On the Well-Posedness of the Boussinesq Equation in the Triebel-Lizorkin-Lorentz Spaces

Abstract

We establish the local well-posedness and obtain a blow-up criterion of smooth solutions for the Boussinesq equations in the framework of Triebel-Lizorkin-Lorentz spaces. The main ingredients of our proofs are Littlewood-Paley decomposition and the paradifferential calculus.

1. Introduction

In this paper, we consider the following inviscid Boussinesq equations: 𝜕_𝑡𝑢+𝑢⋅∇𝑢+∇𝑝=𝜃𝑒_𝑑,inℝ^𝑑×(),𝜕0,𝑇_𝑡𝜃+𝑢⋅∇𝜃=0,inℝ^𝑑×(0,𝑇),div𝑢=0,inℝ^𝑑×(),𝑢()0,𝑇𝑥,0=𝑢₀(𝑥),𝜃(𝑥,0)=𝜃₀(𝑥),inℝ^𝑑, (1.1) with div𝑢₀=0 and 𝑒_𝑑=(0,…,0,1). This system describes the motion of lighter or denser incompressible fluid under the influence of gravitational forces, for instance, in the atmospheric sciences, where the vector 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑑) is the velocity, the scalar function 𝜃 denotes the temperature, and 𝑝 stands for the pressure in the fluid (see [1]). The term 𝜃𝑒_𝑑 takes into account the influence of the gravity and the stratification on the motion of the fluid. In case 𝑑=2, it is also used as a simplified model for the 3D axisymmetric Euler equations with swirl away from the symmetric axis𝑟=0.

The regularity or singularity questions of (1.1) are an outstanding open problem in the mathematical fluid mechanics. Over the past few years, (1.1) has been studied extensively both theoretically (see [1–6] and references therein) and numerically (see [7–9]). In particular, the local well-posedness and some blow-up criteria of (1.1) are established by Chae and Nam [3] in the Sobolev spaces, Liu et al. [5, 6] and Xiang [10] in the Besov spaces, and Cui et al. [11] in the Hölder spaces. The purpose of this paper is to establish the local well-posedness for the Boussineq equations (1.1) and to obtain a blow-up criterion of the smooth solutions in the framework of Triebel-Lizorkin-Lorentz spaces, which contain the classical Triebel-Lizorkin spaces and Lorentz spaces. Indeed, it is more natural to consider the well-posedness in the Triebel-Lizorkin-type spaces than in the Besov spaces in the sense that 𝐹^𝑠_𝑝,2=𝑊^𝑠,𝑝 for 1<𝑝<∞.

Before proceeding further, we mention several local well-posedness results for other fluid equations in the Triebel-Lizorkin spaces. Recently, Chae [12] introduced a family trajectory mapping {𝑋_𝑗(𝛼,𝑡)} satisfying d𝑋d𝑡^𝑗(𝛼,𝑡)=𝑆_𝑗−2𝑣𝑋_𝑗(),𝑋𝛼,𝑡,𝑡_𝑗(𝛼,0)=𝛼, (1.2) where 𝑆_𝑗−2 is a frequency projection to the ball {𝜉∈ℝ²||𝜉|≲2^𝑗} and used the following equivalent relation: 2^𝑗𝑠Δ_𝑗𝜃𝑋_𝑗(𝛼,𝑡),𝑡_{𝐿𝑞(ℤ)𝐿𝑝(d𝛼)}∼2^𝑗𝑠Δ_𝑗𝜃(𝑥,𝑡)_{𝐿𝑞(ℤ)𝐿𝑝(d𝑥)}=‖𝜃(𝑡)‖_̇𝐹^𝑠_𝑝,𝑞, (1.3) to estimate the frequency-localized solutions of the Euler equations in the Triebel-Lizorkin spaces. However, it seems difficult to give a strict proof for (1.3)-type equivalent relation due to the lack of the uniform change of the coordinates independent of 𝑗. To avoid this trouble, Chen et al. [13] introduced a particle trajectory mapping 𝑋(𝛼,𝑡) independent of 𝑗 defined by d𝑋()(𝑋()),𝑋()d𝑡𝛼,𝑡=𝑣𝛼,𝑡,𝑡𝛼,0=𝛼, (1.4) and then they established a new commutator estimate to obtain the local well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces.

In this paper, we will adapt the method of Chen et al. [13] to establish the local well-posedness and to obtain a blow-up criterion of the smooth solutions for the Boussineq equations (1.1) in the framework of Triebel-Lizorkin-Lorentz spaces. Precisely, we first define the particle trajectory mapping𝑋(𝛼,𝑡) by (1.4). Then to show the well-posedness in the new framework, we have to establish the commutator estimate (Proposition 2.5) and the product estimate (Proposition 2.6). For the blow-up criterion, we also need the logarithmic inequality (Proposition 2.7). Fortunately, these preliminary estimates and inequalities have been obtained in our recent work [14].

Our main results can be formulated in the following way.

Theorem 1.1. (i) Local Existence. Let 1<𝑟≤𝑝<∞,1<𝑞<∞, and 𝑠>𝑑/𝑝+1. If (𝑢₀,𝜃₀)∈(𝐹^𝑠,𝑟_𝑝,𝑞(ℝ^𝑑))^𝑑+1, then there exists 𝑇₁=𝑇₁(‖𝑢₀‖_𝐹^𝑠,𝑟_𝑝,𝑞,‖𝜃₀‖_𝐹^𝑠,𝑟_𝑝,𝑞)>0 such that the Boussinesq equation (1.1) has a unique solution (𝑢,𝜃)∈(𝐶([0,𝑇₁];𝐹^𝑠,𝑟_𝑝,𝑞(ℝ^𝑑)))^𝑑+1.
(ii) Blow-Up Criterion. The local-in-time solution (𝑢,𝜃)∈(𝐶([0,𝑇₁];𝐹^𝑠,𝑟_𝑝,𝑞(ℝ^𝑑)))^𝑑+1 constructed in (i) blows up at finite time𝑇^∗>𝑇₁, that is, limsup_{𝑡→𝑇∗−}‖𝑢(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞=∞, (1.5) if and only if ^𝑇∗₀‖𝑢(𝑡)‖_{̇𝐹1∞,∞}+‖𝜃(𝑡)‖_{̇𝐹1∞,∞}d𝑡=∞. (1.6)

Remark 1.2. In case 𝜃₀≡0, we have 𝜃≡0 and thus get the local well-posedness and blow-up criterion for the incompressible Euler equations in the Triebel-Lizorkin-Lorentz spaces, which generalize the corresponding results in [12] because of 𝐹^𝑠,𝑝_𝑝,𝑞=𝐹^𝑠_𝑝,𝑞. For general 𝜃₀, we also extend the results of [3] from Sobolev spaces to Triebel-Lizorkin-Lorentz spaces.

Remark 1.3. For the MHD equations, we can also generalize the local well-posedness and blow-up criterion of [13] from the Triebel-Lizorkin spaces to the Triebel-Lizorkin-Lorentz spaces by using similar arguments.

The rest of this paper is organized as follows. We state some preliminaries on functional settings in Section 2 and then prove the local well-posedness and blow-up criterion in Section 3.

2. Preliminaries

In this section, we give some definitions and basic estimates (see [14, 15] for details). First of all, we let 𝒮(ℝ^𝑑) be the Schwartz class of rapidly decreasing functions. For a fixed radially symmetric bump function 𝜒∈𝒞^∞₀(𝐵(0,4/3)) with value 1 over the ball𝐵(0,3/4), we set 𝜑(𝜉)∶=𝜒(𝜉/2)−𝜒(𝜉) and then have the following dyadic partition of unity: 𝜒(𝜉)+_𝑗≥0𝜑2^−𝑗𝜉=1,𝜉∈ℝ^𝑑,_𝑗∈ℤ𝜑2^−𝑗𝜉=1,𝜉∈ℝ^𝑑⧵{0}. (2.1) The frequency localization operators Δ_𝑗 and 𝑆_𝑗 can be defined as follows: Δ_𝑗𝑢∶=𝜑2^−𝑗𝐷𝑢=ℱ⁻¹𝜑2^−𝑗⋅ℱ(𝑢)=2^𝑗𝑑ℱ⁻¹𝜑2^𝑗⋅𝑆∗𝑢,𝑗∈ℤ,_𝑗𝑢∶=_{𝑗′≤𝑗−1}Δ_𝑗′𝑢=𝜒2^−𝑗𝐷𝑢=2^𝑗𝑑ℱ⁻¹𝜒2^𝑗⋅∗𝑢,𝑗∈ℤ, (2.2) where ℱ is the Fourier transform and 𝐷 is the Fourier multiplier with symbol |𝜉|. The frequency localization operators Δ_𝑗 and 𝑆_𝑗 have nice almost orthogonal properties in 𝐿²: Δ_𝑗Δ_𝑗′𝑢≡0,if𝑗−𝑗^′≥2,Δ_𝑗𝑆_𝑗′−1𝑢Δ_𝑗′𝑣≡0,if𝑗−𝑗^′≥5 (2.3) for any 𝑢,𝑣∈𝒮^′(ℝ^𝑑).

Next we recall several function spaces used in this paper. The first one is the Triebel-Lizorkin spaces. For 𝑠∈ℝ, (𝑝,𝑞)∈[1,+∞)×[1,+∞], we define the homogeneous Triebel-Lizorkin spaces 𝐹̇^𝑠_𝑝,𝑞(ℝ^𝑑) as the set of tempered distributions 𝑢∈𝒮^′/𝒫(ℝ^𝑑) such that ‖𝑢‖_̇𝐹^𝑠_{𝑝,𝑞(ℝ𝑑)}∶=2^𝑗𝑠Δ_𝑗𝑢_{𝑙𝑞(ℤ)𝐿𝑝(ℝ𝑑)}<∞, (2.4) where 𝒫 is the polynomial space. Then for 𝑠>0, (𝑝,𝑞)∈[1,+∞)×[1,+∞], we can define the inhomogeneous Triebel-Lizorkin spaces 𝐹^𝑠_𝑝,𝑞(ℝ^𝑑) as the set of tempered distributions 𝑢 such that ‖𝑢‖_𝐹^𝑠_{𝑝,𝑞(ℝ𝑑)}∶=‖𝑢‖_{𝐿𝑝(ℝ𝑑)}+‖𝑢‖_̇𝐹^𝑠_{𝑝,𝑞(ℝ𝑑)}<∞. (2.5) The inhomogeneous Triebel-Lizorkin spaces 𝐹^𝑠_𝑝,𝑞 are Banach spaces with norms ‖⋅‖_𝐹^𝑠_𝑝,𝑞.

The second one is the Lorentz spaces. Let 𝑢_∗(𝜏) be the distribution function of a function 𝑢(𝑥) and let 𝑢^∗(𝑡) be its nonincreasing rearrangement, that is, 𝑢_∗(𝜏)=meas𝑥||𝑢(𝑥)|>𝜏},𝑢^∗(𝜏)=inf{𝜏>0|𝑢_∗(𝜏)≤𝑡. (2.6) Then for 1<𝑝,𝑟<∞, the Lorentz spaces 𝐿^𝑝,𝑟(ℝ^𝑑) are defined as the set of those measurable functions 𝑢 satisfying ‖𝑢‖_{𝐿𝑝,𝑟}=𝑟𝑝^∞₀𝑡^1/𝑝𝑢^∗∗(𝑡)^𝑟d𝑡𝑡^1/𝑟<∞, (2.7) where 𝑢^∗∗(𝑡)∶=(1/𝑡)∫^𝑡₀𝑢^∗(𝑠)d𝑠 is the average rearrangement. The Lorentz spaces 𝐿^𝑝,𝑟 are Banach spaces with norms ‖⋅‖_{𝐿𝑝,𝑟}. Since 𝑢_∗(𝜏) is nonincreasing in 𝜏, it is easy to verify that the following equivalent characterization holds: ‖𝑢‖_{𝐿𝑝,𝑟}∼2^𝑗𝑢_∗^1/𝑝2^𝑗_𝑙𝑟. (2.8) Moreover, it is well known that there exist 1≤𝑝₁and𝑝₂<∞ such that 𝐿^𝑝,𝑟 are the interpolations spaces between 𝐿^𝑝1 and 𝐿^𝑝2. Thus we can deduce the following basic fact, which is standard in the usual 𝐿^𝑝 spaces.

Lemma 2.1. Assume 𝑢∈𝐿^𝑝,𝑟 for 1<𝑝,𝑟<∞. If 𝑋∶𝛼→𝑋(𝛼) is a volume-preserving diffeomorphism, then ‖𝑢(𝛼)‖_{𝐿𝑝,𝑟}=‖𝑢(𝑋(𝛼))‖_{𝐿𝑝,𝑟}. (2.9)

The last one is called the Triebel-Lizorkin-Lorentz spaces and is introduced recently by Yang et al. [16] to unify the Triebel-Lizorkin spaces and the Lorentz spaces.

Definition 2.2. For 𝑠∈ℝ and 1<𝑝,𝑞,𝑟<∞, a distribution 𝑢∈𝒮^′ is said to be in the homogeneous Triebel-Lizorkin-Lorentz spaces 𝐹̇^𝑠,𝑟_𝑝,𝑞(ℝ^𝑑) if 2^𝑘meas𝑥∈ℝ^𝑑∣2^𝑗𝒮Δ_𝑗𝑢_{𝑙𝑞𝑗(ℤ)}>2^𝑘1/𝑝_{𝑙𝑟𝑘(ℤ)}<∞. (2.10) Indeed we can deduce that 𝑢∈𝐹̇^𝑠,𝑟_𝑝,𝑞(ℝ^𝑑)  if and only if ‖‖2^𝑗𝑠|Δ_𝑗𝑢|‖_{𝑙𝑞(ℤ)}‖_{𝐿𝑝,𝑟}<∞ by the equivalent characterization (2.8) of Lorentz space 𝐿^𝑝,𝑟. Then 𝐹̇^𝑠_𝑝,𝑞(ℝ^𝑑)⋍𝐹̇^𝑠,𝑝_𝑝,𝑞(ℝ^𝑑) and 𝐿^𝑝,𝑟(ℝ^𝑑)⋍𝐹̇^0,𝑟_𝑝,2(ℝ^𝑑) (see [16, Theorem  5]). For 𝑠>0, the inhomogeneous Triebel-Lizorkin-Lorentz spaces 𝐹^𝑠,𝑟_𝑝,𝑞(ℝ^𝑑) are defined by 𝐹^𝑠,𝑟_𝑝,𝑞ℝ^𝑑=𝐿^𝑝,𝑟∩𝐹̇^𝑠,𝑟_𝑝,𝑞ℝ^𝑑 (2.11) with norm ‖𝑢‖_𝐹^𝑠,𝑟_{𝑝,𝑞(ℝ𝑑)}=‖𝑢‖_{𝐿𝑝,𝑟}+‖𝑢‖_̇𝐹^𝑠,𝑟_{𝑝,𝑞(ℝ𝑑)}. (2.12) Clearly, 𝐹^𝑠,𝑟_𝑝,𝑞(ℝ^𝑑) is a Banach space. It has been proved that there exist 1<𝑝₁and𝑝₂<∞ such that 𝐹^𝑠,𝑟_𝑝,𝑞(ℝ^𝑑) are the interpolation spaces between 𝐹^𝑠_𝑝1,𝑞(ℝ^𝑑) and 𝐹^𝑠_𝑝2,𝑞(ℝ^𝑑) (see [16, Theorem  6]). Thus by the boundedness of Riesz transform on the Triebel-Lizorkin spaces (see Frazier et al. [17]) we have the following.

Lemma 2.3. Riesz transform is bounded from the Triebel-Lizorkin-Lorentz space 𝐹^𝑠,𝑟_𝑝,𝑞(ℝ^𝑑) into itself.
Also by 𝐿^𝑝,𝑟↪𝐿^𝑝 for 𝑟≤𝑝, we have the following Sobolev’s embedding theorem: 𝐹^𝑠,𝑟_𝑝,𝑞↪𝐹^𝑠,p_𝑝,𝑞=𝐹^𝑠_𝑝,𝑞↪𝐿^∞𝑑,for𝑠>𝑝. (2.13)

The following lemma is referred as Bernstein’s inequality, which describes the way derivatives act on spectrally localized functions.

Lemma 2.4 (Bernstein’s inequality). Let 1<𝑝,𝑟<∞, and 𝑢∈𝐿^𝑝,𝑟(ℝ^𝑑). Then suppℱ(𝑢)⊂𝜉∈ℝ^𝑑∶𝜉∼2^𝑗⟹𝜕^𝛼𝑢_{𝐿𝑝,𝑟}∼2^𝑗|𝛼|‖𝑢‖_{𝐿𝑝,𝑟}. (2.14)

Here and thereafter, we use 𝐴≲𝐵 and 𝐴∼𝐵 to denote 𝐴≤𝐶𝐵 and 𝑐𝐵≤𝐴≤𝐶𝐵 for some constants 𝐶>𝑐>0, respectively. Thus Bernstein’s inequality together with the Littlewood-Paley decomposition gives that𝜕^𝛼𝑢_{̇𝐹𝑠,𝑟𝑝,𝑞}∼‖𝑢‖_̇𝐹^{𝑠+|𝛼|,𝑟}_𝑝,𝑞,1<𝑝,𝑟<∞ (2.15) for any 𝑢∈𝐹̇^{𝑠+|𝛼|,𝑟}_𝑝,𝑞(ℝ^𝑑).

To conclude this section, we recall three important propositions in the Triebel-Lizorkin-Lorentz spaces, which will be frequently used in the proof of Theorem 1.1. For their proofs, we refer to our recent work [14] (for the case in the Triebel-Lizorkin spaces, see also Chae [12] and Chen et al. [13]). The first one is related to the product estimates.

Proposition 2.5. Let 1<𝑝,𝑞,𝑟<∞. There exists a constant 𝐶>0 such that for 𝑠>0‖𝑢𝜃‖_̇𝐹^𝑠,𝑟_𝑝,𝑞≤𝐶‖𝑢‖_𝐿∞‖𝜃‖_̇𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃‖_𝐿∞‖𝑢‖_̇𝐹^𝑠,𝑟_𝑝,𝑞, (2.16) and for 𝑠>−1‖𝑢𝜃‖_𝐹^𝑠,𝑟_𝑝,𝑞≤𝐶‖𝑢‖_𝐿∞‖𝜃‖_𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃‖_𝐿∞‖𝑢‖_𝐹^𝑠,𝑟_𝑝,𝑞, (2.17) as long as the right-hand sides of (2.16) and (2.17) are finite.

The next one concerns the commutator estimates.

Proposition 2.6. Let 𝑠>0, 1<𝑝,𝑞,𝑟<∞, and let the vector field 𝑣 be divergence free. There exists a constant 𝐶>0 such that 2^𝑗𝑠𝑣,Δ_𝑗⋅∇𝜃_{𝑙𝑞(ℤ)𝐿𝑝,𝑟}≤𝐶‖∇𝑣‖_𝐿∞‖𝜃‖_̇𝐹^𝑠,𝑟_𝑝,𝑞+‖∇𝜃‖_𝐿∞‖𝑣‖_̇𝐹^𝑠,𝑟_𝑝,𝑞,2^𝑗𝑠𝑣,Δ_𝑗⋅∇𝜃_{𝑙𝑞(ℤ)𝐿𝑝,𝑟}≤𝐶‖∇𝑣‖_𝐿∞‖𝜃‖_̇𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃‖_𝐿∞‖∇𝑣‖_̇𝐹^𝑠,𝑟_𝑝,𝑞, (2.18)(2.19) as long as the right-hand sides of (2.18) and (2.19) are finite.

The last one is the logarithmic Triebel-Lizorkin-Lorentz inequality with features similar to the classical logarithmic Sobolev inequality [18] and to the logarithmic Triebel-Lizorkin inequality [12].

Proposition 2.7. Let 1<𝑟≤𝑝<∞,1<𝑞<∞, and 𝑠>𝑑/𝑝. There exists a constant 𝐶>0 such that ‖𝑢‖_𝐿∞≤𝐶1+‖𝑢‖_{̇𝐹0∞,∞}log⁺‖𝑢‖_𝐹^𝑠,𝑟_𝑝,𝑞+1. (2.20)

3. Proof of Theorem 1.1

In this section, we prove the local well-posedness of solutions to (1.1) by constructing the successive approximations and using the a priori estimate and establish the blow-up criterion of smooth solutions by the logarithmic Triebel-Lizorkin-Lorentz inequality. The proof is divided into five steps.

Step 1 (a priori estimate). For a given fluid velocity 𝑢(𝑥,𝑡), we let 𝑋(𝛼,𝑡) be the particle-trajectory mapping defined by the solutions of the following ordinary differential equations: d𝑋()(𝑋()),𝑋()d𝑡𝛼,𝑡=𝑢𝛼,𝑡,𝑡𝛼,0=𝛼. (3.1)
Then from the Boussinesq equations (1.1) and the ODEs (3.1), we have d𝑢(𝑋())(𝑋())(𝑋())𝑒d𝑡𝛼,𝑡,𝑡=−∇𝑝𝛼,𝑡,𝑡+𝜃𝛼,𝑡,𝑡_𝑑,d𝜃(𝑋())𝑢(𝑋())d𝑡𝛼,𝑡,𝑡=0,𝛼,0,0=𝑢₀(𝛼),𝜃(𝑋(𝛼,0),0)=𝜃₀(𝛼), (3.2) from which we get 𝑢(𝑋(𝛼,𝑡),𝑡)=𝑢₀(𝛼)−^𝑡₀∇𝑝(𝑋(𝛼𝜏),𝜏)d𝜏+^𝑡₀𝜃(𝑋(𝛼,𝜏),𝜏)𝑒_𝑑𝜃(𝑋())d𝜏,𝛼,𝑡,𝑡=𝜃₀(𝛼). (3.3) It follows from div𝑢=0 that the mapping 𝛼→𝑋(𝛼,𝑡) is a volume-preserving diffeomorphism for any given 𝑡>0. Thus by Lemma 2.1 we have ‖𝑢(𝑡)‖_{𝐿𝑝,𝑟}+‖𝜃(𝑡)‖_{𝐿𝑝,𝑟}=‖𝑢(𝑋(𝛼,𝑡),𝑡)‖_{𝐿𝑝,𝑟}+‖𝜃(𝑋(𝛼,𝑡),𝑡)‖_{𝐿𝑝,𝑟}≤𝑢_{0𝐿𝑝,𝑟}+𝜃_{0𝐿𝑝,𝑟}+^𝑡₀∇𝑝(𝑋(𝛼,𝜏),𝜏)_{𝐿𝑝,𝑟}d𝜏+^𝑡₀‖𝜃(𝑋(𝛼,𝜏),𝜏)‖_{𝐿𝑝,𝑟}=𝑢d𝜏_{0𝐿𝑝,𝑟}+𝜃_{0𝐿𝑝,𝑟}+^𝑡₀∇𝑝(𝜏)_{𝐿𝑝,𝑟}d𝜏+^𝑡₀‖𝜃(𝜏)‖_{𝐿𝑝,𝑟}d𝜏. (3.4) To deal with the pressure term, we take the divergence on both sides of (1.1)₁ and then obtain the representation of the pressure: 𝑝=(−Δ)⁻¹𝜕_𝑖𝑢_𝑗𝜕_𝑗𝑢_𝑖−(−Δ)⁻¹𝜕_𝑑𝜃=(−Δ)⁻¹𝜕_𝑖𝜕_𝑗𝑢_𝑖𝑢_𝑗−(−Δ)⁻¹𝜕_𝑑𝜃, (3.5) where we used the Einstein convention on the summation over repeated indices and the fact that div𝑢=0. Thus by the boundedness of Riesz transform on 𝐿^𝑝,𝑟, we have ∇𝑝(𝜏)_{𝐿𝑝,𝑟}≤∇(−Δ)⁻¹𝜕_𝑖𝜕_𝑗𝑢_𝑖𝑢_𝑗(𝜏)_{𝐿𝑝,𝑟}+∇(−Δ)⁻¹𝜕_𝑑𝜃(𝜏)_{𝐿𝑝,𝑟}≲‖𝑢⋅∇𝑢(𝜏)‖_{𝐿𝑝,𝑟}+‖𝜃(𝜏)‖_{𝐿𝑝,𝑟}≤‖𝑢(𝜏)‖_{𝐿𝑝,𝑟}‖∇𝑢(𝜏)‖_∞+‖𝜃(𝜏)‖_{𝐿𝑝,𝑟}. (3.6) Substituting the last inequality into (3.4), we obtain ‖𝑢(𝑡)‖_{𝐿𝑝,𝑟}+‖𝜃(𝑡)‖_{𝐿𝑝,𝑟}≲𝑢_{0𝐿𝑝,𝑟}+𝜃_{0𝐿𝑝,𝑟}+^𝑡₀‖𝑢(𝜏)‖_{𝐿𝑝,𝑟}‖∇𝑢(𝜏)‖_∞d𝜏+^𝑡₀‖𝜃(𝜏)‖_{𝐿𝑝,𝑟}d𝜏. (3.7)
On the other hand, we apply the frequency projection Δ_𝑗 to both sides of (1.1) to obtain 𝜕_𝑡Δ_𝑗𝑢+𝑢⋅∇Δ_𝑗𝑢=𝑢,Δ_𝑗⋅∇𝑢−∇Δ_𝑗𝑝+Δ_𝑗𝜃𝑒_𝑑,𝜕_𝑡Δ_𝑗𝜃+𝑢⋅∇Δ_𝑗𝜃=𝑢,Δ_𝑗⋅∇𝜃. (3.8) Then by (3.1), we have dΔd𝑡^𝑗𝑢(𝑋(𝛼,𝑡),𝑡)=𝑢,Δ_𝑗⋅∇𝑢(𝑋(𝛼,𝑡),𝑡)−∇Δ_𝑗𝑝(𝑋(𝛼,𝑡),𝑡)+Δ_𝑗𝜃(𝑋(𝛼,𝑡),𝑡)𝑒_𝑑,dΔd𝑡^𝑗𝜃(𝑋(𝛼,𝑡),𝑡)=𝑢,Δ_𝑗⋅∇𝜃(𝑋(𝛼,𝑡),𝑡). (3.9) Integrating the previous two equations from 0 to 𝑡 gives Δ_𝑗𝑢(𝑋(𝛼,𝑡),𝑡)+Δ_𝑗𝜃(𝑋(𝛼,𝑡),𝑡)=Δ_𝑗𝑢₀(𝛼)+Δ_𝑗𝜃₀(𝛼)+^𝑡₀𝑢,Δ_𝑗⋅∇𝑢(𝑋(𝛼,𝜏),𝜏)+𝑢,Δ_𝑗(𝑋())−⋅∇𝜃𝛼,𝜏,𝜏d𝜏_𝑡0∇Δ_𝑗𝑝(𝑋(𝛼,𝜏),𝜏)d𝜏+^𝑡₀𝜃(𝑋(𝛼,𝜏),𝜏)𝑒_𝑑d𝜏. (3.10) We first multiply both sides by 2^𝑗𝑠, take the 𝑙^𝑞(ℤ) norm, and then use Minkowski’s inequality to obtain 2^𝑗𝑠Δ_𝑗𝑢(𝑋(𝛼,𝑡),𝑡)_{𝑙𝑞(ℤ)}+2^𝑗𝑠Δ_𝑗𝜃(𝑋(𝛼,𝑡),𝑡)_{𝑙𝑞(ℤ)}≤2^𝑗𝑠Δ_𝑗𝑢₀(𝛼)_{𝑙𝑞(ℤ)}+2^𝑗𝑠Δ_𝑗𝜃₀(𝛼)_{𝑙𝑞(ℤ)}+^𝑡₀2^𝑗𝑠𝑢,Δ_𝑗⋅∇𝑢(𝑋(𝛼,𝜏),𝜏)_{𝑙𝑞(ℤ)}+2^𝑗𝑠𝑢,Δ_𝑗⋅∇𝜃(𝑋(𝛼,𝜏),𝜏)_{𝑙𝑞(ℤ)}+d𝜏_𝑡02^𝑗𝑠∇Δ_𝑗𝑝(𝑋(𝛼,𝜏),𝜏)_{𝑙𝑞(ℤ)}d𝜏+^𝑡₀2^𝑗𝑠|𝜃(𝑋(𝛼,𝜏),𝜏)|_{𝑙𝑞(ℤ)}d𝜏, (3.11) and then we take the 𝐿^𝑝,𝑟 norm with respect to 𝛼 and use Minkowski’s inequality to get 2^𝑗𝑠Δ_𝑗𝑢(𝑋(𝛼,𝑡),𝑡)_{𝑙𝑞(ℤ)𝐿𝑝,𝑟}+2^𝑗𝑠Δ_𝑗𝜃(𝑋(𝛼,𝑡),𝑡)_{𝑙𝑞(ℤ)𝐿𝑝,𝑟}≤𝑢_{0̇𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0̇𝐹𝑠,𝑟𝑝,𝑞}+^𝑡₀2^𝑗𝑠𝑢,Δ_𝑗⋅∇𝑢(𝑋(𝛼,𝜏),𝜏)_{𝑙𝑞(ℤ)𝐿𝑝,𝑟}+d𝜏_𝑡02^𝑗𝑠𝑢,Δ_𝑗⋅∇𝜃(𝑋(𝛼,𝜏),𝜏)_{𝑙𝑞(ℤ)𝐿𝑝,𝑟}+d𝜏_𝑡02^𝑗𝑠∇Δ_𝑗𝑝(𝑋(𝛼,𝜏),𝜏)_{𝑙𝑞(ℤ)𝐿𝑝,𝑟}d𝜏+^𝑡₀2^𝑗𝑠|𝜃(𝑋(𝛼,𝜏),𝜏)|_{𝑙𝑞(ℤ)𝐿𝑝,𝑟}d𝜏. (3.12) Recalling the fact that 𝛼→𝑋(𝛼,𝑡) is a volume-preserving diffeomorphism for any given 𝑡>0 and using Lemma 2.1, we have ‖𝑢(𝑡)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝑡)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞≲𝑢_{0̇𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0̇𝐹𝑠,𝑟𝑝,𝑞}+^𝑡₀2^𝑗𝑠𝑢,Δ_𝑗⋅∇𝑢(𝜏)_{𝑙𝑞(ℤ)𝐿𝑝,𝑟}+2^𝑗𝑠𝑢,Δ_𝑗⋅∇𝜃(𝜏)_{𝑙𝑞(ℤ)𝐿𝑝,𝑟}+d𝜏_𝑡0∇𝑝(𝜏)_{̇𝐹𝑠,𝑟𝑝,𝑞}d𝜏+^𝑡₀‖𝜃(𝜏)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞d𝜏. (3.13) Thanks to the commutator estimates (2.18), the integrand in the first integral is dominated by ‖∇𝑢(𝜏)‖_𝐿∞‖𝑢(𝜏)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞+‖∇𝑢(𝜏)‖_𝐿∞‖𝜃(𝜏)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞+‖∇𝜃(𝜏)‖_𝐿∞‖𝑢(𝜏)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞, (3.14) and thus ‖𝑢(𝑡)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝑡)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞≲𝑢_{0̇𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0̇𝐹𝑠,𝑟𝑝,𝑞}+^𝑡₀∇𝑝(𝜏)_{̇𝐹𝑠,𝑟𝑝,𝑞}+d𝜏_𝑡01+‖∇𝑢(𝜏)‖_𝐿∞+‖∇𝜃(𝜏)‖_𝐿∞‖𝑢(𝜏)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝜏)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞d𝜏. (3.15) For the term related to the pressure, we use (3.5) to obtain 𝜕_𝑘𝜕_𝑙𝑝=𝜕_𝑘𝜕_𝑙(−Δ)⁻¹𝜕_𝑖𝑢_𝑗𝜕_𝑗𝑢_𝑖−𝜕_𝑘𝜕_𝑙(−Δ)⁻¹𝜕_𝑑𝜃∶=𝑅_𝑘𝑅_𝑙𝜕_𝑖𝑢_𝑗𝜕_𝑗𝑢_𝑖−𝑅_𝑘𝑅_𝑙𝜕_𝑑𝜃, (3.16) where 𝑅_𝑘and𝑅_𝑙 are Riesz transform, and then ∇𝑝_{̇𝐹𝑠,𝑟𝑝,𝑞}≤𝜕_𝑘𝜕_𝑙𝑝_{̇𝐹𝑠−1,𝑟𝑝,𝑞}≤𝜕_𝑖𝑢_𝑗𝜕_𝑗𝑢_{𝑖̇𝐹𝑠−1,𝑟𝑝,𝑞}+𝜕_𝑑𝜃_{̇𝐹𝑠−1,𝑟𝑝,𝑞}≤‖𝑢‖_̇𝐹^𝑠,𝑟_𝑝,𝑞‖∇𝑢‖+‖𝜃‖_̇𝐹^𝑠,𝑟_𝑝,𝑞, (3.17) where we used the boundedness of Riesz transform on 𝐹̇^𝑠,𝑟_𝑝,𝑞 and the product estimate (2.16). Substituting the last inequality into (3.15), we have ‖𝑢(𝑡)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝑡)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞≲𝑢_{0̇𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0̇𝐹𝑠,𝑟𝑝,𝑞}+^𝑡₀1+‖∇𝑢(𝜏)‖_𝐿∞+‖∇𝜃(𝜏)‖_𝐿∞‖𝑢(𝜏)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝜏)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞d𝜏. (3.18) This together with Gronwall’s inequality gives the a priori estimates in the homogeneous Triebel-Lizorkin-Lorentz spaces: ‖𝑢(𝑡)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝑡)‖_̇𝐹^𝑠,𝑟_𝑝,𝑞≤𝐶𝑢_{0̇𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0̇𝐹𝑠,𝑟𝑝,𝑞}exp𝐶^𝑡₀1+‖∇𝑣(𝜏)‖_𝐿∞+‖∇𝜃(𝜏)‖_𝐿∞d𝜏. (3.19)
Similarly, by (3.7) and (3.18), we get ‖𝑢(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞≲𝑢_{0𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0𝐹𝑠,𝑟𝑝,𝑞}+^𝑡₀1+‖∇𝑢(𝜏)‖_𝐿∞+‖∇𝜃(𝜏)‖_𝐿∞‖𝑢(𝜏)‖_𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝜏)‖_𝐹^𝑠,𝑟_𝑝,𝑞d𝜏, (3.20) and thus Gronwall’s inequality yields the a priori estimates in the inhomogeneous spaces: ‖𝑢(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞≤𝐶𝑢_{0𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0𝐹𝑠,𝑟𝑝,𝑞}exp𝐶^𝑡₀1+‖∇𝑢(𝜏)‖_𝐿∞+‖∇𝜃(𝜏)‖_𝐿∞d𝜏. (3.21)

Step 2 (approximate solutions and uniform estimates). In order to construct the approximate solutions of (1.1), we first set (𝑢⁽⁰⁾,𝜃⁽⁰⁾)=(0,0) and then define {(𝑢^(𝑛),𝜃^(𝑛))}_𝑛∈ℕ as the solutions of the linear equations: 𝜕_𝑡𝑢^(𝑛+1)+𝑢^(𝑛)⋅∇𝑢^(𝑛+1)+∇𝑝^(𝑛+1)=𝜃^(𝑛+1)𝑒_𝑑,inℝ^𝑑×(),𝜕0,∞_𝑡𝜃^(𝑛+1)+𝑢^(𝑛)⋅∇𝜃^(𝑛+1)=0,inℝ^𝑑×(0,∞),div𝑢^(𝑛)=0,inℝ^𝑑×(0,∞), (3.22) with initial data𝑢^(𝑛+1)(𝑥,0)=𝑢₀^(𝑛+1)(𝑥)∶=𝑆_𝑛+2𝑢₀(𝑥),𝜃^(𝑛+1)(𝑥,0)=𝜃₀^(𝑛+1)(𝑥)∶=𝑆_𝑛+2𝜃₀(𝑥).
For each 𝑛∈ℕ, similar to Step 1, we let the particle-trajectory mapping 𝑋^(𝑛)(𝛼,𝑡) be the solution of the following ordinary differential equation: d𝑋d𝑡^(𝑛)(𝛼,𝑡)=𝑢^(𝑛)𝑋^(𝑛)(),𝑋𝛼,𝑡,𝑡_(𝑛)(𝛼,0)=𝛼. (3.23) Then following the same procedure of estimate leading to (3.20), we obtain 𝑢^(𝑛+1)(𝑡)_{𝐹𝑠,𝑟𝑝,𝑞}+𝜃^(𝑛+1)(𝑡)_{𝐹𝑠,𝑟𝑝,𝑞}≲𝑆_𝑛+2𝑢_{0𝐹𝑠,𝑟𝑝,𝑞}+𝑆_𝑛+2𝜃_{0𝐹𝑠,𝑟𝑝,𝑞}+^𝑡₀1+∇𝑢^(𝑛)(𝜏)_𝐿∞×𝑢^(𝑛+1)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}+𝜃^(𝑛+1)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}+d𝜏d𝜏._𝑡0∇𝑢^(𝑛+1)(𝜏)_𝐿∞+∇𝜃^(𝑛+1)(𝜏)_𝐿∞𝑢^(𝑛)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}. (3.24) Note that 𝑆_𝑛+2𝑢_{0𝐹𝑠,𝑟𝑝,𝑞}≤𝑢_{0𝐹𝑠,𝑟𝑝,𝑞},𝑆_𝑛+2𝜃_{0𝐹𝑠,𝑟𝑝,𝑞}≤𝜃_{0𝐹𝑠,𝑟𝑝,𝑞}. (3.25) This together with Sobolev embedding theorem 𝐹^{𝑠−1,𝑟}_𝑝,𝑞↪𝐿^∞ for 𝑠−1>𝑑/𝑝 yields that 𝑢^(𝑛+1)(𝑡)_{𝐹𝑠,𝑟𝑝,𝑞}+𝜃^(𝑛+1)(𝑡)_{𝐹𝑠,𝑟𝑝,𝑞}≲𝑢_{0𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0𝐹𝑠,𝑟𝑝,𝑞}+^𝑡₀1+𝑢^(𝑛)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}𝑢^(𝑛+1)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}+𝜃^(𝑛+1)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}d𝜏. (3.26) It follows from Gronwall’s inequality that 𝑢^(𝑛+1)(𝑡)_{𝐹𝑠,𝑟𝑝,𝑞}+𝜃^(𝑛+1)(𝑡)_{𝐹𝑠,𝑟𝑝,𝑞}≤𝐶𝑢_{0𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0𝐹𝑠,𝑟𝑝,𝑞}exp𝐶^𝑡₀1+𝑢^(𝑛)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}d𝜏 (3.27) for some 𝐶>0 independent of 𝑛. Choosing 𝑇₀∶=𝑇₀(‖𝑢₀‖_𝐹^𝑠,𝑟_𝑝,𝑞,‖𝜃₀‖_𝐹^𝑠,𝑟_𝑝,𝑞)>0 such that 𝐶1+2𝐶𝑢_{0𝐹𝑠,𝑟𝑝,𝑞}+2𝐶𝜃_{0𝐹𝑠,𝑟𝑝,𝑞}𝑇₀≤log2, (3.28) and using the standard induction arguments, we obtain for any 𝑛∈ℕ₀sup_{0≤𝑡≤𝑇0}𝑢^(𝑛+1)(𝑡)_{𝐹𝑠,𝑟𝑝,𝑞}+𝜃^(𝑛+1)(𝑡)_{𝐹𝑠,𝑟𝑝,𝑞}≤2𝐶𝑢_{0𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0𝐹𝑠,𝑟𝑝,𝑞}. (3.29)

Step 3 (existence). To prove the local existence, we show that there exists 𝑇₁∈(0,𝑇₀] independent of 𝑛 such that (𝑢^(𝑛),𝜃^(𝑛)) is a Cauchy sequence in (𝐶([0,𝑇₁];𝐹^{𝑠−1,𝑟}_𝑝,𝑞(ℝ^𝑑)))^𝑑+1. For this purpose, we denote 𝛿𝑢^(𝑛+1)∶=𝑢^(𝑛+1)−𝑢^(𝑛),𝛿𝜃^(𝑛+1)∶=𝜃^(𝑛+1)−𝜃^(𝑛),𝛿𝑝^(𝑛+1)∶=𝑝^(𝑛+1)−𝑝^(𝑛). (3.30) It follows that (𝛿𝑢^(𝑛+1),𝛿𝜃^(𝑛+1),𝛿𝑝^(𝑛+1)) satisfies the following: 𝜕_𝑡𝛿𝑢^(𝑛+1)+𝑢^(𝑛)⋅∇𝛿𝑢^(𝑛+1)+∇𝛿𝑝^(𝑛+1)=−𝛿𝑢^(𝑛)⋅∇𝑢^(𝑛)+𝛿𝜃^(𝑛+1)𝑒_𝑑,inℝ^𝑑×(),𝜕0,∞_𝑡𝛿𝜃^(𝑛+1)+𝑢^(𝑛)⋅∇𝛿𝜃^(𝑛+1)=−𝛿𝑢^(𝑛)⋅∇𝜃^(𝑛),inℝ^𝑑×(0,∞),div𝑢^(𝑛)=0,inℝ^𝑑×(0,∞),𝛿𝑢^(𝑛+1)(𝑥,0)=Δ_𝑛+1𝑢₀(𝑥),𝛿𝜃^(𝑛+1)(𝑥,0)=Δ_𝑛+1𝜃₀(𝑥),inℝ^𝑑. (3.31) Since 𝑋^(𝑛)(𝛼,𝑡) is a solution of the ordinary differential equations (3.23), we have dd𝑡𝛿𝑢^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡=−𝛿𝑢^(𝑛)⋅∇𝑢^(𝑛)𝑋^(𝑛)(𝛼,𝑡),𝑡−∇𝛿𝑝^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡+𝛿𝜃^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡𝑒_𝑑,dd𝑡𝛿𝜃^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡=−𝛿𝑢^(𝑛)⋅∇𝜃^(𝑛)𝑋^(𝑛)(𝛼,𝑡),𝑡. (3.32) Integrating the previous equations on [0,𝑡], we have 𝛿𝑢^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡+𝛿𝜃^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡=Δ_𝑛+1𝑢₀(𝛼)+Δ_𝑛+1𝜃₀(𝛼)−^𝑡₀𝛿𝑢^(𝑛)⋅∇𝑢^(𝑛)+𝜃^(𝑛)𝑋^(𝑛)()−𝛼,𝜏,𝜏d𝜏_𝑡0∇𝛿𝑝^(𝑛+1)𝑋^(𝑛)(𝛼,𝜏),𝜏d𝜏+^𝑡₀𝛿𝜃^(𝑛+1)𝑋^(𝑛)(𝛼,𝜏),𝜏𝑒_𝑑d𝜏. (3.33) Similar to Step 1, we take the 𝐿^𝑝,𝑟 norm on both sides, use Minkowski’s inequality and Hölder’s inequality in Lorentz spaces, and then obtain 𝛿𝑢^(𝑛+1)(𝑡)_{𝐿𝑝,𝑟}+𝛿𝜃^(𝑛+1)(𝑡)_{𝐿𝑝,𝑟}≲Δ_𝑛+1𝑢_{0𝐿𝑝,𝑟}+Δ_𝑛+1𝜃_{0𝐿𝑝,𝑟}+^𝑡₀𝛿𝑢^(𝑛)⋅∇𝑢^(𝑛)+𝜃^(𝑛)(𝜏)_{𝐿𝑝,𝑟}+d𝜏_𝑡0∇𝛿𝑝^(𝑛+1)_{𝐿𝑝,𝑟}d𝜏+^𝑡₀𝛿𝜃^(𝑛+1)(𝜏)_{𝐿𝑝,𝑟}≲Δd𝜏_𝑛+1𝑢_{0𝐿𝑝,𝑟}+Δ_𝑛+1𝜃_{0𝐿𝑝,𝑟}+^𝑡₀𝛿𝑢^(𝑛)_{𝐿𝑝,𝑟}∇𝑢^(𝑛)(𝜏)_∞+∇𝜃^(𝑛)(𝜏)_𝐿∞+d𝜏_𝑡01+∇𝑢^(𝑛)(𝜏)_𝐿∞𝛿𝑢^(𝑛+1)(𝜏)_{𝐿𝑝,𝑟}+𝛿𝜃^(𝑛+1)(𝜏)_{𝐿𝑝,𝑟}≲Δd𝜏_𝑛+1𝑢_{0𝐿𝑝,𝑟}+Δ_𝑛+1𝜃_{0𝐿𝑝,𝑟}+^𝑡₀𝛿𝑢^(𝑛)_{𝐿𝑝,𝑟}𝑢^(𝑛)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}+𝜃^(𝑛)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}d𝜏+^𝑡₀1+𝑢^(𝑛)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}𝛿𝑢^(𝑛+1)(𝜏)_{𝐿𝑝,𝑟}+𝛿𝜃^(𝑛+1)(𝜏)_{𝐿𝑝,𝑟}d𝜏, (3.34) where we used 𝐹^{𝑠−1,𝑟}_𝑝,𝑞↪𝐿^∞ for 𝑠−1>𝑑/𝑝 in the last inequality.
Next, we apply Δ_𝑗 to the first equation of (3.31) to obtain 𝜕_𝑡Δ_𝑗𝛿𝑢^(𝑛+1)+𝑢^(𝑛)⋅∇Δ_𝑗𝛿𝑢^(𝑛+1)=𝑢^(𝑛),Δ_𝑗⋅∇𝛿𝑢^(𝑛+1)−Δ_𝑗𝛿𝑢^(𝑛)⋅∇𝑢^(𝑛)−∇Δ_𝑗𝛿𝑝^(𝑛+1)+Δ_𝑗𝛿𝜃^(𝑛+1)𝑒_𝑑,𝜕_𝑡Δ_𝑗𝛿𝜃^(𝑛+1)+𝑢^(𝑛)⋅∇Δ_𝑗𝛿𝜃^(𝑛+1)=𝑢^(𝑛),Δ_𝑗⋅∇𝛿𝜃^(𝑛+1)−Δ_𝑗𝛿𝑢^(𝑛)⋅∇𝜃^(𝑛). (3.35) It follows from the definition of 𝑋^(𝑛)(𝛼,𝑡) that dΔd𝑡^𝑗𝛿𝑢^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡=𝑢^(𝑛),Δ_𝑗⋅∇𝛿𝑢^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡−Δ_𝑗𝛿𝑢^(𝑛)⋅∇𝑢^(𝑛)𝑋^(𝑛)(𝛼,𝑡),𝑡−∇Δ_𝑗𝛿𝑝^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡+Δ_𝑗𝛿𝜃^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡𝑒_𝑑,dΔd𝑡^𝑗𝛿𝜃^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡=𝑢^(𝑛),Δ_𝑗⋅∇𝛿𝜃^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡−Δ_𝑗𝛿𝑢^(𝑛)⋅∇𝜃^(𝑛)𝑋^(𝑛)(𝛼,𝑡),𝑡. (3.36) Integrating the previous equations on [0,𝑡] yields Δ_𝑗𝛿𝑢^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡+Δ_𝑗𝛿𝜃^(𝑛+1)𝑋^(𝑛)(𝛼,𝑡),𝑡=Δ_𝑗Δ_𝑛+1𝑢₀(𝛼)+Δ_𝑗Δ_𝑛+1𝜃₀(𝛼)+^𝑡₀𝑢^(𝑛),Δ_𝑗⋅∇𝛿𝑢^(𝑛+1)𝑋^(𝑛)()+𝛼,𝜏,𝜏d𝜏_𝑡0𝑢^(𝑛),Δ_𝑗⋅∇𝛿𝜃^(𝑛+1)𝑋^(𝑛)(𝛼,𝜏),𝜏d𝜏−^𝑡₀Δ_𝑗𝛿𝑢^(𝑛)⋅∇𝑢^(𝑛)𝑋^(𝑛)()−𝛼,𝜏,𝜏d𝜏_𝑡0Δ_𝑗𝛿𝑢^(𝑛)⋅∇𝜃^(𝑛)𝑋^(𝑛)(𝛼,𝜏),𝜏d𝜏−^𝑡₀∇Δ_𝑗𝛿𝑝^(𝑛+1)𝑋^(𝑛)()+𝛼,𝜏,𝜏d𝜏_𝑡0Δ_𝑗𝛿𝜃^(𝑛+1)𝑋^(𝑛)(𝛼,𝜏),𝜏𝑒_𝑑d𝜏. (3.37) Then following the similar procedure of estimate leading to (3.20), we get 𝛿𝑢^(𝑛+1)(𝑡)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}+𝛿𝜃^(𝑛+1)(𝑡)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}≲Δ_𝑛+1𝑢_{0̇𝐹𝑠−1,𝑟𝑝,𝑞}+Δ_𝑛+1𝜃_{0̇𝐹𝑠−1,𝑟𝑝,𝑞}+^𝑡₀2^{𝑗(𝑠−1)}𝑢^(𝑛),Δ_𝑗⋅∇𝛿𝑢^(𝑛+1)(𝜏)_{𝑙𝑞(ℤ)𝐿𝑝,𝑟}+d𝜏_𝑡02^{𝑗(𝑠−1)}𝑢^(𝑛),Δ_𝑗⋅∇𝛿𝜃^(𝑛+1)(𝜏)_{𝑙𝑞(ℤ)𝐿𝑝,𝑟}d𝜏+^𝑡₀𝛿𝑢^(𝑛)⋅∇𝑢^(𝑛)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}+d𝜏_𝑡0𝛿𝑢^(𝑛)⋅∇𝜃^(𝑛)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}d𝜏+^𝑡₀∇𝛿𝑝^(𝑛+1)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}d𝜏+^𝑡₀𝛿𝜃^(𝑛+1)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}≲Δd𝜏_𝑛+1𝑢_{0̇𝐹𝑠−1,𝑟𝑝,𝑞}+Δ_𝑛+1𝜃_{0̇𝐹𝑠−1,𝑟𝑝,𝑞}+^𝑡₀1+∇𝑢^(𝑛)(𝜏)_∞𝛿𝑢^(𝑛+1)(𝜏)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}+𝛿𝜃^(𝑛+1)(𝜏)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}+d𝜏_𝑡0𝛿𝑢^(𝑛+1)(𝜏)_∞+𝛿𝜃^(𝑛+1)(𝜏)_∞∇𝑢^(𝑛)(𝜏)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}+d𝜏_𝑡0𝛿𝑢^(𝑛)(𝜏)_∞∇𝑢^(𝑛)(𝜏)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}+∇𝜃^(𝑛)(𝜏)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}+d𝜏_𝑡0𝛿𝑢^(𝑛)(𝜏)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}∇𝑢^(𝑛)(𝜏)_∞+∇𝜃^(𝑛)(𝜏)_∞d𝜏, (3.38) where we used the commutator estimates (2.19) and the product estimates (2.16) in last inequality. This together with (3.34), Sobolev embedding theorem, and the boundedness of Riesz transform gives that 𝛿𝑢^(𝑛+1)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}+𝛿𝜃^(𝑛+1)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}≲Δ_𝑛+1𝑢_{0𝐹𝑠−1,𝑟𝑝,𝑞}+Δ_𝑛+1𝜃_{0𝐹𝑠−1,𝑟𝑝,𝑞}+^𝑡₀1+𝑢^(𝑛)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}𝛿𝑢^(𝑛+1)(𝜏)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}+𝛿𝜃^(𝑛+1)(𝜏)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}+d𝜏_𝑡0𝛿𝑢^(𝑛)(𝜏)_{𝐹𝑠−1,𝑟𝑝,𝑞}𝑢^(𝑛)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}+𝜃^(𝑛)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}d𝜏≲2^−(𝑛+1)𝑢_{0̇𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0̇𝐹𝑠,𝑟𝑝,𝑞}+𝑇₁1+sup_{𝑡∈[0,𝑇1]}𝑢^(𝑛)(𝑡)_{𝐹𝑠,𝑟𝑝,𝑞}sup_{𝑡∈[0,𝑇1]}𝛿𝑢^(𝑛+1)(𝑡)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}+𝛿𝜃^(𝑛+1)(𝑡)_{̇𝐹𝑠−1,𝑟𝑝,𝑞}+𝑇₁sup_{𝑡∈[0,𝑇1]}𝑢^(𝑛)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}+𝜃^(𝑛)(𝜏)_{𝐹𝑠,𝑟𝑝,𝑞}sup_{𝑡∈[0,𝑇1]}𝛿𝑢^(𝑛)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}. (3.39) It follows from the uniform bounds (3.29) that sup_{𝑡∈[0,𝑇1]}𝛿𝑢^(𝑛+1)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}+𝛿𝜃^(𝑛+1)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}≤𝐶2^−(𝑛+1)𝑢_{0̇𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0̇𝐹𝑠,𝑟𝑝,𝑞}+𝐶𝑇₁sup_{𝑡∈[0,𝑇1]}𝛿𝑢^(𝑛+1)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}+𝛿𝜃^(𝑛+1)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}+𝐶𝑇₁sup_{𝑡∈[0,𝑇1]}𝛿𝑢^(𝑛)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}+𝛿𝜃^(𝑛)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}. (3.40) If we choose 𝑇₁ small enough such that 𝐶𝑇₁≤1/2, then sup_{𝑡∈[0,𝑇1]}𝛿𝑢^(𝑛+1)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}+𝛿𝜃^(𝑛+1)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}≤𝐶2^−(𝑛+1)𝑢_{0̇𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0̇𝐹𝑠,𝑟𝑝,𝑞}+2𝐶𝑇₁sup_{𝑡∈[0,𝑇1]}𝛿𝑢^(𝑛)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}+𝛿𝜃^(𝑛)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}. (3.41) By the standard induction arguments, we obtain sup_{𝑡∈[0,𝑇1]}𝛿𝑢^(𝑛+1)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}+𝛿𝜃^(𝑛+1)(𝑡)_{𝐹𝑠−1,𝑟𝑝,𝑞}≤2𝐶2^−𝑛𝑢_{0̇𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0̇𝐹𝑠,𝑟𝑝,𝑞}, (3.42) which implies that (𝑢^(𝑛)(𝑥,𝑡),𝜃^(𝑛)(𝑥,𝑡)) is a Cauchy sequence in (𝐶([0,𝑇₁];𝐹^{𝑠−1,𝑟}_𝑝,𝑞))^𝑑+1 with 𝑇₁≤min{𝑇₀,1/2𝐶}. Thus 𝑢^(𝑛)(𝑥,𝑡)→𝑢(𝑥,𝑡) and 𝜃^(𝑛)(𝑥,𝑡)→𝜃(𝑥,𝑡) in (𝐶([0,𝑇₁];𝐹^𝑠,𝑟_𝑝,𝑞))^𝑑+1 as 𝑛→∞. From (3.22) and the uniform estimates (3.29), we have actually (𝑢(𝑥,𝑡),𝜃(𝑥,𝑡))∈(𝐶([0,𝑇₁];𝐹^𝑠,𝑟_𝑝,𝑞))^𝑑+1 solving the Boussinesq equation (1.1).

Step 4 (uniqueness). To get the uniqueness we closely follow the arguments in last step. Assume that (𝑢,𝜃) and (𝑢,𝜃) are two solutions of the Boussinesq equations (1.1) with the same initial data (𝑢₀,𝜃₀) in the class 𝐿^∞([0,𝑇₁];𝐹^𝑠,𝑟_𝑝,𝑞(ℝ^𝑑)). Denote 𝛿𝑢∶=𝑢−𝑢,𝛿𝜃∶=𝜃−𝜃,𝛿𝑝∶=𝑝−𝑝. (3.43) Then it follows that 𝛿𝜃 satisfies the following: 𝜕_𝑡𝛿𝑢+𝑢⋅∇𝛿𝑢+∇𝛿𝑝=−𝛿𝑢⋅∇𝑢+𝛿𝜃𝑒_𝑑,inℝ^𝑑×0,𝑇₁,𝜕_𝑡𝛿𝜃+𝑢⋅∇𝛿𝜃=−𝛿𝑢⋅∇𝜃,inℝ^𝑑×0,𝑇₁,div𝑢=0,inℝ^𝑑×0,𝑇₁,𝛿𝑢(𝑥,0)=0,𝛿𝜃(𝑥,0)=0,inℝ^𝑑. (3.44) We follow the strategy used to derive (3.40) to obtain sup_{[]𝑡∈0,𝑇}‖𝛿𝑢(𝑡)‖_𝐹^{𝑠−1,𝑟}_𝑝,𝑞+‖𝛿𝜃(𝑡)‖_𝐹^{𝑠−1,𝑟}_𝑝,𝑞≤𝐶𝑇sup_{[]𝑡∈0,𝑇}‖𝛿𝑢(𝑡)‖_𝐹^{𝑠−1,𝑟}_𝑝,𝑞+‖𝛿𝜃(𝑡)‖_𝐹^{𝑠−1,𝑟}_𝑝,𝑞 (3.45) for any 𝑇≤𝑇₁. Whenever 𝑇 is small enough such that 𝐶𝑇<1, we have 𝛿𝑢(𝑥,𝑡)≡𝛿𝜃(𝑥,𝑡)≡0 for any 𝑡≤𝑇, that is, 𝑢(𝑥,𝑡)≡𝑢(𝑥,𝑡) and 𝜃(𝑥,𝑡)≡𝜃(𝑥,𝑡). By using the standard continuity argument, we see 𝑢(𝑥,𝑡)≡𝑢(𝑥,𝑡) and 𝜃(𝑥,𝑡)≡𝜃(𝑥,𝑡) for any 𝑡≤𝑇₁.

Step 5 (blow-up criterion). By the a priori estimate (3.21), we only need to dominate ‖∇𝑢‖_∞ and ‖∇𝜃‖_∞. Indeed, it follows from the logarithmic Triebel-Lizorkin-Lorentz inequality (2.20) with 𝑠−1>𝑑/𝑝 that ‖∇𝑢‖_𝐿∞≲1+‖∇𝑢‖_{̇𝐹0∞,∞}log⁺‖∇𝑢‖_𝐹^{𝑠−1,𝑟}_𝑝,𝑞+1≲1+‖𝑢‖_{̇𝐹1∞,∞}log⁺‖𝑢‖_𝐹^𝑠,𝑟_𝑝,𝑞,‖‖+1∇𝜃_𝐿∞≲1+‖∇𝜃‖_{̇𝐹0∞,∞}log⁺‖∇𝜃‖_𝐹^{𝑠−1,𝑟}_𝑝,𝑞+1≲1+‖𝜃‖_{̇𝐹1∞,∞}log⁺‖𝜃‖_𝐹^𝑠,𝑟_𝑝,𝑞+1. (3.46) Thus, the a priori estimate (3.21) gives that ‖𝑢(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞≤𝐶𝑢_{0𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0𝐹𝑠,𝑟𝑝,𝑞}exp𝐶^𝑡₀‖𝑢(𝜏)‖_{̇𝐹1∞,∞}+‖𝜃(𝜏)‖_{̇𝐹1∞,∞}×log⁺‖𝑢(𝜏)‖_𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝜏)‖_𝐹^𝑠,𝑟_𝑝,𝑞+1d𝜏. (3.47) By using Gronwall’s inequality, we have ‖𝑢(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞≤𝐶𝑢_{0𝐹𝑠,𝑟𝑝,𝑞}+𝜃_{0𝐹𝑠,𝑟𝑝,𝑞}exp𝐶exp𝐶^𝑡₀‖𝑢(𝜏)‖_{̇𝐹1∞,∞}+‖𝜃(𝜏)‖_{̇𝐹1∞,∞}d𝜏. (3.48) Thus if limsup_{𝑡→𝑇∗−}(‖𝑢(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞)=∞, then ∫^𝑇∗₀(‖𝑢(𝑡)‖_{̇𝐹1∞,∞}+‖𝜃(𝑡)‖_{̇𝐹1∞,∞})d𝑡=∞.
On the other hand, it follows from Sobolev embedding 𝐹^𝑠,𝑟_𝑝,𝑞↪𝑊^1,∞↪𝐹̇¹_∞,∞ for 𝑠−1>𝑑/𝑝 that ^𝑇∗₀‖𝑢(𝑡)‖_{̇𝐹1∞,∞}+‖𝜃(𝑡)‖_{̇𝐹1∞,∞}d𝑡≤𝑇^∗sup_{0≤𝜏≤𝑇∗}‖𝑢(𝜏)‖_{̇𝐹1∞,∞}+‖𝜃(𝜏)‖_{̇𝐹1∞,∞}≲𝑇^∗sup_{0≤𝜏≤𝑇∗}‖∇𝑢(𝜏)‖_𝐿∞+‖∇𝜃(𝜏)‖_𝐿∞≲𝑇^∗sup_{0≤𝜏≤𝑇∗}‖𝑢(𝜏)‖_𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝜏)‖_𝐹^𝑠,𝑟_𝑝,𝑞. (3.49) Thus ∫^𝑇∗₀(‖𝑢(𝑡)‖_{̇𝐹1∞,∞}+‖𝜃(𝑡)‖_{̇𝐹1∞,∞})d𝑡=∞ implies limsup_{𝑡→𝑇∗−}(‖𝑢(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞+‖𝜃(𝑡)‖_𝐹^𝑠,𝑟_𝑝,𝑞)=∞.

Acknowledgments

This work was completed while the first author visited the Beijing Institute of Applied Physics and Computational Mathematics. He is very grateful to this institution for its hospitality and financial support and to Professor Changxing Miao for his useful suggestions. Z. Xiang was partially supported by NNSF of China (11101068), the Sichuan Youth Science & Technology Foundation (2011JQ0003), the Fundamental Research Funds for the Central Universities (ZYGX2009X019), and by SRF for ROCS, SEM. W. Yan was partially supported by NNSF of China (11071025), the Foundation of CAEP 2010A0202010, and the Foundation of LCP.