Logarithmically Improved Regularity Criteria for a Fluid System with the Linear Soret Effect

and Applied Analysis 3 Thus the method in 7, 10 cannot be used here. We need new interpolation inequalities. Remark 1.3. In 10 , authors researched the regularity for the 3D viscous Boussinesq equations under condition 1.4 . However, when φ 0, 1.1 is the well-known Boussinesq system with zero viscosity, so our results improve result of 10 . 2. Preliminaries and Lemmas First, we recall some definitions and lemmas, which play an important role in studying the regularity of solution to partial differential equations. Definition 2.1. BMO denotes the space of functions of bounded mean oscillation of John and Nirenberg associated with the norm ∥f ∥ BMO sup x∈R3,R>0 1 |BR x | ∫ BR x ∣∣∣∣ f ( y ) − 1 |BR x | ∫ BR x f z dz ∣∣∣∣ dy. 2.1 In order to prove Theorem 1.1, we need the following Gagliardo-Nirenberg inequality. Lemma 2.2. There exists a uniform positive constant C > 0 such that ∥∥∥∇if ∥∥∥ L2m/i C ∥f ∥ 1−i/m L∞ ∥∇mf∥i/m L2 , 0 i m 2.2 holds for all f ∈ L∞ R ∩Hm R . Lemma 2.3 see 9 . There exists a uniform positive constant C > 0 such that ∥∥∇f∥∥L∞ C ( 1 ∥f ∥ L2 ∥∇ × f∥∥BMO √ ln ( e ∥f ∥ H3 ) ) . 2.3 Lemma 2.4 see 11 . The following calculus inequality holds: ∥Λs fg − fΛsg∥∥Lp (∥∇f∥Lp1 ∥∥∥Λs−1g ∥∥ Lq1 ∥Λsf ∥ Lp2 ∥g ∥ Lq2 ) , 2.4 with s > 0, Λ −Δ s/2 and 1/p 1/p1 1/q1 1/p2 1/q2. Lemma 2.5 see 12 . In three-dimensional space, the following inequalities ∥∥∇f∥∥L2 C ∥f ∥2/3 L2 ∥∥∇3f ∥∥ 1/3 L2 , ∥f ∥ L∞ C ∥f ∥1/4 L2 ∥∥∥∇2f ∥∥∥ 3/4 L2 , ∥f ∥ L4 C ∥f ∥3/4 L2 ∥∥∥∇3f ∥∥ 1/4 L2 . 2.5


Introduction and Main Results
In this paper, we consider the regularity of the following fluid system with the linear Soret effect: u t u · ∇u ∇P θ φ e 3 , x, t ∈ R 3 × 0, ∞ , θ t u · ∇θ − Δθ 0, φ t u · ∇φ − Δφ Δθ, where u u 1 x, t , u 2 x, t , u 3 x, t denotes the fluid velocity vector field, P P x, t is the scalar pressure, θ x, t is the scalar temperature, and φ is the concentration field, e 3 0, 0, 1 T , while u 0 , θ 0 , and φ 0 are the given initial velocity, initial temperature, and initial concentration, respectively, with ∇ · u 0 0. The term Δθ in 1.1 is the linear Soret effect see page 102 in 1 , 2-4 .

Abstract and Applied Analysis
The question of global existence or blow-up in finite time of smooth solutions for the 3D incompressible Euler or Navier-Stokes equations has been one of the most outstanding open problems in applied analysis, as well as that for the 3D incompressible magnetohydrodynamics MHD equations. This challenging problem has attracted significant attention. In the absence of the global well-posedness, the development of blow-up or non-blow-up theory is of major importance for both theoretical and practical purposes.

1.2
When φ 0, 1.1 is the well-known Boussinesq system with zero viscosity, Fan and Zhou 7 proved the following blow-up criterion: HereḂ 0 ∞,∞ denotes the homogeneous Besov space. Recently, Chan and Vasseur 8 and Zhou and Lei 9 proved some logarithmically improved regularity criterion for the 3D Navier-Stokes equations. Qiu et al. 10 obtained Serrin-type blow-up criteria of smooth solution for the 3D viscous Boussinesq equations. They showed that smooth solution u ·, t , θ ·, t for 0 t < T remains smooth at time t T , provided that the following condition holds: Motivated by the previous results on the regularity criteria of the fluid dynamics equations, the purpose of this paper is to establish a logarithmically improved regularity criterion in terms of the vorticity field for 1.1 in the BMO space, which is defined in Section 2. Now we state our main results as follows.
Then the solution u, θ, φ can be smoothly extended after time t T provided that 1.4 is satisfied.
In the process of proof in 7, 10 , they used the important fact that Due to the linear Soret effect, we cannot prove Abstract and Applied Analysis 3 Thus the method in 7, 10 cannot be used here. We need new interpolation inequalities. Remark 1.3. In 10 , authors researched the regularity for the 3D viscous Boussinesq equations under condition 1.4 . However, when φ 0, 1.1 is the well-known Boussinesq system with zero viscosity, so our results improve result of 10 .

Preliminaries and Lemmas
First, we recall some definitions and lemmas, which play an important role in studying the regularity of solution to partial differential equations.
In order to prove Theorem 1.1, we need the following Gagliardo-Nirenberg inequality.

Lemma 2.2.
There exists a uniform positive constant C > 0 such that

Lemma 2.3 see 9 .
There exists a uniform positive constant C > 0 such that

Lemma 2.5 see 12 .
In three-dimensional space, the following inequalities 2.5 hold.

Proofs of the Main Results
In this section, we prove Theorem 1.1.
Proof of Theorem 1.1. Multiplying both the sides of the second equation of 1.1 by θ and the third equation of 1.1 by φ, respectively, and integrating by parts over R 3 , we obtain Combining 3.1 with 3.2 , and using Gronwall's inequality, we infer that

3.3
Multiplying both the sides of the first equation of 1.1 by u, and integrating by parts over R 3 , by 3.3 , we get which implies u L ∞ 0,T ;L 2 C.

3.5
Next we go to estimate L 2 -norm of ∇u, ∇θ and ∇φ. Multiplying the first equation, the second equation and the third equation of 1.1 by −Δu, −Δθ, and −Δφ, respectively, then integrating by parts over R 3 , we deduce that

3.8
Abstract and Applied Analysis 5 Now combining 3.6 , 3.7 with 3.8 and using Gronwall's inequality, we obtain Noting 1.4 , one concludes that for any small constant ε > 0, there exists T * < T such that where Λ s −Δ s/2 . It follows from 3.9 , 3.10 , and Lemma 2.3 that

3.12
where C 1 depends on ∇u T * 2 ∇φ T * 2 L 2 , and C 0 is an absolute positive constant.
Finally we go to estimate for H 3 -norm of u, θ, and φ. Applying the operation Λ 3 on both the sides of the first equation of 1.1 , then multiplying Λ 3 u, and integrating by parts over R 3 , by 2.4 , 3.12 , Hölder's inequality, and Young's inequality, we have

3.16
Inserting the above estimates A 2 and A 3 into 3.14 , we get 3.17 provided that C 0 ε 1/5, which can be achieved by the absolute continuous property of integral 1.4 . 7 Similarly, for the φ-equation, we have