Blow-Up Criteria of Smooth Solutions for the Cahn-Hilliard-Boussinesq System with Zero Viscosity in a Bounded Domain

Abstract

We prove that a smooth solution of the 3D Cahn-Hilliard-Boussinesq system with zero viscosity in a bounded domain breaks down if a certain norm of vorticity blows up at the same time. Here, this norm is weaker than bmo-norm.

1. Introduction

Let Ω⊆ℝ³ be a bounded, simply connected domain with smooth boundary 𝜕Ω, and 𝑛 is the unit outward normal vector to 𝜕Ω. We consider the following Cahn-Hilliard-Boussinesq system with zero viscosity in Ω×(0,∞) [1]:𝜕_𝑡𝑢+(𝑢⋅∇)𝑢+∇𝜋=𝜇∇𝜙+𝜃𝑒₃,𝜕div𝑢=0,_𝑡𝜕𝜃+𝑢⋅∇𝜃=Δ𝜃,_𝑡𝜙+𝑢⋅∇𝜙=Δ𝜇,−Δ𝜙+𝑓^′𝜙=𝜇,𝑢⋅𝑛=0,𝜕𝜃=𝜕𝑛𝜕𝜙=𝜕𝑛𝜕𝜇())=𝑢𝜕𝑛=0on𝜕Ω×0,∞,𝑢,𝜃,𝜙(𝑥,0₀,𝜃₀,𝜙₀(𝑥),𝑥∈Ω, (1.1)(1.2)(1.3)(1.4)(1.5)(1.6)(1.7) where 𝑢, the fluid velocity field, 𝜃, the temperature, 𝜙, the order parameter, 𝜋, and the pressure are the unknowns. 𝑒₃∶=(0,0,1)^𝑡. 𝜇 is a chemical potential. 𝑓(𝜙)∶=(1/4)(𝜙²−1)² is the double well potential.

When 𝜃=𝜙≡0, (1.1) and (1.2) are Euler equations. Ogawa-Taniuchi [3] proved that a smooth solution breaks down if a certain norm of vorticity blows up at the same time. Here this norm is weaker than bmo-norm.

Before presenting our results, we introduce some function spaces and some notations.

Let 𝜂,𝜙_𝑗,𝑗=0,±1,±2,±3,… be the Littlewood-Paley dyadic decomposition of unity that satisfies𝜂∈𝐶^∞₀(𝐵(0,1)),𝜙∈𝐶^∞₀𝐵()10,2⧵𝐵0,2,𝜙_𝑗(𝜉)=𝜙2^−𝑗𝜉,𝜂(𝜉)+∑^∞_𝑗=0𝜙_𝑗(𝜉)=1, (1.8) for any 𝜉∈ℝ³, where 𝐵(𝑥,𝑟) denotes the ball centered at 𝑥 of radius 𝑟.

We first recall the space of Besov type introduced by Vishik [2].

Definition 1.1 (see [2]). Let Θ(𝛼)(≥1) be a nondecreasing function on [1,∞). 𝑉_Θ∶={𝑓∈𝒮^′;‖𝑓‖_𝑉Θ<∞} is introduced by the norm 𝑓_𝑉Θ∶=sup_𝑁=1,2,…𝜂𝑓^∨_𝐿∞+∑^𝑁_𝑗=0𝜙_𝑗𝑓^∨_𝐿∞Θ(𝑁), (1.9) where 𝑓 and 𝑓̌ denote the Fourier and inverse Fourier transforms.

We note that𝑓_𝑉Θ≤𝐶𝑓_{̇𝐵0∞,∞}≤𝐶𝑓_bmo≤𝐶𝑓_𝐿∞,ifΘ(𝑁)≥𝑁. (1.10)

Now let us introduce the space of bmo type in [3].

Definition 1.2. Let 𝛽(𝑟) be a positive function on (0,1], and Ω⊂ℝ³ is a domain with 𝜕Ω∈𝐶^∞.(1)bmo_𝛽(ℝ³) is the space defined as a set for an 𝐿¹_loc(ℝ³) function 𝑓 such that 𝑓_{bmo𝛽(ℝ3)}∶=sup_{0<𝑟<1,𝑥∈ℝ3}1|𝐵(𝑥,𝑟)|𝛽(𝑟)_{𝐵()𝑥,𝑟}𝑓𝑦−𝑓_{𝐵(𝑥,𝑟)}𝑑𝑦+sup_𝑥∈ℝ31|𝐵(𝑥,1)|_𝐵()𝑥,1𝑓𝑦𝑑𝑦<∞, (1.11) where 𝑓_𝐵∶=(1/|𝐵|)∫_𝐵𝑓(𝑦)𝑑𝑦.(2)On Ω⊂ℝ³ we define bmo_𝛽 as restrictions of the above space bmo_𝛽(ℝ³):bmo_𝛽(Ω)∶=𝑓∣_Ω;𝑓∈bmo_𝛽ℝ³, (1.12) where 𝑓|_Ω is the restriction of 𝑓 on Ω. The norm of this space is defined by 𝑓_bmo𝛽(Ω)∶=inf𝑓_{bmo𝛽(ℝ3)};𝑓∈bmo_𝛽ℝ³with𝑓=𝑓inΩ. (1.13)

In particular if 𝛽(𝑟)=1, we write bmo_𝛽(ℝ³)=bmo(ℝ³) and bmo_𝛽(Ω)=bmo(Ω). Obviously, bmo⊂bmo_𝛽 if 𝛽≥1.

Definition 1.3. Let Θ(𝛼)(≥1) be a nondecreasing function on [1,∞)𝑌_Θ(Ω)∶=𝑓∈𝐿¹(Ω);𝑓_𝑌Θ(Ω)<∞, (1.14) where 𝑓_𝑌Θ(Ω)∶=sup_𝑝≥1𝑓_𝐿𝑝Θ𝑝,𝑀_Θ(Ω)∶=𝑓∈𝐿¹(Ω);𝑓_𝑀Θ(Ω)<∞, (1.15) where 𝑓_𝑀Θ(Ω)∶=sup_𝑝≥11Θ𝑝sup_{0<𝑟<1,𝑥∈ℝ3}𝑟^−3+3/𝑝_{𝐵(𝑥,𝑟)∩Ω}𝑓𝑦𝑑𝑦. (1.16)

We note that these spaces have the following relations: 𝑓_𝑀Θ(Ω)≤𝐶𝑓_𝑌Θ(Ω)≤𝐶𝑓_bmo(Ω). (1.17)

From now on we impose the following assumptions.

Assumption 1.4. Let 𝛽(𝑟)∶=Θ(log(𝑒+1/𝑟))/log(𝑒+1/𝑟).(H1)Θ(𝛼) is a positive and nondecreasing function on [0,∞) satisfying^+∞𝑑𝛼Θ(𝛼)=∞,Θ(𝛼)≥𝛼. (1.18)(H2) For all 𝑠≥1 there exists 𝐶(𝑠) such thatΘ(𝑠𝛼)≤𝐶(𝑠)Θ(𝛼),∀𝛼≥1. (1.19)(H3)𝛽(𝑟) is a nonincreasing function on (0,1].
Then Ogawa-Taniuchi [3] proved the following blowup criterion: ^𝑇₀‖𝜔(𝑡)‖_bmo𝛽(Ω)+‖𝜔(𝑡)‖_{𝑀Θ(Ω𝜖)}𝑑𝑡=∞, (1.20) where 𝜔∶=curl𝑢 and for all 𝜖>0 and Ω_𝜖∶={𝑥∈Ω;dist(𝑥,𝜕Ω)<𝜖} or ^𝑇₀‖𝜔(𝑡)‖_{bmo𝛽(Ω3𝜖)}+‖𝜔(𝑡)‖_{𝑀Θ(Ω3𝜖)}+𝜌𝜔(𝑡)_𝑉Θ𝑑𝑡=∞, (1.21) for all 0<𝜖<𝜖₀ and all 𝜌∈𝐶^∞(ℝ³) with 𝜌≡1 inΩ⧵Ω_𝜖 and 𝜌≡0 in ℝ³⧵Ω. 𝜖₀ is a small positive constant depending only on Ω.
Since 𝛽(𝑟)≥1, we see 𝑓_bmo𝛽(Ω)≤𝑓_bmo(Ω). (1.22) By this inequality and (1.17), (1.20) implies ^𝑇₀‖𝜔(𝑡)‖_bmo(Ω)𝑑𝑡=∞. (1.23)
The aim of this paper is to prove a similar result for the problem (1.1)–(1.7). It is easy to show that the problem (1.1)–(1.7) has a unique local smooth solution. Thus, we omit the details here. However, the global regularity is still open, which this paper aims to study. We will prove that.

Theorem 1.5. Let 𝑢₀∈𝐻³,𝜃₀∈𝐻²,𝜙₀∈𝐻⁴,div𝑢₀=0 in Ω, 𝑢₀⋅𝑛=𝜕𝜃₀/𝜕𝑛=𝜕𝜙₀/𝜕𝑛=𝜕𝜇₀/𝜕𝑛=0 on 𝜕Ω. Suppose that (𝑢,𝜃,𝜙) is a local smooth solution to the problem (1.1)–(1.7) on [0,𝑇). If 𝑇 is maximal, then (1.20) and (1.21) hold true.

In Section 2, we will give some preliminaries. Section 3 is devoted to the proof of Theorem 1.5.

2. Preliminaries

Lemma 2.1 (see [4]). For any 𝑢∈𝑊^𝑠,𝑝 with div𝑢=0 in Ω and 𝑢⋅𝑛=0 on 𝜕Ω, there holds ‖𝑢‖_{𝑊𝑠,𝑝}≤𝐶‖𝑢‖_𝐿𝑝+‖curl𝑢‖_{𝑊𝑠−1,𝑝}, (2.1) for any 𝑠≥1 and 𝑝∈(1,∞).

Lemma 2.2 (see [5]). Let 𝑠≥1.(1) If 𝑓,𝑔∈𝐻^𝑠(Ω)∩𝐶(Ω), then 𝑓𝑔_𝐻𝑠(Ω)≤𝐶𝑓_𝐻𝑠(Ω)𝑔_𝐿∞(Ω)+𝑓_𝐿∞(Ω)𝑔_𝐻𝑠(Ω). (2.2)(2)If 𝑓∈𝐻^𝑠(Ω)∩𝐶¹(Ω) and 𝑔∈𝐻^𝑠−1(Ω)∩𝐶(Ω), then for |𝛼|≤𝑠, 𝐷^𝛼𝑓𝑔−𝑓𝐷^𝛼𝑔_𝐿2(Ω)≤𝐶𝑓_𝐻𝑠(Ω)𝑔_𝐿∞(Ω)+𝑓_{𝑊1,∞(Ω)}𝑔_{𝐻𝑠−1(Ω)}. (2.3)

Lemma 2.3 (see [3]). For all 𝜖>0, there holds ‖∇𝑢‖_𝐿∞(Ω)≤𝐶1+‖𝑢‖_𝐿2(Ω)+‖curl𝑢‖_bmo𝛽(Ω)+‖curl𝑢‖_{𝑀Θ(Ω𝜖)}×Θlog𝑒+‖𝑢‖_𝐻3(Ω), (2.4) for all 𝑢∈𝐻³(Ω) with div𝑢=0 in Ω and 𝑢⋅𝑛=0 on 𝜕Ω.

Lemma 2.4 (see [3]). There exists a constant 𝜖₀ depending only on Ω such that the following holds.
For all 0<𝜖<𝜖₀, and for all 𝜌∈𝐶^∞(ℝ³) with 𝜌≡1 in Ω⧵Ω_𝜖 and 𝜌≡0 in ℝ³⧵Ω, there exists constant 𝐶 depending only on 𝜖,𝜌,Ω and Θ such that ‖∇𝑢‖_𝐿∞(Ω)≤𝐶1+‖𝑢‖_𝐿2(Ω)+‖curl𝑢‖_{bmo𝛽(Ω3𝜖)}+‖curl𝑢‖_{𝑀Θ(Ω3𝜖)}+𝜌curl𝑢_𝑉ΘΘlog𝑒+‖𝑢‖_𝐻3(Ω), (2.5) for all 𝑢∈𝐻³(Ω) with div𝑢=0 in Ω and 𝑢⋅𝑛=0 on 𝜕Ω.

Lemma 2.5 (see [6]). Let 𝜓 be nonnegative function on (0,𝑇) with ∫^𝑇₀𝜓(𝑡)𝑑𝑡<∞, let Θ(𝛼) be a positive and nondecreasing for 𝛼≥1 and ∫^+∞(𝑑𝛼/Θ(𝛼))=∞. Assume that 𝑣∈𝐶([0,𝑇)) and 0≤𝑣(𝑡)≤𝑣(0)+^𝑡₀𝜓(𝑠)Θ(𝑣(𝑠))𝑑𝑠∀0≤𝑡<𝑇. (2.6) Then, sup_{0≤𝑡≤𝑇}𝑣(𝑡)<∞.

3. Proof of Theorem 1.5

In this section, all the integrations with respect to spacial variable are on the domain 𝜔 (we omit it for simplicity).

Since the proof of (1.21) is similar to that of (1.20), we only need to prove (1.20). By the standard argument of continuation of local solutions, it suffices to prove that if^𝑇₀‖𝜔(𝑡)‖_bmo𝛽(Ω)+‖𝜔(𝑡)‖_{𝑀Θ(Ω𝜖)}𝑑𝑡<∞forsome𝜖>0, (3.1) then𝑢∈𝐿^∞0,𝑇;𝐻³,𝜃∈𝐿^∞0,𝑇;𝐻²∩𝐿²0,𝑇;𝐻³,𝜙∈𝐿^∞0,𝑇;𝐻⁴. (3.2)

First, by the maximum principle, it follows from (1.2) and (1.3) that‖𝜃‖_{𝐿∞(0,𝑇;𝐿∞)}≤𝐶. (3.3)

Testing (1.3) by 𝜃, using (1.2), we see that12𝑑𝜃𝑑𝑡²𝑑𝑥+|∇𝜃|²𝑑𝑥=0, (3.4) whence‖𝜃‖_{𝐿2(0,𝑇;𝐻1)}≤𝐶. (3.5)

Testing (1.4) by 𝜙, using (1.2) and (1.6), we find that12𝑑𝜙𝑑𝑡²𝑑𝑥+Δ𝜙²𝑑𝑥=𝑓^′𝜙Δ𝜙𝑑𝑥=𝜙³−𝜙Δ𝜙𝑑𝑥=−3𝜙²∇𝜙²𝜙𝑑𝑥−𝜙Δ𝜙𝑑𝑥≤−𝜙Δ𝜙𝑑𝑥≤_𝐿2Δ𝜙_𝐿2≤12Δ𝜙²_𝐿2+12𝜙²_𝐿2, (3.6) which gives𝜙_{𝐿∞(0,𝑇;𝐿2)}+𝜙_{𝐿2(0,𝑇;𝐻2)}≤𝐶. (3.7)

Testing (1.1) and (1.4) by 𝑢 and 𝜇, respectively, using (1.2), (1.5), (1.6), and (3.3), we infer that 𝑑1𝑑𝑡2∇𝜙²𝜙+1+𝑓2𝑢²𝑑𝑥+∇𝜇²𝑑𝑥=𝜃𝑒₃𝑢𝑑𝑥≤‖𝜃‖_𝐿2‖𝑢‖_𝐿2≤𝐶‖𝑢‖_𝐿2, (3.8) which yields𝜙_{𝐿∞(0,𝑇;𝐻1)}‖𝑢‖≤𝐶,_{𝐿∞(0,𝑇;𝐿2)}≤𝐶,∇𝜇_{𝐿2(0,𝑇;𝐿2)}≤𝐶. (3.9)(3.10)(3.11)

In the following calculations, we will use the following Gagliardo-Nirenberg inequality:𝜙²_𝐿∞≤𝐶𝜙_𝐻1𝜙_𝐻2. (3.12)

It follows from (1.5), (3.11), (3.7), (3.9) and (3.12) that^𝑇₀∇Δ𝜙²=𝑑𝑥𝑑𝑡_𝑇0∇𝑓^′𝜙−𝜇²𝑑𝑥𝑑𝑡≤𝐶^𝑇₀∇𝜇²𝑑𝑥𝑑𝑡+𝐶^𝑇₀∇𝑓^′𝜙²𝑑𝑥𝑑𝑡≤𝐶+𝐶^𝑇₀∇𝜙³−𝜙²𝑑𝑥𝑑𝑡≤𝐶+𝐶^𝑇₀𝜙²∇𝜙²𝑑𝑥𝑑𝑡≤𝐶+𝐶∇𝜙²_{𝐿∞0,𝑇;𝐿2}^𝑇₀𝜙⁴_𝐿∞𝑑𝑡≤𝐶+𝐶^𝑇₀𝜙⁴_𝐿∞𝑑𝑡≤𝐶+𝐶^𝑇₀𝜙²_𝐻1𝜙²_𝐻2𝜙𝑑𝑡≤𝐶+𝐶²_{𝐿∞0,𝑇;𝐻1}^𝑇₀𝜙²_𝐻2𝑑𝑡≤𝐶, (3.13) which implies ∇𝜙_{𝐿2(0,𝑇;𝐿∞)}𝜙≤𝐶,_{𝐿4(0,𝑇;𝐿∞)}≤𝐶. (3.14)

Testing (1.4) by Δ²𝜙, using (3.9), (3.10), and (3.14), we deduce that12𝑑𝑑𝑡Δ𝜙²𝑑𝑥+Δ²𝜙²𝑑𝑥=Δ𝑓^′𝜙⋅Δ²𝜙𝑑𝑥−(𝑢⋅∇)𝜙⋅Δ²≤𝜙𝑑𝑥Δ𝑓^′𝜙_𝐿2+‖𝑢‖_𝐿2∇𝜙_𝐿∞Δ²𝜙_𝐿2≤𝐶𝜙²_𝐿∞Δ𝜙_𝐿2+𝜙_𝐿∞∇𝜙_𝐿∞∇𝜙_𝐿2+∇𝜙_𝐿∞Δ²𝜙_𝐿2≤12Δ²𝜙²_𝐿2+𝐶𝜙⁴_𝐿∞Δ𝜙²_𝐿2+𝐶𝜙²_𝐻2∇𝜙²_𝐿∞+𝐶∇𝜙²_𝐿∞, (3.15) which leads to𝜙_{𝐿∞(0,𝑇;𝐻2)}+𝜙_{𝐿2(0,𝑇;𝐻4)}≤𝐶. (3.16) Testing (1.3) by −Δ𝜃, using (1.2) and (1.6), we infer that12𝑑||𝑑𝑡∇𝜃²𝑑𝑥+(Δ𝜃)²()=𝑑𝑥=𝑢⋅∇𝜃⋅Δ𝜃𝑑𝑥_𝑖𝜕_𝑖(𝑢⋅∇𝜃)𝜕_𝑖𝜃𝑑𝑥−_𝑖𝜕_𝑖(𝑢⋅∇𝜃)⋅𝜕_𝑖=))𝜃𝑑𝑥div((𝑢⋅∇𝜃∇𝜃𝑑𝑥−_𝑖𝜕_𝑖𝑢⋅∇𝜃⋅𝜕_𝑖1𝜃𝑑𝑥−2_𝑖𝑢⋅∇𝜕_𝑖𝜃²𝑑𝑥=−_𝑖𝜕_𝑖𝑢⋅∇𝜃⋅𝜕_𝑖‖‖𝜃𝑑𝑥≤𝐶∇𝑢_𝐿∞‖∇𝜃‖²_𝐿2. (3.17) Equations (1.3) and (1.6) can be rewritten asΔ𝜃=𝑔∶=𝜕_𝑡𝜃+𝑢⋅∇𝜃asΩ×(0,∞),𝜕𝜃().𝜕𝑛=0,on𝜕Ω×0,∞ (3.18)

By the classical regularity theory of elliptic equation, using (3.10), we get‖𝜃‖_𝐻3≤𝐶𝑔_𝐻1≤𝐶𝜕_𝑡𝜃_𝐻1+𝐶‖𝑢⋅∇𝜃‖_𝐻1≤𝐶𝜕_𝑡𝜃_𝐻1+𝐶‖𝑢‖_𝐿2‖∇𝜃‖_𝐿∞+𝐶‖𝑢‖_𝐿6‖Δ𝜃‖_𝐿3+𝐶‖∇𝑢‖_𝐿∞‖∇𝜃‖_𝐿2≤𝐶𝜕_𝑡𝜃_𝐻1+𝐶‖∇𝜃‖_𝐿∞+𝐶‖𝑢‖_𝐿6‖Δ𝜃‖_𝐿3+𝐶‖∇𝑢‖_𝐿∞‖∇𝜃‖_𝐿2. (3.19)

Now using the following Gagliardo-Nirenberg inequalities:‖∇𝜃‖_𝐿∞≤𝐶‖𝜃‖_𝐿^1/3_∞‖𝜃‖_𝐻^2/3₃,‖Δ𝜃‖_𝐿3≤𝐶‖𝜃‖_𝐿^1/3_∞‖𝜃‖_𝐻^2/3₃,‖𝑢‖³_𝐿6≤𝐶‖𝑢‖²_𝐿2‖𝑢‖_𝐻3, (3.20) we obtain‖𝜃‖_𝐻3≤𝐶𝜕_𝑡𝜃_𝐻1+𝐶+𝐶‖𝑢‖_𝐻3+𝐶‖∇𝑢‖_𝐿∞‖∇𝜃‖_𝐿2. (3.21)

Taking curlto (1.1), using (1.2) and (1.5), we have𝜕_𝑡𝜔+𝑢⋅∇𝜔=𝜔⋅∇𝑢−curlΔ𝜙∇𝜙+curl𝜃𝑒₃. (3.22)

Taking Δ to (3.22), testing by Δ𝜔, using (1.2) and (1.6), we derive12𝑑||𝑑𝑡Δ𝜔²𝑑𝑥=−(Δ(𝑢⋅∇𝜔)−𝑢∇Δ𝜔)Δ𝜔𝑑𝑥+Δ(𝜔⋅∇𝑢)⋅Δ𝜔𝑑𝑥−ΔcurlΔ𝜙⋅∇𝜙⋅Δ𝜔𝑑𝑥+Δcurl𝜃𝑒₃≤‖Δ()‖⋅Δ𝜔𝑑𝑥𝑢⋅∇𝜔−𝑢∇Δ𝜔_𝐿2+‖Δ(𝜔⋅∇𝑢)‖_𝐿2+ΔcurlΔ𝜙∇𝜙_𝐿2+Δcurl𝜃𝑒_3𝐿2‖Δ𝜔‖_𝐿2∶=𝐼₁+𝐼₂+𝐼₃+𝐼₄‖Δ𝜔‖_𝐿2. (3.23)

Using (1.2), (1.6), Lemma 2.2, 𝐼₁ and 𝐼₂ can be bounded as follows: 𝐼₁=_𝑖Δ𝜕_𝑖𝑢_𝑖𝜔−𝑢_𝑖𝜕_𝑖Δ𝜔_𝐿2≤𝐶‖∇𝑢‖_𝐿∞‖Δ𝜔‖_𝐿2+𝐶‖𝜔‖_𝐿∞∇³𝑢_𝐿2≤𝐶‖∇𝑢‖_𝐿∞‖𝑢‖_𝐻3,𝐼₂≤𝐶‖𝜔‖_𝐿∞‖𝑢‖_𝐻3+𝐶‖∇𝑢‖_𝐿∞‖𝜔‖_𝐻2≤𝐶‖∇𝑢‖_𝐿∞‖𝑢‖_𝐻3. (3.24)

Using Lemma 2.2, 𝐼₃ can be bonded as follows𝐼₃=_𝑗Δcurl𝜕_𝑗𝜕_𝑗𝜙∇𝜙_𝐿2≤𝐶∇𝜙_𝐿∞∇𝜙_𝐻4≤𝐶∇𝜙_𝐿∞𝜙_𝐻5. (3.25)

Inserting the above estimates into (3.23), we obtain12𝑑||𝑑𝑡Δ𝜔²𝑑𝑥≤𝐶‖∇𝑢‖_𝐿∞‖𝑢‖_𝐻3+∇𝜙_𝐿∞𝜙_𝐻5+‖𝜃‖_𝐻3‖Δ𝜔‖_𝐿2≤𝐶‖∇𝑢‖_𝐿∞‖𝑢‖³_𝐻3+𝐶∇𝜙²_𝐿∞‖Δ𝜔‖²_𝐿2+𝐶‖Δ𝜔‖²_𝐿2+𝛿𝜙²_𝐻5+𝛿‖𝜃‖²_𝐻3, (3.26) for any 0<𝛿<1.

Testing (1.1) by 𝜕_𝑡𝑢, using (1.2), (1.6), (3.3), (3.10), and (3.16), and noting that𝜇⋅∇𝜙⋅𝜕_𝑡𝑢𝑑𝑥=−Δ𝜙∇𝜙⋅𝜕_𝑡𝑢𝑑𝑥, (3.27) we reach𝜕_𝑡𝑢_𝐿2≤‖𝜃‖_𝐿2+‖𝑢⋅∇𝑢‖_𝐿2+Δ𝜙∇𝜙_𝐿2≤𝐶+𝐶‖𝑢‖_𝐿6‖∇𝑢‖_𝐿3+𝐶Δ𝜙_𝐿2∇𝜙_𝐿∞≤𝐶+𝐶‖𝑢‖_𝐿^2/3₂‖𝑢‖_𝐻^1/3₃⋅‖𝑢‖_𝐿^1/2₂‖𝑢‖_𝐻^1/2₃+𝐶∇𝜙_𝐿∞≤𝐶+𝐶‖𝑢‖_𝐻^5/6₃+𝐶∇𝜙_𝐿∞. (3.28) Here, we have used the Gagliardo-Nirenberg inequality:‖∇𝑢‖²_𝐿3≤𝐶‖𝑢‖_𝐿2‖𝑢‖_𝐻3. (3.29)

Taking 𝜕_𝑡 to (1.3), we see that𝜕²_𝑡𝜃+𝑢⋅∇𝜕_𝑡𝜃−Δ𝜕_𝑡𝜃=−𝜕_𝑡𝑢⋅∇𝜃. (3.30)

Testing the above equation by 𝜕_𝑡𝜃, using (1.2), (1.6) and (3.3), we have12𝑑𝜕𝑑𝑡^𝑡𝜃²𝑑𝑥+∇𝜕_𝑡𝜃²𝑑𝑥=−𝜕_𝑡𝑢⋅∇𝜃⋅𝜕_𝑡𝜃𝑑𝑥=𝜕_𝑡𝑢⋅𝜃∇𝜕_𝑡≤‖𝜃‖𝜃𝑑𝑥_𝐿∞𝜕_𝑡𝑢_𝐿2∇𝜕_𝑡𝜃_𝐿2≤𝐶𝜕_𝑡𝑢_𝐿2∇𝜕_𝑡𝜃_𝐿2≤12∇𝜕_𝑡𝜃²_𝐿2+𝐶𝜕_𝑡𝑢²_𝐿2, (3.31) whence𝑑𝜕𝑑𝑡^𝑡𝜃²𝑑𝑥+∇𝜕_𝑡𝜃²𝑑𝑥≤𝐶+𝐶‖𝑢‖²_𝐻3+𝐶∇𝜙²_𝐿∞. (3.32)

By the classical regularity theory of elliptic equation, it follows from (1.4), (1.5), (1.6), (3.16), and (3.10) that𝜙_𝐻5≤𝐶𝑓^′𝜙−𝜇_𝐻3≤𝐶𝑓^′𝜙_𝐻3+𝐶𝜇_𝐻3≤𝐶𝑓^′𝜙_𝐻3+𝐶𝜕_𝑡𝜙+𝑢⋅∇𝜙_𝐻1≤𝐶+𝐶𝜙_𝐻3+𝐶𝜕_𝑡𝜙_𝐻1+𝐶‖∇𝑢‖_𝐿∞∇𝜙_𝐿2+𝐶‖𝑢‖_𝐿2Δ𝜙_𝐿∞≤𝐶+𝐶𝜙_𝐻4+𝐶𝜕_𝑡𝜙_𝐻1+𝐶‖𝑢‖_𝐻3≤𝐶+𝐶𝜙_𝐻^1/3₂𝜙_𝐻^2/3₅+𝐶𝜕_𝑡𝜙_𝐻1+𝐶‖𝑢‖_𝐻3, (3.33) which implies𝜙_𝐻5≤𝐶+𝐶𝜕_𝑡𝜙_𝐻1+𝐶‖𝑢‖_𝐻3. (3.34) Here, we have used the Gagliardo-Nirenberg inequality:𝜙_𝐻4≤𝐶𝜙_𝐻^1/3₂𝜙_𝐻^2/3₅. (3.35)

Taking 𝜕_𝑡 to (1.4) and (1.5), we have𝜕²_𝑡𝜙+𝑢⋅∇𝜕_𝑡𝜙=Δ𝜕_𝑡𝜇−𝜕_𝑡𝑢⋅∇𝜙,−Δ𝜕_𝑡𝜙+𝜕_𝑡𝜙³−𝜙=𝜕_𝑡𝜇. (3.36)(3.37)

Testing (3.36) by 𝜕_𝑡𝜙, using (3.37), (1.4), (1.6), (3.16), and (3.28), we arrive at12𝑑𝜕𝑑𝑡^𝑡𝜙²𝑑𝑥+Δ𝜕_𝑡𝜙²𝑑𝑥=𝜕_𝑡𝜙³−𝜙⋅Δ𝜕_𝑡𝜙𝑑𝑥−𝜕_𝑡𝑢⋅∇𝜙⋅𝜕_𝑡=𝜙𝑑𝑥3𝜙²−1𝜕_𝑡𝜙⋅Δ𝜕_𝑡𝜙𝑑𝑥+𝜕_𝑡𝑢⋅𝜙∇𝜕_𝑡≤𝜙𝑑𝑥3𝜙²−1_𝐿∞𝜕_𝑡𝜙_𝐿2Δ𝜕_𝑡𝜙_𝐿2+𝜕_𝑡𝑢_𝐿2𝜙_𝐿∞∇𝜕_𝑡𝜙_𝐿2≤𝐶𝜕_𝑡𝜙_𝐿2Δ𝜕_𝑡𝜙_𝐿2+𝐶𝜕_𝑡𝑢_𝐿2∇𝜕_𝑡𝜙_𝐿2≤𝐶𝜕_𝑡𝜙_𝐿2Δ𝜕_𝑡𝜙_𝐿2+𝐶1+‖𝑢‖_𝐻^5/6₃+∇𝜙_𝐿∞∇𝜕_𝑡𝜙_𝐿2, (3.38) whence𝑑𝜕𝑑𝑡^𝑡𝜙²𝑑𝑥+Δ𝜕_𝑡𝜙²𝑑𝑥≤𝐶𝜕_𝑡𝜙²_𝐿2+𝐶+𝐶‖𝑢‖²_𝐻3+𝐶∇𝜙²_𝐿∞. (3.39) Here, we have used∇𝜕_𝑡𝜙²_𝐿2≤𝐶𝜕_𝑡𝜙_𝐿2Δ𝜕_𝑡𝜙_𝐿2. (3.40)

Combining (3.17), (3.21), (3.26), (3.32), (3.34), and (3.39), using Lemmas 2.1, 2.3, and 2.5, we conclude that‖𝑢‖_{𝐿∞(0,𝑇;𝐻3)}≤𝐶, (3.41)‖𝜃‖_{𝐿∞(0,𝑇;𝐻1)}+𝜕_𝑡𝜃_{𝐿∞(0,𝑇;𝐿2)}+𝜕_𝑡𝜃_{𝐿2(0,𝑇;𝐻1)}≤𝐶, (3.42)𝜕_𝑡𝜙_{𝐿∞(0,𝑇;𝐿2)}≤𝐶. (3.43) It follows from (1.3), (3.41), and (3.42) that𝜃∈𝐿^∞0,𝑇;𝐻²∩𝐿²0,𝑇;𝐻³. (3.44) It follows from (1.4), (1.5), (3.41), and (3.43) that𝜙_{𝐿∞(0,𝑇;𝐻4)}≤𝐶. (3.45)

This completes the proof.

Acknowledgment

This paper is partially supported by the Zhejiang Innovation Project (Grant no. T200905), the ZJNSF (Grant no. R6090109), and the NSFC (Grant no. 11171154).