Blow-Up Criteria for Three-Dimensional Boussinesq Equations in Triebel-Lizorkin Spaces

Abstract

We establish a new blow-up criteria for solution of the three-dimensional Boussinesq equations in Triebel-Lizorkin spaces by using Littlewood-Paley decomposition.

1. Introduction and Main Results

In this paper, we consider the regularity of the following three-dimensional incompressible Boussinesq equations: 𝑢_𝑡−𝜇Δ𝑢+𝑢⋅∇𝑢+∇𝑃=𝜃𝑒₃,(𝑥,𝑡)∈ℝ³×(),𝜃0,∞_𝑡𝑢()−𝜅Δ𝜃+𝑢⋅∇𝜃=0,∇⋅𝑢=0,𝑥,0=𝑢₀,𝜃(𝑥,0)=𝜃₀, (1.1) where 𝑢=(𝑢¹(𝑥,𝑡),𝑢²(𝑥,𝑡),𝑢³(𝑥,𝑡)) denotes the fluid velocity vector field, 𝑃=𝑃(𝑥,𝑡) is the scalar pressure, 𝜃(𝑥,𝑡) is the scalar temperature, 𝜇>0 is the constant kinematic viscosity, 𝜅>0 is the thermal diffusivity, and 𝑒₃=(0,0,1)^𝑇, while 𝑢₀ and 𝜃₀ are the given initial velocity and initial temperature, respectively, with ∇⋅𝑢₀=0. Boussinesq systems are widely used to model the dynamics of the ocean or the atmosphere. They arise from the density-dependent fluid equations by using the so-called Boussinesq approximation which consists in neglecting the density dependence in all the terms but the one involving the gravity. This approximation can be justified from compressible fluid equations by a simultaneous low Mach number/Froude number limit; we refer to [1] for a rigorous justification. It is well known that the question of global existence or finite-time blow-up of smooth solutions for the 3D incompressible Boussinesq equations. This challenging problem has attracted significant attention. Therefore, it is interesting to study the blow-up criterion of the solutions for system (1.1).

Recently, Fan and Zhou [2] and Ishimura and Morimoto [3] proved the following blow-up criterion, respectively:curl𝑢∈𝐿¹0,𝑇;𝐵̇⁰_∞,∞ℝ³,∇𝑢∈𝐿¹0,𝑇;𝐿^∞ℝ³. (1.2)(1.3) Subsequently, Qiu et al. [4] obtained Serrin-type regularity condition for the three-dimensional Boussinesq equations under the incompressibility condition. Furthermore, Xu et al. [5] obtained the similar regularity criteria of smooth solution for the 3D Boussinesq equations in the Morrey-Campanato space.

Our purpose in this paper is to establish a blow-up criteria of smooth solution for the three-dimensional Boussinesq equations under the incompressibility condition ∇⋅𝑢₀=0 in Triebel-Lizorkin spaces.

Now we state our main results as follows.

Theorem 1.1. Let (𝑢₀,𝜃₀)∈𝐻¹(ℝ³), (𝑢(⋅,𝑡),𝜃(⋅,𝑡)) be the smooth solution to the problem (1.1) with the initial data (𝑢₀,𝜃₀) for 0⩽𝑡<𝑇. If the solution 𝑢 satisfies the following condition ∇𝑢∈𝐿^𝑝0,𝑇;𝐹̇_{𝑞,(2𝑞/3)}ℝ³,2𝑝+3𝑞3=2,2<𝑞⩽∞, (1.4) then the solution (𝑢,𝜃) can be extended smoothly beyond 𝑡=𝑇.

Corollary 1.2. Let (𝑢₀,𝜃₀)∈𝐻¹(ℝ³), (𝑢(⋅,𝑡),𝜃(⋅,𝑡)) be the smooth solution to the problem (1.1) with the initial data (𝑢₀,𝜃₀) for 0⩽𝑡<𝑇. If the solution 𝑢 satisfies the following condition curl𝑢∈𝐿¹0,𝑇;𝐵̇_∞,∞ℝ³, (1.5) then the solution (𝑢,𝜃) can be extended smoothly beyond 𝑡=𝑇.

Remark 1.3. By Corollary 1.2, we can see that our main result is an improvement of (1.2).

2. Preliminaries and Lemmas

The proof of the results presented in this paper is based on a dyadic partition of unity in Fourier variables, the so-called homogeneous Littlewood-Paley decomposition. So, we first introduce the Littlewood-Paley decomposition and Triebel-Lizorkin spaces.

Let 𝒮(ℝ³) be the Schwartz class of rapidly decreasing function. Given 𝑓∈𝒮(ℝ³), its Fourier transform ℱ𝑓=𝑓 is definedby 𝑓(𝜉)=_ℝ3𝑒^{−𝑖𝑥⋅𝜉}𝑓(𝑥)𝑑𝑥. (2.1) Let (𝜒,𝜑) be a couple of smooth functions valued in [0,1] such that 𝜒 is supported in the ball {𝜉∈ℝ³∶|𝜉|⩽4/3}, 𝜑 is supported in the shell {𝜉∈ℝ³∶3/4⩽|𝜉|⩽8/3}, and 𝜒(𝜉)+_𝑗⩾0𝜑2^−𝑗𝜉=1,∀𝜉∈ℝ³,_𝑗∈𝑍𝜑2^−𝑗𝜉=1,∀𝜉∈ℝ³⧵{0}. (2.2) Denoting 𝜑_𝑗=𝜑(2^−𝑗𝜉), ℎ=𝐹⁻¹𝜑, and ℎ=𝐹⁻¹𝜒, we define the dyadic blocks asΔ̇_𝑗𝑓=𝜑2^−𝑗𝐷=2^3𝑗_ℝ3ℎ2^𝑗𝑦𝑓𝑆̇𝑥−𝑦𝑑𝑦,𝑗∈ℤ,_𝑗𝑓=_{𝑘⩽𝑗−1}Δ_𝑘𝑓=2^3𝑗_ℝ3ℎ2^𝑗𝑦𝑓𝑥−𝑦𝑑𝑦,𝑗∈ℤ. (2.3)

Definition 2.1. Let 𝒮^′_ℎ be the space of temperate distribution 𝑢 such that lim_{𝑗→−∞}𝑆̇_𝑗𝑓=0,in𝒮^′. (2.4) The formal equality𝑓=_𝑗∈ℤΔ̇_𝑗𝑓 (2.5) holds in 𝑆^′_ℎ and is called the homogeneous Littlewood-Paley decomposition. It has nice properties of quasi-orthogonality Δ̇_𝑗Δ̇_𝑞𝑓≡0,𝑗−𝑞⩾2. (2.6)

Let us now define the homogeneous Besov spaces and Triebel-Lizorkin spaces; we refer to [6, 7] for more detailed properties.

Definition 2.2. Letting 𝑠∈ℝ,𝑝,𝑞∈[1,∞], the homogeneous Besov space 𝐵̇^𝑠_𝑝,𝑞 is defined by𝐵̇^𝑠_𝑝,𝑞=𝑓∈𝒵^′𝑅³∣𝑓_{̇𝐵𝑠𝑝,𝑞}<∞. (2.7) Here 𝑓_{̇𝐵𝑠𝑝,𝑞}=^∞_𝑗=∑_−∞2^𝑗𝑠𝑞Δ̇_𝑗𝑓^𝑞_𝑝^1/𝑞,𝑞<∞,sup_𝑗∈ℤ2^𝑗𝑠Δ̇_𝑗𝑓_𝑝,𝑞=∞, (2.8) and 𝒵^′(ℝ³) denotes the dual space of 𝒵(ℝ³)={𝑓∈𝒮(ℝ³)∣𝐷^𝛼𝑓(0)=0,∀𝛼∈ℕ³multi-index}.

Definition 2.3. Let 𝑠∈ℝ,  𝑝∈[1,∞),  and  𝑞∈[1,∞], and for 𝑠∈ℝ,  𝑝=∞,  and  𝑞=∞, the homogeneous Triebel-Lizorkin space 𝐹̇^𝑠_𝑝,𝑞 is defined by 𝐹̇^𝑠_𝑝,𝑞=𝑓∈𝒵^′ℝ³∣𝑓_{̇𝐹𝑠𝑝,𝑞}<∞. (2.9) Here 𝑓_{̇𝐹𝑠𝑝,𝑞}=^∞_𝑗=∑_−∞2^𝑗𝑠𝑞Δ̇_𝑗𝑓^𝑞1/𝑞_𝐿𝑝,𝑞<∞,sup_𝑗∈ℤ2^𝑗𝑠Δ̇_𝑗𝑓_𝑝,𝑞=∞, (2.10) for 𝑝=∞ and 𝑞∈[1,∞), the space 𝐹̇^𝑠_𝑝,𝑞 is defined by means of Carleson measures which is not treated in this paper. Notice that by Minkowski’s inequality, we have the following inclusions: 𝐵̇^𝑠_𝑝,𝑞⊂𝐹̇^𝑠_𝑝,𝑞𝐹̇,if𝑞⩽𝑝,_{𝑠𝑝,𝑞}⊂𝐵̇^𝑠_𝑝,𝑞,if𝑞⩾𝑝. (2.11) Also it is well known that 𝐵̇^𝑠_𝑝,𝑝=𝐹̇^𝑠_𝑝,𝑝,𝐿^∞⊂𝐵̇⁰_∞,∞=𝐹̇⁰_∞,∞,𝐵̇^𝑠_2,2=𝐹̇^𝑠_2,2=𝐻̇^𝑠. (2.12)

Throughout the proof of Theorem 1.1 in Section 3, we will use the following interpolation inequality frequently: 𝑓_𝐿𝑞⩽𝐶𝑓_𝐿^3/𝑞−1/2₂𝐶∇𝑓_𝐿^3/2−3/𝑞₂,2⩽𝑞⩽6,𝑓∈𝐿²ℝ³∩𝐻̇¹ℝ³. (2.13)

Lemma 2.4. Let 𝑘∈𝑁. Then there exists a constant 𝐶 independent of 𝑓,𝑗 such that for 1⩽𝑝⩽𝑞⩽∞sup_|𝛼|=𝑘𝜕^𝛼Δ̇_𝑗𝑓_𝑞⩽𝐶2^{𝑗𝑘+3𝑗(1/𝑝−1/𝑞)}Δ̇_𝑗𝑓_𝑝. (2.14)

Remark 2.5. From the above Beinstein estimate, we easily know thatΔ̇_𝑗𝑓_𝑞⩽𝐶2^{3𝑗(1/𝑝−1/𝑞)}Δ̇_𝑗𝑓_𝑝. (2.15)

3. Proofs of the Main Results

In this section, we prove Theorem 1.1. For simplicity, without loss of generality, we assume 𝜇=𝜅=1.

Proof of Theorem 1.1. Differentiating the first equation and the second equation of (1.1) with respect to 𝑥_𝑘(1⩽𝑘⩽3), and multiplying the resulting equations by 𝜕𝑢/𝜕𝑥_𝑘=𝜕_𝑘𝑢 and 𝜕𝜃/𝜕𝑥_𝑘=𝜕_𝑘𝜃, respectively, then by integrating by parts over ℝ³ we get 12𝑑𝜕𝑑𝑡^𝑘𝑢²_𝐿2+∇𝜕_𝑘𝑢²_𝐿2=−𝜕_𝑘[(𝑢⋅∇)𝑢]⋅𝜕_𝑘𝑢𝑑𝑥−𝜕_𝑘∇𝑃⋅𝜕_𝑘𝑢𝑑𝑥+𝜕_𝑘𝜃𝑒₃𝜕_𝑘1𝑢𝑑𝑥,2𝑑𝜕𝑑𝑡^𝑘𝜃²_𝐿2+∇𝜕_𝑘𝜃²_𝐿2=−𝜕_𝑘[(𝑢⋅∇)𝜃]⋅𝜕_𝑘𝜃𝑑𝑥. (3.1) Noting the incompressibility condition ∇⋅𝑢=0, since 𝜕_𝑘[(𝑢⋅∇)𝑢]⋅𝜕_𝑘𝑢𝑑𝑥=𝜕_𝑘𝑢⋅∇𝑢⋅𝜕_𝑘𝜕𝑢𝑑𝑥,_𝑘∇𝑃⋅𝜕_𝑘𝜕𝑢𝑑𝑥=0,_𝑘[(𝑢⋅∇)𝜃]⋅𝜕_𝑘𝜃𝑑𝑥=𝜕_𝑘𝑢⋅∇𝜃⋅𝜕_𝑘𝜃𝑑𝑥, (3.2) then the above equations (3.1) can be rewritten as12𝑑𝜕𝑑𝑡^𝑘𝑢²_𝐿2+∇𝜕_𝑘𝑢²_𝐿2=−𝜕_𝑘𝑢⋅∇𝑢⋅𝜕_𝑘𝑢𝑑𝑥+𝜕_𝑘𝜃𝑒₃𝜕_𝑘1𝑢𝑑𝑥,2𝑑𝜕𝑑𝑡^𝑘𝜃²_𝐿2+∇𝜕_𝑘𝜃²_𝐿2=−𝜕_𝑘𝑢⋅∇𝜃⋅𝜕_𝑘𝜃𝑑𝑥. (3.3) Adding up (3.3), then we have 12𝑑𝜕𝑑𝑡^𝑘𝑢²_𝐿2+𝜕_𝑘𝜃²_𝐿2+∇𝜕_𝑘𝑢²_𝐿2+∇𝜕_𝑘𝜃²_𝐿2=−𝜕_𝑘𝑢⋅∇𝑢⋅𝜕_𝑘𝑢𝑑𝑥−𝜕_𝑘𝑢⋅∇𝜃⋅𝜕_𝑘𝜃𝑑𝑥+𝜕_𝑘𝜃𝑒₃⋅𝜕_𝑘𝑢𝑑𝑥≜𝐼₁+𝐼₂+𝐼₃. (3.4) Firstly, for the third term 𝐼₃, by Hölder’s inequality and Young’s inequality, we get𝐼₃=𝜕_𝑘𝜃𝑒₃⋅𝜕_𝑘1𝑢𝑑𝑥⩽2‖∇𝜃‖²_𝐿2+12‖∇𝑢‖²_𝐿2. (3.5) The other terms are bounded similarly. For simplicity, we detail the term 𝐼₂. Using the Littlewood-Paley decomposition (2.5), we decompose ∇𝑢 as follows: ∇𝑢=_𝑗∈ℤΔ̇_𝑗(∇𝑢)=_𝑗<−𝑁Δ̇_𝑗(∇𝑢)+^𝑗=𝑁_𝑗=−𝑁Δ̇_𝑗(∇𝑢)+_𝑗>𝑁Δ̇_𝑗(∇𝑢). (3.6) Here 𝑁 is a positive integer to be chosen later. Plugging (3.6) into 𝐼₂ produces that 𝐼₂=_{𝑗<−𝑁ℝ3}Δ̇_𝑗(∇𝑢)|∇𝜃|²+𝑑𝑥_{𝑗=𝑁𝑗=−𝑁ℝ3}Δ̇_𝑗(∇𝑢)|∇𝜃|²𝑑𝑥+_{𝑗>𝑁ℝ3}Δ̇_𝑗(∇𝑢)|∇𝜃|²𝑑𝑥≡𝐼¹₂+𝐼²₂+𝐼³₂. (3.7) For 𝐼¹₂, using the Hölder inequality, (2.12), and (2.15), we obtain that 𝐼¹₂⩽‖∇𝜃‖²_{𝐿2𝑗<−𝑁}Δ̇_𝑗∇𝑢_𝐿∞⩽𝐶‖∇𝜃‖²_{𝐿2𝑗<−𝑁}2^(3/2)𝑗Δ̇_𝑗∇𝑢_𝐿2⩽𝐶2^−(3/2)𝑁‖∇𝑢‖_𝐿²‖∇𝜃‖²_𝐿2⩽𝐶2^−(3/2)𝑁‖∇𝑢‖²_𝐿2+‖∇𝜃‖²_𝐿2^3/2. (3.8) For 𝐼²₂, from the Hölder inequality and (2.15), it follows that 𝐼²₂=^𝑗=𝑁_{𝑗=−𝑁ℝ3}|∇𝜃|²Δ̇_𝑗(∇𝑢)𝑑𝑥=_ℝ3|∇𝜃|^2𝑗=𝑁_𝑗=−𝑁Δ̇_𝑗(∇𝑢)𝑑𝑥⩽_ℝ3|∇𝜃|^2𝑗=𝑁_𝑗=−𝑁Δ̇_𝑗(∇𝑢)^{2𝑞/33/2𝑞}𝑁^1−3/2𝑞𝑑𝑥⩽𝐶𝑁^{(2𝑞−3)/2𝑞}_ℝ3|∇𝜃|^2𝑗=𝑁_𝑗=−𝑁Δ̇_𝑗(∇𝑢)^{2𝑞/33/2𝑞}𝑑𝑥⩽𝐶𝑁^{(2𝑞−3)/2𝑞}‖∇𝜃‖²_𝐿′2𝑞‖∇𝑢‖_̇𝐹⁰_{𝑞,(2𝑞/3)}. (3.9) Here 𝑞^′ denotes the conjugate exponent of 𝑞. Since 2𝑞>3 by the Gagliardo-Nirenberg inequality and the Young inequality, we have 𝐼²₂⩽𝐶𝑁^{(2𝑞−3)/2𝑞}‖∇𝜃‖_𝐿^{(2𝑞−3)/𝑞}₂∇²𝜃_𝐿^3/𝑞₂‖∇𝑢‖_̇𝐹⁰_{𝑞,(2𝑞/3)}⩽12∇²𝜃²_𝐿2+𝐶𝑁‖∇𝜃‖²_𝐿2‖∇𝑢‖^𝑝_{̇𝐹0𝑞,2𝑞/3}. (3.10) For 𝐼³₂, from the Hölder and Young inequalities, (2.12), (2.15), and Gagliardo-Nirenberg inequality, we have 𝐼³₂=_{𝑗>𝑁ℝ3}Δ̇_𝑗(∇𝑢)|∇𝜃|²⩽‖‖𝑑𝑥∇𝜃²_{𝐿3𝑗>𝑁}Δ̇_𝑗(∇𝑢)_𝐿3⩽𝐶‖∇𝜃‖²_{𝐿3𝑗>𝑁}2^(𝑗/2)Δ̇_𝑗(∇𝑢)_𝐿2,𝐶⩽‖∇𝜃‖_𝐿²∇²𝜃_{𝐿2𝑗>𝑁}2^−𝑗1/2_𝑗>𝑁2^2𝑗Δ̇_𝑗(∇𝑢)²₂^1/2⩽𝐶2^−(𝑁/2)‖∇𝜃‖_𝐿²∇²𝜃_𝐿2∇²𝑢₂⩽𝐶2^−(𝑁/2)‖∇𝜃‖_𝐿²∇²𝜃²_𝐿2+∇²𝑢²_𝐿2. (3.11) Plugging (3.8), (3.10), and (3.11) into (3.7) yields 𝐼₂⩽𝐶2^−(3/2)𝑁‖∇𝑢‖²₂+‖∇𝜃‖²₂^3/2+12∇²𝜃²₂+𝐶𝑁‖∇𝜃‖²_𝐿2‖∇𝑢‖^𝑝_{̇𝐹0𝑞,2𝑞/3}+𝐶2^−𝑁/2‖∇𝜃‖_𝐿²∇²𝜃²_𝐿2+∇²𝑢²_𝐿2. (3.12) Similarly, we also obtain the estimate 𝐼₁⩽𝐶2^−(3/2)𝑁‖∇𝑢‖²₂+‖∇𝜃‖²₂^3/2+12∇²𝑢²₂+𝐶𝑁‖∇𝑢‖²_𝐿2‖∇𝑢‖^𝑝_{̇𝐹0𝑞,2𝑞/3}+𝐶2^−𝑁/2‖∇𝑢‖_𝐿2∇²𝜃²_𝐿2+∇²𝑢²_𝐿2. (3.13) Putting (3.5), (3.12), and (3.13) into (3.4) yields 12𝑑‖‖𝑑𝑡∇𝑢,∇𝜃²_𝐿2+12∇²𝑢,∇²𝜃²_𝐿2⩽𝐶2^−𝑁‖(∇𝑢,∇𝜃)‖²₂^3/2+𝐶𝑁‖(∇𝑢,∇𝜃)‖²_𝐿2‖∇𝑢‖^𝑝_{̇𝐹0𝑞,2𝑞/3}+𝐶2^−𝑁‖(∇𝑢,∇𝜃)‖²_𝐿2^1/2∇²𝑢,∇²𝜃²_𝐿2. (3.14) Now we take 𝑁 in (3.14) such that 𝐶2^−𝑁‖(∇𝑢,∇𝜃)‖²_𝐿2⩽1,16 (3.15) that is,𝑁⩾𝐶log𝑒+‖(∇𝑢,∇𝜃)‖²_𝐿2log2+4. (3.16) Then (3.14) implies that 𝑑𝑑𝑡‖(∇𝑢,∇𝜃)‖²_𝐿2⩽𝐶+𝐶log𝑒+‖(∇𝑢,∇𝜃)‖²_𝐿2‖(∇𝑢,∇𝜃)‖²_𝐿2‖∇𝑢‖^𝑝_{̇𝐹0𝑞,2𝑞/3}. (3.17) Applying the Gronwall inequality twice, we have‖(∇𝑢,∇𝜃)‖²_𝐿2⩽𝐶expexp𝐶^𝑇₀‖∇𝑢‖^𝑝_{̇𝐹0𝑞,2𝑞/3}(𝑠)𝑑𝑠, (3.18) for all 𝑡∈(0,𝑇). This completes the proof of Theorem 1.1.

Proof of Corollary 1.2. In Theorem 1.1, taking 𝑝=1, and combining (2.12) with the classical Riesz transformation is bounded in 𝐵̇_∞,∞(ℝ³), we can prove it.