Regularity Criteria for a Coupled Navier-Stokes and Q-Tensor System

Abstract

We study a system describing the evolution of a nematic liquid crystal flow. The system couples a forced Navier-Stokes system describing the flow with a parabolic-type system describing the evolution of the nematic crystal director fields (Q-tensors). We prove some regularity criteria for the local strong solutions. However, we do not provide estimates on the rates of increase of high norms.

1. Introduction

We consider the following coupled Navier-Stokes and 𝑄-tensor system [1–4]: 𝜕_𝑡𝑄𝑄+𝑢⋅∇𝑄+𝑄Ω−Ω𝑄=Δ𝑄−𝑎𝑄+𝑏²−𝕀𝑑tr𝑄²−𝑐𝑄tr𝑄²,𝜕_𝑡()(),()=𝑄𝑢+𝑢⋅∇𝑢+∇𝜋=Δ𝑢−div∇𝑄⊙∇𝑄+div𝑄Δ𝑄−Δ𝑄𝑄div𝑢=0,𝑄,𝑢)(𝑥,0₀,𝑢₀inℝ^𝑑(𝑑=2,3). (1)(2)(3)(4) Here the unknowns 𝑢,𝜋, and 𝑄 denote the velocity field of the fluid, the pressure, and the order parameter, respectively. A 𝑄-tensor is a symmetric and traceless 𝑑×𝑑-matrix, Ω∶=(1/2)(∇𝑢−(∇𝑢)^𝑇),𝑎∈ℝ,𝑏>0 and 𝑐>0 are physical constants, 𝑑 is the space dimension, (∇𝑄⊙∇𝑄)_𝑖𝑗∶=tr(𝜕_𝑖𝑄𝜕_𝑗𝑄),tr𝑄∶=∑^𝑑_𝑖=1𝑄_𝑖𝑖, and thus tr(𝑄²)=∑^𝑑_𝑖,𝑗=1𝑄²_𝑖𝑗.

When 𝑄≡0, (2) and (3) are the well-known Navier-Stokes system, for which Kozono et al. [5] and Kozono and Shimada [6] proved the well-known regularity criteria 𝑢∈𝐿^2/(1−𝑠)0,𝑇;𝐵̇^−𝑠_∞,∞,0<𝑠<1,𝑢∈𝐿²0,𝑇;𝐵̇⁰_∞,∞,𝜔∶=curl𝑢∈𝐿¹0,𝑇;𝐵̇⁰_∞,∞, (5)(6)(7) where 𝐵̇^𝑠_𝑝,𝑞 denotes the homogeneous Besov spaces [7].

Very recently, Paicu and Zarnescu [8] proved the existence of global-in-time weak solutions in 3-dimensional space and of smooth solutions in 2-dimensional space. The aim of this paper is to study the regularity criteria.

If one formally takes 𝑄∶=𝑠₊(𝑛⊗𝑛−(1/3)𝕀), with 𝑠₊ a constant, then the equations reduce to the generally accepted equations of Leslie [9], which have been studied in [10–15]. We will prove the following.

Theorem 1. Let 𝑢₀∈𝐻¹,𝑄₀∈𝐻²,div𝑢₀=0 in ℝ^𝑑 with 𝑑=2,3. Let (𝑢,𝑄) be a unique strong solution in 𝐿^∞(0,𝑇₁;𝐻¹)∩𝐿²(0,𝑇₁;𝐻²)×𝐿^∞(0,𝑇₁;𝐻²)∩𝐿²(0,𝑇₁;𝐻³) with 0<𝑇₁<𝑇<∞.(i)If 𝑑=2,3 and 𝑢 satisfies one of the conditions (5), (6), or (7) and ∇𝑄 satisfies ∇𝑄∈𝐿^𝑝0,𝑇;𝐿^𝑞2𝑤𝑖𝑡ℎ𝑝+𝑑𝑞=1,𝑑<𝑞≤∞, (8) for some finite 𝑇<∞, then the solution (𝑢,𝑄) can be extended beyond 𝑇>0.(ii)If 𝑑=3 and 𝑢 satisfies ∇𝑢∈𝐿^𝑝0,𝑇;𝐿^𝑞2𝑤𝑖𝑡ℎ𝑝+3𝑞=2,2<𝑞≤3, (9) for some finite 𝑇<∞, then the solution (𝑢,𝑄) can be extended beyond 𝑇>0.

Remark 2. By the well-known inequality ‖curl𝑢‖_𝐿^𝑞≤‖∇𝑢‖_𝐿^𝑞≤𝐶‖curl𝑢‖_𝐿^𝑞, the condition (9) can be replaced by 𝜔∶=curl𝑢∈𝐿^𝑝0,𝑇;𝐿^𝑞2with𝑝+3𝑞3=2,2<𝑞≤3. (10)

It has been proved in [8] that the system (1)–(4) has a Lyapunov functional: 𝐸(𝑡)1∶=2𝑢²+12|∇𝑄|²+𝑎2tr𝑄²−𝑏3tr𝑄³+𝑐4tr²𝑄²𝑑𝑥, (11) which satisfies 𝑑||𝑑𝑡𝐸=−∇𝑢²−𝑄𝑑𝑥trΔ𝑄−𝑎𝑄−𝑏²−𝕀𝑑tr𝑄²−𝑐𝑄tr𝑄²²𝑑𝑥≤0, (12) from which we easily obtain [8] 𝑢∈𝐿^∞0,𝑇;𝐿²∩𝐿²0,𝑇;𝐻¹,𝑄∈𝐿^∞0,𝑇;𝐻¹∩𝐿²0,𝑇;𝐻². (13) When 𝑑=2, (13) give [8] 𝑢∈𝐿⁴0,𝑇;𝐿⁴⊂𝐿^2/(1−1/2)0,𝑇;𝐵̇^−1/2_∞,∞,∇𝑄∈𝐿⁴0,𝑇;𝐿⁴, (14) thus (5) and (8) hold true; this proves the existence of global-in-time strong solutions when 𝑑=2. In [8], this result was proven by complicated Littlewood-Paley theory, Bony’s paraproduct decomposition, and the logarithmic Sobolev inequality. The purpose of this paper is to make the argument in [8] much simpler. However, in [8], they obtained in addition the rate of increase of high norms.

Our proof uses an energy method and relies on a simple 𝐿^∞ estimate of 𝑄 and the following cancellation property:

Lemma 3 (see [8]). Let 𝑄^∼,𝑄∶ℝ^𝑑→ℝ^𝑑×𝑑 be symmetric matrix-valued functions and let Ω_𝛼𝛽∶=(1/2)(𝜕_𝛽𝑢_𝛼−𝜕_𝛼𝑢_𝛽)∈ℝ^𝑑×𝑑 be smooth and decaying and sufficiently fast at infinity (so that one can integrate by parts without boundary terms). Then trΩ𝑄^∼−𝑄^∼ΩΔ𝑄𝑑𝑥−_{𝛼,𝛽,𝛾}𝜕𝑄_𝛽^∼_𝛼𝛾Δ𝑄_𝛾𝛽−Δ𝑄^∼_𝛼𝛾𝑄_𝛾𝛽𝑢_𝛼𝑑𝑥=0. (15)

2. Proof of Theorem 1

This section is devoted to the proof of Theorem 1. Since it is easy to prove that there are 𝑇₀>0 and a unique strong solution (𝑢,𝑄) to the problem (1)–(4) in (0,𝑇₀], we only need to prove a priori estimates.

First, we prove the following key estimate: ‖𝑄‖_𝐿^∞_(0,𝑇;𝐿^∞₎≤𝐶. (16) To prove (16), we multiply (1) by 2𝑝𝑄tr^𝑝−1(𝑄²) and take the trace to obtain 𝜕_𝑡+𝑢⋅∇tr^𝑝𝑄²=2𝑝Δ𝑄_𝛼𝛽𝑄_𝛼𝛽tr^𝑝−1𝑄²−2𝑝𝑎tr^𝑝𝑄²+2𝑝𝑏tr𝑄³tr^𝑝−1𝑄²−2𝑝𝑐tr^𝑝+1𝑄². (17)

Let us observe that for 𝑄, a traceless, symmetric, 3×3 matrix, we have tr𝑄³≤3𝜖8tr²𝑄²+1𝜖tr𝑄²,∀𝜖>0. (18)

Integrating over ℝ^𝑑, integrating by parts, and using (3), (18), and the assumption 𝑐>0, we obtain 𝑑𝑑𝑡tr^𝑝𝑄²𝑑𝑥≤−2𝑝∇𝑄_𝛼𝛽∇𝑄_𝛼𝛽tr^𝑝−1𝑄²𝜕𝑑𝑥−4𝑝𝑝−1_𝛾𝑄_𝛼𝛽𝑄_𝛼𝛽𝜕_𝛾𝑄_𝛿𝜆𝑄_𝛿𝜆tr^𝑝−2𝑄²𝑑𝑥+𝑐𝑝tr^𝑝𝑄²𝑑𝑥≤𝑐𝑝tr^𝑝𝑄²𝑑𝑥, (19) which gives tr^𝑝(𝑄²)𝑑𝑥^1/(2𝑝)≤𝑒^𝐶𝑇tr^𝑝(𝑄²₀)𝑑𝑥^1/(2𝑝) (20) with 𝐶 independent of 𝑝.

Thanks to a simple lemma in [16, Page 102], we take 𝑝→∞ in (20), this proves (16).

Taking Δ to (1), testing scalarly by Δ𝑄, and using (3); we find that 12𝑑||𝑑𝑡Δ𝑄²𝑑𝑥+|∇Δ𝑄|²=()+()𝑑𝑥ΔΩ𝑄−𝑄ΔΩ∶Δ𝑄𝑑𝑥ΩΔ𝑄−Δ𝑄Ω∶Δ𝑄𝑑𝑥+2_𝑖𝜕_𝑖Ω𝜕_𝑖𝑄−𝜕_𝑖𝑄𝜕_𝑖Ω−∶Δ𝑄𝑑𝑥Δ𝑢⋅∇𝑄∶Δ𝑄𝑑𝑥+2_𝑖,𝑗𝜕_𝑖𝑢_𝑗𝜕_𝑗𝜕_𝑖+Δ𝑄𝑄∶Δ𝑄𝑑𝑥−𝑎𝑄+𝑏²−𝕀𝑑tr𝑄²−𝑐𝑄tr𝑄²∶Δ𝑄𝑑𝑥=∶𝐼₁+𝐼₂+𝐼₃+𝐼₄+𝐼₅+𝐼₆. (21) Here “testing scalarly by Δ𝑄” means multiplying with respect to the Frobenius inner product of matrices, 𝐴∶𝐵=tr(𝐴𝐵) and integrating over ℝ^𝑑.

Testing (2) by −Δ𝑢 and using (3), we infer that 12𝑑||𝑑𝑡∇𝑢²𝑑𝑥+|Δ𝑢|²=()()−()𝑑𝑥𝑢⋅∇𝑢⋅Δ𝑢𝑑𝑥+div∇𝑄⊙∇𝑄Δ𝑢𝑑𝑥div𝑄Δ𝑄−Δ𝑄𝑄Δ𝑢𝑑𝑥=∶𝐽₁+𝐽₂+𝐽₃. (22)

Summing (21) and (22) and using the cancellation 𝐼₁+𝐽₃=0, due to Lemma 3, we have 12𝑑||𝑑𝑡Δ𝑄²+|∇𝑢|²𝑑𝑥+|∇Δ𝑄|²+|Δ𝑢|²𝑑𝑥=𝐼₂+𝐼₃+𝐼₄+𝐼₅+𝐼₆+𝐽₁+𝐽₂. (23)

(i) Let (8) hold true

Using the integration by parts, 𝐼₂ can be bounded as 𝐼₂=_𝑖𝜕_𝑖𝑄𝜕_𝑖(ΩΔ𝑄)−𝜕_𝑖Ω𝜕_𝑖𝑄Δ𝑄−Ω𝜕_𝑖𝑄𝜕_𝑖‖‖Δ𝑄𝑑𝑥≤𝐶∇𝑄_𝐿^𝑞‖Ω‖_𝐿^{2𝑞/(𝑞−2)}‖∇Δ𝑄‖_𝐿²+𝐶‖∇𝑄‖_𝐿^𝑞‖∇Ω‖_𝐿²‖Δ𝑄‖_𝐿^{2𝑞/(𝑞−2)}≤𝐶‖∇𝑄‖_𝐿^𝑞‖∇𝑢‖_𝐿^{1−𝑑/𝑞}₂‖Δ𝑢‖_𝐿^𝑑/𝑞₂‖∇Δ𝑄‖_𝐿²+𝐶‖∇𝑄‖_𝐿^𝑞‖Δ𝑢‖_𝐿²‖Δ𝑄‖_𝐿^{1−𝑑/𝑞}₂‖∇Δ𝑄‖_𝐿^𝑑/𝑞₂≤𝐶‖∇𝑄‖_𝐿^𝑞‖∇𝑢‖_𝐿²+‖Δ𝑄‖_𝐿^{21−𝑑/𝑞}×‖Δ𝑢‖_𝐿²+‖∇Δ𝑄‖_𝐿^21+𝑑/𝑞≤1‖‖32Δ𝑢²_𝐿2+1‖‖32∇Δ𝑄²_𝐿2+𝐶‖∇𝑄‖^𝑝_𝐿𝑞×‖∇𝑢‖²_𝐿2+‖Δ𝑄‖²_𝐿2. (24) Here 𝑝 and 𝑞 satisfy the relation (8), and we have used the Gagliardo-Nirenberg inequality ‖𝑤‖_𝐿^{2𝑞/(𝑞−2)}≤𝐶‖𝑤‖_𝐿^{1−𝑑/𝑞}₂‖∇𝑤‖_𝐿^𝑑/𝑞₂. (25)

Similarly, we get 𝐼₃,𝐼₄,𝐼₅,𝐽₂≤1‖‖32Δ𝑢²_𝐿2+1‖‖32∇Δ𝑄²_𝐿2+𝐶‖∇𝑄‖^𝑝_𝐿𝑞‖∇𝑢‖²_𝐿2+‖Δ𝑄‖²_𝐿2. (26)

By using (16), 𝐼₆ is simply bounded as 𝐼₆≤𝐶‖Δ𝑄‖²_𝐿2. (27) Here we treat the term 𝑏∇𝑄∇𝑄Δ𝑄𝑑𝑥≤𝐶‖∇𝑄‖²_𝐿4‖Δ𝑄‖_𝐿²≤𝐶‖𝑄‖_𝐿^∞‖Δ𝑄‖²_𝐿2≤𝐶‖Δ𝑄‖²_𝐿2 (28) by the Gagliardo-Nirenberg inequality ‖∇𝑄‖²_𝐿4≤𝐶‖𝑄‖_𝐿^∞‖Δ𝑄‖_𝐿². (29)

Inserting the above estimates into (23), we derive 12𝑑||𝑑𝑡Δ𝑄²+|∇𝑢|²1𝑑𝑥+2|∇Δ𝑄|²+|Δ𝑢|²𝑑𝑥≤𝐽₁+𝐶‖∇𝑄‖^𝑝_𝐿𝑞‖∇𝑢‖²_𝐿2+‖Δ𝑄‖²_𝐿2+𝐶‖Δ𝑄‖²_𝐿2. (30)

Now we estimate 𝐽₁ as follows.

(1) Let (5) hold true

We will use the following inequality [6]: ‖𝑢⋅∇𝑢‖_𝐿²≤𝐶‖𝑢‖_̇𝐵^−𝑠_∞,∞‖∇𝑢‖_̇𝐻^𝑠, (31) and the Gagliardo-Nirenberg inequality ‖∇𝑢‖_̇𝐻^𝑠≤𝐶‖∇𝑢‖_𝐿^1−𝑠₂‖Δ𝑢‖^𝑠_𝐿2. (32)

Using (31) and (32), we bound 𝐽₁ as follows: 𝐽₁≤‖𝑢⋅∇𝑢‖_𝐿²‖Δ𝑢‖_𝐿²≤𝐶‖𝑢‖_̇𝐵^−𝑠_∞,∞‖∇𝑢‖_𝐿^1−𝑠₂‖Δ𝑢‖^𝑠_𝐿2≤1‖‖16Δ𝑢²_𝐿2+𝐶‖𝑢‖_̇𝐵^2/(1−𝑠)_{−𝑠∞,∞}‖∇𝑢‖²_𝐿2. (33) Substituting the above estimates into (30), we reach ‖𝑢‖_𝐿^∞_(0,𝑇;𝐻¹₎+‖𝑢‖_𝐿²_(0,𝑇;𝐻²₎‖𝑄‖≤𝐶,_{𝐿∞(0,𝑇;𝐻2)}+‖𝑄‖_𝐿²_(0,𝑇;𝐻³₎≤𝐶. (34)

This completes the proof.

(2) Let (6) hold true

Using the following elegant inequality [17, 18]: ‖∇𝑢‖²_𝐿4≤𝐶‖𝑢‖_̇𝐵⁰_∞,∞‖Δ𝑢‖_𝐿², (35) we bound 𝐽₁ as follows: 𝐽₁=_𝑖,𝑗𝑢_𝑖𝜕_𝑖𝑢𝜕²_𝑗𝑢𝑑𝑥=−_𝑖,𝑗𝜕_𝑗𝑢_𝑖𝜕_𝑖𝑢𝜕_𝑗‖‖𝑢𝑑𝑥≤𝐶∇𝑢²_𝐿4‖∇𝑢‖_𝐿²≤𝐶‖𝑢‖_̇𝐵⁰_∞,∞‖∇𝑢‖_𝐿²‖Δ𝑢‖_𝐿²≤1‖‖16Δ𝑢²_𝐿2+𝐶‖𝑢‖²_{̇𝐵0∞,∞}‖∇𝑢‖²_𝐿2. (36)

Substituting the above estimates into (30), we have (34).

This completes the proof.

(3) Let (7) hold true

Let {𝜙_𝑗}_𝑗∈ℤ be the Littlewood-Paley dyadic decomposition of unity that satisfies 𝜙∈𝐶^∞₀(𝐵₂⧵𝐵_1/2),𝜙_𝑗(𝜉)=𝜙(2^−𝑗𝜉), and ∑_𝑗∈ℤ𝜙_𝑗(𝜉)=1 for any 𝜉≠0, where is the Fourier transform and 𝐵_𝑟 is the ball with radius 𝑟 centered at the origin.

We decompose 𝜕_𝑗𝑢 as follows: 𝜕_𝑗𝑢=^∞_𝑘=−∞𝜙_𝑘∗𝜕_𝑗𝑢=_𝑘<−𝑁𝜙_𝑘∗𝜕_𝑗𝑢+^𝑁_𝑘=−𝑁𝜙_𝑘∗𝜕_𝑗𝑢+_𝑘>𝑁𝜙_𝑘∗𝜕_𝑗𝑢, (37) where 𝑁 is a positive integer to be chosen later. Plugging this decomposition into 𝐽₁, we derive 𝐽₁=−_𝑖,𝑗𝜕_𝑗𝑢_𝑖𝜕_𝑖𝑢_𝑘<−𝑁𝜙_𝑘∗𝜕_𝑗𝑢+^𝑁_𝑘=−𝑁𝜙_𝑘∗𝜕_𝑗𝑢+_𝑘>𝑁𝜙_𝑘∗𝜕_𝑗𝑢𝑑𝑥=∶𝐽₁₁+𝐽₁₂+𝐽₁₃. (38)

Recalling Bernstein’s inequality, 𝜙_𝑘∗𝑓_𝐿𝑞≤𝐶2^{𝑑𝑘(1/𝑝−1/𝑞)}𝜙_𝑘∗𝑓_𝐿𝑝,1≤𝑝≤𝑞≤∞, (39) with 𝐶 being a positive constant independent of 𝑓 and 𝑘, we apply Hölder’s inequality to deduce that 𝐽₁₁≤_𝑖,𝑗𝜕_𝑗𝑢_𝑖𝐿2𝜕_𝑖𝑢_{𝐿2𝑘<−𝑁}𝜙_𝑘∗𝜕_𝑗𝑢_𝐿∞≤𝐶‖∇𝑢‖²_{𝐿2𝑘<−𝑁}2^(𝑑/2)𝑘𝜙_𝑘∗∇𝑢_𝐿2≤𝐶2^{−(𝑑/2)𝑁}‖∇𝑢‖³_𝐿2,𝐽₁₂≤_𝑖,𝑗𝜕_𝑗𝑢_𝑖𝐿2𝜕_𝑖𝑢_𝐿2^𝑁_𝑘=−𝑁𝜙_𝑘∗𝜕_𝑗𝑢_𝐿∞≤𝐶‖∇𝑢‖²_𝐿2⋅𝑁‖∇𝑢‖_̇𝐵⁰_∞,∞,𝐽₁₃≤_𝑖,𝑗𝜕_𝑗𝑢_𝑖𝐿3𝜕_𝑖𝑢_{𝐿3𝑘>𝑁}𝜙_𝑘∗𝜕_𝑗𝑢_𝐿3≤𝐶‖∇𝑢‖²_{𝐿3𝑘>𝑁}2^(𝑑/6)𝑘𝜙_𝑘∗∇𝑢_𝐿2≤𝐶‖∇𝑢‖²_{𝐿3𝑘>𝑁}2^{2(𝑑/6−1)𝑘1/2}×_𝑘>𝑁2^2𝑘𝜙_𝑘∗∇𝑢²_𝐿2^1/2≤𝐶‖∇𝑢‖²_𝐿32^{−(1−(𝑑/6)𝑁)}‖Δ𝑢‖_𝐿²≤𝐶2^{−(1−𝑑/6)𝑁}‖∇𝑢‖_𝐿^{2(1−𝑑/6)}₂‖Δ𝑢‖_𝐿^1+𝑑/3₂. (40)

Now we choose 𝑁 so that 𝐶2^{−(1−𝑑/6)𝑁}‖∇𝑢‖_𝐿^{2(1−𝑑/6)}₂≤1/16 and 2^{−(𝑑/2)𝑁}‖∇𝑢‖_𝐿²≤1/16 to conclude that 𝐽₁≤𝐶‖∇𝑢‖²_𝐿2+𝐶‖∇𝑢‖²_𝐿2‖∇𝑢‖_̇𝐵⁰_∞,∞log𝑒+‖∇𝑢‖_𝐿²+‖Δ𝑄‖_𝐿²+1‖‖16Δ𝑢²_𝐿2+𝐶. (41) Substituting the above estimates into (30), we arrive at (34).

This completes the proof of part (i).

(ii) Let (9) hold true

We still have (23).

𝐼₂ is simply bounded as 𝐼₂≤𝐶‖Ω‖_𝐿^𝑞‖Δ𝑄‖²_{𝐿2𝑞/(𝑞−1)}≤𝐶‖∇𝑢‖_𝐿^𝑞‖Δ𝑄‖_𝐿^{2(1−3/2𝑞)}₂∇³𝑄_𝐿^3/𝑞₂≤1∇32³𝑄²_𝐿2+𝐶‖∇𝑢‖^𝑝_𝐿𝑞‖Δ𝑄‖²_𝐿2. (42) Here we have used the Gagliardo-Nirenberg inequality ‖𝑤‖_𝐿^{2𝑞/(𝑞−1)}≤𝐶‖𝑤‖_𝐿^1−3/2𝑞₂‖∇𝑤‖_𝐿^3/2𝑞₂. (43)

𝐼₃ is simply bounded as 𝐼₃=−2_𝑖Ω𝜕_𝑖𝜕_𝑖𝑄Δ𝑄−𝜕²_𝑖𝑄ΩΔ𝑄−𝜕_𝑖𝑄Ω𝜕_𝑖‖Ω‖Δ𝑄𝑑𝑥≤𝐶_𝐿^𝑞‖Δ𝑄‖²_{𝐿2𝑞/(𝑞−1)}+‖∇𝑄‖_𝐿^{2𝑞/(𝑞−2)}∇³𝑄_𝐿2≤𝐶‖∇𝑢‖_𝐿^𝑞‖Δ𝑄‖_𝐿^{2(1−3/2𝑞)}₂∇³𝑄_𝐿^3/𝑞₂≤1∇32³𝑄²_𝐿2+𝐶‖∇𝑢‖^𝑝_𝐿𝑞‖Δ𝑄‖²_𝐿2. (44) Here we have used (43) and the Gagliardo-Nirenberg inequality ‖𝑤‖_𝐿^{2𝑞/(𝑞−2)}≤𝐶‖∇𝑤‖_𝐿^2−3/𝑞₂‖Δ𝑤‖_𝐿^3/𝑞−1₂. (45) Similarly, 𝐼₄,𝐼₅,𝐽₂, and 𝐽₁ can be bounded as follows: 𝐼₄+𝐼₅+𝐽₂≤𝐶‖∇𝑢‖_𝐿^𝑞‖Δ𝑄‖²_{𝐿2𝑞/(𝑞−1)}+‖∇𝑄‖_𝐿^{2𝑞/(𝑞−2)}∇³𝑄_𝐿2≤𝐶‖∇𝑢‖_𝐿^𝑞‖Δ𝑄‖_𝐿^2−3/𝑞₂∇³𝑄_𝐿^3/𝑞₂≤1∇32³𝑄²_𝐿2+𝐶‖∇𝑢‖^𝑝_𝐿𝑞‖Δ𝑄‖²_𝐿2,𝐽₁=−_𝑖,𝑗𝜕_𝑖𝑢_𝑗𝜕_𝑖𝑢𝜕_𝑗‖‖𝑢𝑑𝑥≤𝐶∇𝑢_𝐿^𝑞‖∇𝑢‖²_{𝐿2𝑞/(𝑞−1)}≤𝐶‖∇𝑢‖_𝐿^𝑞‖∇𝑢‖_𝐿^{2(1−3/2𝑞)}₂‖Δ𝑢‖_𝐿^3/𝑞₂≤1‖‖32Δ𝑢²_𝐿2+𝐶‖∇𝑢‖^𝑝_𝐿𝑞‖∇𝑢‖²_𝐿2. (46)

𝐼₆ is bounded as above.

Inserting the above estimates into (23), using the Gronwall inequality, we arrive at (34).

This completes the proof.

Acknowledgments

This paper is supported by NSFC (no. 11171154). The authors are indebted to the referee for nice suggestions which improved the paper.