×ℝ+,∂tu+u⋅∇u+∇q=ηe⟶2,∇⋅u=0.The unknown functionsη=ηx,t,u=u1x,t,u2x,t,q=qx,t stand for the temperature, the velocity field, and the scalar pressure, respectively. e⟶2=0,1 is the unit vector in the vertical direction. They can describe the natural convection in the 2D inviscid incompressible fluids such as the dynamics of the ocean or the atmosphere (see, e.g., [1–3]). Besides the physical significance, there is also a strong mathematical motivation for studying these equations. In fact, the 2D Boussinesq equation can be used as a model for the 3D axisymmetric Euler equations (see, e.g., [4]). Therefore, the study of equation (1) can provide us with the useful information to understand the Euler equations.Due to the physical and strong mathematical significance of the 2D Boussinesq equations, many researchers have paid much attention on its study. For the Cauchy problem around the stationary solution0,0,0, global regularity of solutions is known when the classical dissipation is present in at least one of the equations (1) (see [5, 6]) or under a variety of more general conditions on dissipation (see [7]) or under adding a damp term (see [8–11]). In contrast, the global regularity problem on the inviscid 2D Boussinesq equations (1) now is still open. Some attempts have been made for a step forward in this direction, which appears to be out of reach in spite of the progress on the local well-posedness and regularity criteria (see [12, 13]). On the other hand, for the initial boundary problem, Hu et al. [14] obtained the global well-posedness for the Boussinesq equations with non-slip boundary condition, and Lai et al. [2] and Littman [4] obtained the local well-posedness for the Boussinesq equations with slip boundary condition. Recently, some researchers started to consider the global well-posedness around the stationary solution of the strong stratification [11, 15, 16]. In particular, Elgindi and Widmayer [15] proved the long-time existence which is the life span of the associated solutions is ε−4/3 if the initial data are of size ε. In this paper, we will present a new life span which is suitable for general initial data by the method of Strichartz estimate combining with a blowup criterion.To state our result more precisely, we firstly consider the solutions of (1) around a stratified solution and rewrite equation (1). It is well known that equation (1) admits an explicit stationary solution us,qs,ηs of the form(2)us,qs,ηs=0,N2x222,N2x2,satisfying the hydrostatic balance(3)dqsdx2=ηs,where N is called the buoyancy or the Brunt–Vaisala frequency and represents the strength of stable stratification. Setting(4)θ=η−ηs,p=q−qs,we can reformulate (1) into(5)∂tθ+u⋅∇θ=−N2u2,∂tu+u⋅∇u+∇p=θe2,∇⋅u=0.Furthermore, let us set(6)u3=θN,U=u1,u2,u3T,and we get(7)∂tU+U⋅∇˜U+∇p=NJU,∇˜⋅U=0,where ∇˜=∇,0T and J is a constant matrix given by(8)J=0000010−10.Since∇˜⋅U=0, let us introduce the extended Helmholtz projiector of the velocity u onto the divergence-free vector fields which is defined by(9)ℙ≔ℙ2001=δjk+RjRk1≤j,k≤2001,where Rj1≤j≤2 denote the Riesz transforms on ℝ2. Applying the operator ℙ to (7) gives the following equation:(10)∂tU+ℙU⋅∇˜U=NℙJℙU.The initial data to the above equations are given by(11)U|t=0=U0.Now we state the main result.Theorem 1.
Lets∈ℕ satisfy s>3 and(12)U0=u1,u2,u3∈Hsℝ2,∇˜⋅U0=0.
Then, equations (10) and (11) possess a unique solution:(13)U∈C0,T;Hs,with(14)T=ClnN3/4,where the constant C is independent of N.Remark 1.
This result implies that the existence time will be larger as the buoyancy increases and the life span of the classical solutions satisfies(15)T>ClnN3/4.


### 1.1. Plan of the Article

The paper is organized as follows. InSection 2, we first derive the explicit formula of solutions to the linearized equations of (10) and (11), and then we establish the decay estimate and Strichartz estimate of a linear propagator. In Section 3, we establish the blowup criterion of equations (10) and (11). In Section 4, we present the proof of Theorem 1.


### 1.2. Notations

Throughout this paper, we denote byC the constants which may differ from line to line. Lp1≤p≤∞, Hss∈ℝ, and Bp,qs,s∈ℝ,1≤p,q≤∞ denote the Lebesgue spaces, Sobolev spaces, and the inhomogeneous Besov spaces, respectively. Let ℱ,ℱ−1 denote the Fourier transformation and inverse Fourier transformation, respectively. The Littlewood–Paley multipliers Δj∈ℤ are defined by(16)Δjf=ℱ−1ψξ2jℱf,where ψ∈C0∞ℝn,suppψ⊂ξ∈ℝn|1/2≤ξ≤2 such that(17)∑j∈ℤψξ2j=1,∀ξ≠0.The low-frequency multiplierχξ is defined by(18)χξ=1−∑j≥1ψξ2j.


## 1.1. Plan of the Article

The paper is organized as follows. InSection 2, we first derive the explicit formula of solutions to the linearized equations of (10) and (11), and then we establish the decay estimate and Strichartz estimate of a linear propagator. In Section 3, we establish the blowup criterion of equations (10) and (11). In Section 4, we present the proof of Theorem 1.


## 1.2. Notations

Throughout this paper, we denote byC the constants which may differ from line to line. Lp1≤p≤∞, Hss∈ℝ, and Bp,qs,s∈ℝ,1≤p,q≤∞ denote the Lebesgue spaces, Sobolev spaces, and the inhomogeneous Besov spaces, respectively. Let ℱ,ℱ−1 denote the Fourier transformation and inverse Fourier transformation, respectively. The Littlewood–Paley multipliers Δj∈ℤ are defined by(16)Δjf=ℱ−1ψξ2jℱf,where ψ∈C0∞ℝn,suppψ⊂ξ∈ℝn|1/2≤ξ≤2 such that(17)∑j∈ℤψξ2j=1,∀ξ≠0.The low-frequency multiplierχξ is defined by(18)χξ=1−∑j≥1ψξ2j.


## 2. Linearized Equations

In this section, we derive the representation of solutions to the linearized equations of (10) and (11) and establish the decay estimate and Strichartz estimate of the linear propagator e±iNtwD given by(19)e±iNtwDf≔ℱ−1e±iNtξ1/ξf^.


### 2.1. The Representation of Solutions to Linearized Equations

We study the following linearized equations associated to equations (10) and (11):(20)∂tU=NℙJℙU,∇˜⋅U=0,U0,x=U0x.Applying the Fourier transform to (20) yields(21)∂tU^−NPξJPξU^=0,ξ,0⋅U^=0,U^0,x=U^0ξ,where Pξ is the multiplier matrix of the operator ℙ defined by(22)ℙU^ξ=PξU^ξ,which is given explicitly by(23)Pξ≔δjk−ξjξk|ξ|21⩽j,k⩽2001.Then, a direct calculation yields(24)PξJPξ=1ξ2ξ22−ξ1ξ20−ξ1ξ2ξ12000ξ2,and the eigenvalues of PξJPξ are(25)ω0=0,ω±ξ≔±iωξ=±iξ1ξ,and the corresponding eigenvectors are(26)e0=1ξξ1,ξ2,0,e±=1ξ1ξ±iξ1ξ2,∓iξ12,1.Thus, the solution of (21) is(27)U^t,ξ=eNPξJPξt,U^0ξ=U^0ξ,e0e0+eiNωξtU^0ξ,e+e++e−iNωξtU^0ξ,e−e−.Since∇˜⋅U=0, we have U^0ξ,e0=0. Hence, we get(28)U^t,ξ=eNPξJPξt,U^0ξ=eiNωξtU^0ξ,e+e++e−iNωξtU^0ξ,e−e−.Setting(29)P+U0=ℱ−1U^0ξ,e+e+,P−U0=ℱ−1U^0ξ,e−e−,one has(30)Ut,x=eiNωDtP+U0+e−iNωDtP−U0.


### 2.2. Decay Estimate

We derive the following decay estimate of the operatore±itNwD.Lemma 1.
It holds that(31)e±itNwDΦDfL∞≤C1+Nt−1/2fL1,for all f∈L1ℝ2, where the operator ΦD is defined by(32)ΦDf=ℱ−1Φ^ξf^,in which Φ^ξ satisfies(33)suppΦ^⊂ξ∈ℝ2|2−2≤ξ≤22,Φ^ξ=1,∀ξ∈ξ∈ℝ2|2−1≤ξ≤2.Actually, the result of Lemma1 is an immediate consequence of the following lemma.Lemma 2.
There exists a positive constantC independent of t,x∈ℝ1+2 such that(34)∫ℝ2eixξ±iNtξ1/ξΦ^ξdξ≤C1+Nt−1/2.Next, we give the details of the proof of Lemma2. Firstly, we recall an important lemma (see Keel and Tao [17] and Majda [18]).Lemma 3.
Letdσ be a surface measure on a smooth surface S in ℝn and let ϕ∈C0∞ℝn. Suppose that for all x∈S, at least k of the principle curvatures are non-zero. Then, it holds that(35)d^μη=ϕd^ση≤Cη−k/2.


#### 2.2.1. The Proof of Lemma2

By the theory of Fourier transform of measures supported on surfaces [18], we have(36)∫ℝ2eixξ±iNtξ1/ξψξdξ=∫ℝ2eix,Nt⋅ξ,±ξ1/ξψξdξ=d^μη,η=x,Nt,where μ is some measure supported on the surface S which is given by(37)S=ξ,ρ∈ℝ2×ℝ:ρ=±ξ1ξ,14≤ξ≤4.By Lemma3, the decay of d^μη is determined by the number of non-vanishing principle curvatures of the surface S. Equivalently, the number of non-vanishing principle curvatures of the surface S is the rank of the Hessian matrix Hρ=∂2ρ/∂ξj∂ξk1≤j,k≤2. By some computations, we obtain(38)Hρ=∓1ξ5−3ξ1ξ22ξ23ξ12−ξ2ξ23ξ12−ξ2ξ13ξ22−ξ2.From (38), we have(39)DetHρ=±ξ22ξ6.(39) shows that the surface S has a non-vanishing principle curvatures unless ξ2=0. Thus, we decompose(40)∫ℝ2eixξ±iNtξ1/ξψξdξ=I1+I2,where(41)I1=∫ℝ2eixξ±iNtξ1/ξ1−φξ2ψξdξ,I2=∫ℝ2eixξ±iNtξ1/ξφξ2ψξdξ,where φ is a smooth function on ℝ such that φ=1 on −1/2,2 and suppφ⊂−1,1.For the estimate ofI1L∞, by Lemma 3, we have(42)I1L∞≤Cx,Nt−2/2≤CNt−1.For the estimate ofI2L∞, since ξ2=0, we see that(43)Hρ=∓1ξ15000−ξ13.From (43) and the fact ξ∼11/4≤ξ≤4, we see one non-vanishing principle curvature of the surface S. By Lemma 3, we have(44)I2L∞≤Cx,Nt−1/2≤CNt−1/2.Combining (36), (42), and (44), we obtain(45)∫ℝ2eixξ±iNtξ1/ξψξdξ≤CNt−1/2.For smallt, it is trivial that(46)∫ℝ2eixξ±iNtξ1/ξψξdξ≤C.From (45) and (46), we have(47)∫ℝ2eixξ±iNtξ1/ξψξdξ≤C1+Nt−1/2.Thus, we complete the proof of Lemma2.


### 2.3. Strichartz Estimate

Firstly, we recall the following result obtained by Hu et al. [14].Lemma 4.
LetSt,t∈ℝ be a family of operators. Suppose that for all t,s∈ℝ,(48)SsS∗tfxL∞≤1+t−s−σfL1,SsS∗tfxL2≤fL2.Then, the estimates(49)StfxLtqLxr≤fL2hold for all 2≤q,r≤∞ with q,r,σ≠2,∞,1 satisfying(50)1q+σr≤σ2.From Lemmas1 and 4, the fact ΦDΔ0=Δ0, and the scaling in time t↦Nt, we obtain the following.Lemma 5.
For2≤q,r≤∞ satisfying 1/q+1/2r≤1/4, we have the following Strichartz estimate:(51)e±itNwDΔ0fxLtqLxr≤N−1/qΔ0fL2.


## 2.1. The Representation of Solutions to Linearized Equations

We study the following linearized equations associated to equations (10) and (11):(20)∂tU=NℙJℙU,∇˜⋅U=0,U0,x=U0x.Applying the Fourier transform to (20) yields(21)∂tU^−NPξJPξU^=0,ξ,0⋅U^=0,U^0,x=U^0ξ,where Pξ is the multiplier matrix of the operator ℙ defined by(22)ℙU^ξ=PξU^ξ,which is given explicitly by(23)Pξ≔δjk−ξjξk|ξ|21⩽j,k⩽2001.Then, a direct calculation yields(24)PξJPξ=1ξ2ξ22−ξ1ξ20−ξ1ξ2ξ12000ξ2,and the eigenvalues of PξJPξ are(25)ω0=0,ω±ξ≔±iωξ=±iξ1ξ,and the corresponding eigenvectors are(26)e0=1ξξ1,ξ2,0,e±=1ξ1ξ±iξ1ξ2,∓iξ12,1.Thus, the solution of (21) is(27)U^t,ξ=eNPξJPξt,U^0ξ=U^0ξ,e0e0+eiNωξtU^0ξ,e+e++e−iNωξtU^0ξ,e−e−.Since∇˜⋅U=0, we have U^0ξ,e0=0. Hence, we get(28)U^t,ξ=eNPξJPξt,U^0ξ=eiNωξtU^0ξ,e+e++e−iNωξtU^0ξ,e−e−.Setting(29)P+U0=ℱ−1U^0ξ,e+e+,P−U0=ℱ−1U^0ξ,e−e−,one has(30)Ut,x=eiNωDtP+U0+e−iNωDtP−U0.


## 2.2. Decay Estimate

We derive the following decay estimate of the operatore±itNwD.Lemma 1.
It holds that(31)e±itNwDΦDfL∞≤C1+Nt−1/2fL1,for all f∈L1ℝ2, where the operator ΦD is defined by(32)ΦDf=ℱ−1Φ^ξf^,in which Φ^ξ satisfies(33)suppΦ^⊂ξ∈ℝ2|2−2≤ξ≤22,Φ^ξ=1,∀ξ∈ξ∈ℝ2|2−1≤ξ≤2.Actually, the result of Lemma1 is an immediate consequence of the following lemma.Lemma 2.
There exists a positive constantC independent of t,x∈ℝ1+2 such that(34)∫ℝ2eixξ±iNtξ1/ξΦ^ξdξ≤C1+Nt−1/2.Next, we give the details of the proof of Lemma2. Firstly, we recall an important lemma (see Keel and Tao [17] and Majda [18]).Lemma 3.
Letdσ be a surface measure on a smooth surface S in ℝn and let ϕ∈C0∞ℝn. Suppose that for all x∈S, at least k of the principle curvatures are non-zero. Then, it holds that(35)d^μη=ϕd^ση≤Cη−k/2.


### 2.2.1. The Proof of Lemma2

By the theory of Fourier transform of measures supported on surfaces [18], we have(36)∫ℝ2eixξ±iNtξ1/ξψξdξ=∫ℝ2eix,Nt⋅ξ,±ξ1/ξψξdξ=d^μη,η=x,Nt,where μ is some measure supported on the surface S which is given by(37)S=ξ,ρ∈ℝ2×ℝ:ρ=±ξ1ξ,14≤ξ≤4.By Lemma3, the decay of d^μη is determined by the number of non-vanishing principle curvatures of the surface S. Equivalently, the number of non-vanishing principle curvatures of the surface S is the rank of the Hessian matrix Hρ=∂2ρ/∂ξj∂ξk1≤j,k≤2. By some computations, we obtain(38)Hρ=∓1ξ5−3ξ1ξ22ξ23ξ12−ξ2ξ23ξ12−ξ2ξ13ξ22−ξ2.From (38), we have(39)DetHρ=±ξ22ξ6.(39) shows that the surface S has a non-vanishing principle curvatures unless ξ2=0. Thus, we decompose(40)∫ℝ2eixξ±iNtξ1/ξψξdξ=I1+I2,where(41)I1=∫ℝ2eixξ±iNtξ1/ξ1−φξ2ψξdξ,I2=∫ℝ2eixξ±iNtξ1/ξφξ2ψξdξ,where φ is a smooth function on ℝ such that φ=1 on −1/2,2 and suppφ⊂−1,1.For the estimate ofI1L∞, by Lemma 3, we have(42)I1L∞≤Cx,Nt−2/2≤CNt−1.For the estimate ofI2L∞, since ξ2=0, we see that(43)Hρ=∓1ξ15000−ξ13.From (43) and the fact ξ∼11/4≤ξ≤4, we see one non-vanishing principle curvature of the surface S. By Lemma 3, we have(44)I2L∞≤Cx,Nt−1/2≤CNt−1/2.Combining (36), (42), and (44), we obtain(45)∫ℝ2eixξ±iNtξ1/ξψξdξ≤CNt−1/2.For smallt, it is trivial that(46)∫ℝ2eixξ±iNtξ1/ξψξdξ≤C.From (45) and (46), we have(47)∫ℝ2eixξ±iNtξ1/ξψξdξ≤C1+Nt−1/2.Thus, we complete the proof of Lemma2.


## 2.2.1. The Proof of Lemma2

By the theory of Fourier transform of measures supported on surfaces [18], we have(36)∫ℝ2eixξ±iNtξ1/ξψξdξ=∫ℝ2eix,Nt⋅ξ,±ξ1/ξψξdξ=d^μη,η=x,Nt,where μ is some measure supported on the surface S which is given by(37)S=ξ,ρ∈ℝ2×ℝ:ρ=±ξ1ξ,14≤ξ≤4.By Lemma3, the decay of d^μη is determined by the number of non-vanishing principle curvatures of the surface S. Equivalently, the number of non-vanishing principle curvatures of the surface S is the rank of the Hessian matrix Hρ=∂2ρ/∂ξj∂ξk1≤j,k≤2. By some computations, we obtain(38)Hρ=∓1ξ5−3ξ1ξ22ξ23ξ12−ξ2ξ23ξ12−ξ2ξ13ξ22−ξ2.From (38), we have(39)DetHρ=±ξ22ξ6.(39) shows that the surface S has a non-vanishing principle curvatures unless ξ2=0. Thus, we decompose(40)∫ℝ2eixξ±iNtξ1/ξψξdξ=I1+I2,where(41)I1=∫ℝ2eixξ±iNtξ1/ξ1−φξ2ψξdξ,I2=∫ℝ2eixξ±iNtξ1/ξφξ2ψξdξ,where φ is a smooth function on ℝ such that φ=1 on −1/2,2 and suppφ⊂−1,1.For the estimate ofI1L∞, by Lemma 3, we have(42)I1L∞≤Cx,Nt−2/2≤CNt−1.For the estimate ofI2L∞, since ξ2=0, we see that(43)Hρ=∓1ξ15000−ξ13.From (43) and the fact ξ∼11/4≤ξ≤4, we see one non-vanishing principle curvature of the surface S. By Lemma 3, we have(44)I2L∞≤Cx,Nt−1/2≤CNt−1/2.Combining (36), (42), and (44), we obtain(45)∫ℝ2eixξ±iNtξ1/ξψξdξ≤CNt−1/2.For smallt, it is trivial that(46)∫ℝ2eixξ±iNtξ1/ξψξdξ≤C.From (45) and (46), we have(47)∫ℝ2eixξ±iNtξ1/ξψξdξ≤C1+Nt−1/2.Thus, we complete the proof of Lemma2.


## 2.3. Strichartz Estimate

Firstly, we recall the following result obtained by Hu et al. [14].Lemma 4.
LetSt,t∈ℝ be a family of operators. Suppose that for all t,s∈ℝ,(48)SsS∗tfxL∞≤1+t−s−σfL1,SsS∗tfxL2≤fL2.Then, the estimates(49)StfxLtqLxr≤fL2hold for all 2≤q,r≤∞ with q,r,σ≠2,∞,1 satisfying(50)1q+σr≤σ2.From Lemmas1 and 4, the fact ΦDΔ0=Δ0, and the scaling in time t↦Nt, we obtain the following.Lemma 5.
For2≤q,r≤∞ satisfying 1/q+1/2r≤1/4, we have the following Strichartz estimate:(51)e±itNwDΔ0fxLtqLxr≤N−1/qΔ0fL2.


## 3. Blowup Criterion

This section shows a blowup criterion of equations (10) and (11).Lemma 6.
LetUt be a solution of equations (10) and (11) defined on a time interval containing 0,T. Then, for any s>0, we have the bounded estimate(52)UtHs2≤U0Hs2expC∫0T∇UτL∞dτ.Proof.
For the vector variableUt=ut,u3T, we obtain(53)∂tu3+u⋅∇u3=−Nu2,∂tu+u⋅∇u+∇p=Nu3e2,∇⋅u=0.
We takes derivatives of the first and the second equation in (53), multiply by ∇su3 and ∇su, respectively, and integrate over ℝ2 to obtain(54)ddt∇su3L22+∫∇su⋅∇u3∇su3dx=−N∫∇su2∇su3dx,(55)ddt∇suL22+∫∇su⋅∇u∇sudx+∫∇s∇p⋅∇sudx=N∫∇su3e2∇sudx.
Note that(56)−N∫∇su2∇su3dx+N∫∇su3e2∇sudx=0.
Since∇⋅u=0, we have(57)∫∇s∇p⋅∇sudx=0,∫u∇s∇u3∇su3dx=0,∫u∇s∇u∇sudx=0.
The commutator-type estimate (see, e.g., [12, 15]) provides us with(58)∫∇su⋅∇u3∇su3dx≤C∇UtL∞∇sUtL22,∫∇su⋅∇u∇sudx≤C∇UtL∞∇sUtL22.
Thus, adding (54) and (55) gives(59)ddt∇sUtL22≤C∇UtL∞∇sUtL22,which implies that by the Gronwall inequality for t∈0,T,(60)∇sUtL22≤∇sU0L22expC∫0T∇UtL∞dt.
The above equality implies that Lemma6 holds.


## 4. Proof of Theorem1

ApplyingP± to (10) and using the fact P±ℙ=P± gives(61)∂tP±U∓iNωDP±U+P±U⋅∇˜U=0.By the Duhamel principle, we get(62)P±U=e±iNωDtP±U0+∫0te±iNωDt−τP±U⋅∇˜Uτdτ.Due to Lemma5 and scaling, we find that for q≥4,(63)Δje±iNωDtfLtqLx∞≤N−1/q2jΔjfL2.In the following, forα>0, we are going to derive the estimates of(64)P±ULqB∞,∞1+α=χDP±UL∞+supj≥12j1+αΔjP±UL∞Ltq.By (63) and the Bernstein inequality, we have(65)2j1+αΔje±iNωDtP±U0L∞Ltq≤2j2+αN−1/qΔjP±U0L2≤N−1/qP±U0H2+α.On the other hand, we have from (61) that(66)χDe±iNwDtP±U0LtqLx∞≤∑j=−∞2Δje±iNp∇tP±U0Lx∞Ltq≤∑j=−∞2Δje±iNp∇tP±U0LtqLx∞≤N−1/q∑j=−∞22jΔjP±U0L2≤N−1/q∑j=−∞222j1/2∑j=−∞2ΔjP±U0L221/2≤CN−1/qP±U0L2.(65) and (66) give(67)e±iNωDtP±U0LtqB∞,∞1+α≤CN−1/qP±U0H2+α.By (63), we have(68)2j1+αΔj∫0te±iNωDt−τP±U⋅∇˜UτdτL∞Ltq≤∫0t2j1+αΔje±iNωDt−τP±U⋅∇˜UτL∞Ltqdτ≤∫0t2j2+αN−1/qΔjP±U⋅∇˜UτL2dτ≤N−1/q∫0tP±U⋅∇˜UτH2+αdτ≤CN−1/q∫0tUτH3+α2dτ.Similar to (66), we have(69)χ∇∫0te±iNωDt−τP±U⋅∇˜UτdτLtqLx∞≤∑j=−∞2Δj∫0te±iNωDt−τP±U⋅∇˜UτdτLtqLx∞≤N−1/q∫0t∑j=−∞22jΔjP±U⋅∇˜UτL2dτ≤N−1/q∫0tP±U⋅∇˜UτH1dτ≤CN−1/q∫0TuH22dτ.(68) and (69) imply(70)∫0te±iNωDt−τP±U⋅∇˜UτdτLtqB∞,∞1+α≤CN−1/q∫0tUτH3+α2dτ.Combining (67) and (70) yields(71)P±ULqB∞,∞1+α≤CN−1/qP±U0H2+α+∫0tUτH3+α2dτ.Define(72)MT=∫0T∇UL∞dt.We get from (71) and Lemma 6 that(73)MT≤T1−1/q∫0T∇UL∞qdt1/q≤CT1−1/qN−1/qP±U0H2+α+∫0TUtH3+α2dt≤CT1−1/qN−1/qP±U0H2+α+∫0TU0H3+α2eMtdt≤CT1−1/qN−1/qU0H2+α+U0H3+α2TeMT≤C1T1−1/qN−1/qU0H3+α1+U0H3+αTeMT.Letq=4 and suppose that(74)MT≤C1T3/4U0H3+α.We can chooseN sufficiently large such that(75)N−1/41+U0H3+αTeMT≤N−1/41+U0H3+αTeC1T4/3U0H3+α≤12.Combining (73) with (74) gives(76)MT≤12C1T3/4U0H3+α.Finally, the restriction (75) implies T can choose(77)T=ClnN3/4.By the bootstrap principle, we deduce from (76) that (74) actually holds. Thus, from the classical local existence (see [12]) and the blowup criterion Lemma 6, (10) and (11) possess unique classical solutions satisfying u∈C0,T;Hs.



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*Source: 5541403-2021-06-28.xml*