Remarks on the Pressure Regularity Criterion of the Micropolar Fluid Equations in Multiplier Spaces

Abstract

This study is devoted to investigating the regularity criterion of weak solutions of the micropolar fluid equations in ℝ³. The weak solution of micropolar fluid equations is proved to be smooth on (0,𝑇] when the pressure 𝜋(𝑥,𝑡) satisfies the following growth condition in the multiplier spaces 𝑋̇^𝑟, ∫^𝑇₀‖𝜋(𝑠,⋅)‖_̇𝑋^2/(2−𝑟)_𝑟/(1+ln(𝑒+‖𝜋(𝑠,⋅)‖_𝐿²)),𝑑𝑠<∞,and0≤𝑟≤1. The previous results on Lorentz spaces and Morrey spaces are obviously improved.

1. Introduction

Consider the Cauchy problem of the three-dimensional (3D) micropolar fluid equations with unit viscosities 𝜕∇⋅𝑢=0,_𝑡𝜕𝑢−Δ𝑢−∇×𝑤+∇𝜋+𝑢⋅∇𝑢=0,_𝑡𝑤−Δ𝑤−∇∇⋅𝑤+2𝑤−∇×𝑢+𝑢⋅∇𝑤=0 (1.1) associated with the initial condition: 𝑢(𝑥,0)=𝑢₀,𝑤(𝑥,0)=𝑤₀, (1.2) where 𝑢(𝑥,𝑡)=(𝑢₁(𝑥,𝑡),𝑢₂(𝑥,𝑡),𝑢₃(𝑥,𝑡)), and 𝑤(𝑥,𝑡)=(𝑤₁(𝑥,𝑡),𝑤₂(𝑥,𝑡),𝑤₃(𝑥,𝑡)) are the unknown velocity vector field and the microrotation vector field. 𝜋(𝑥,𝑡) is the unknown scalar pressure field. 𝑢₀ and 𝑤₀ represent the prescribed initial data for the velocity and microrotation fields.

Micropolar fluid equations introduced by Eringen [1] are a special model of the non-Newtonian fluids (see [2–6]) which is coupled with the viscous incompressible Navier-Stokes model, microrotational effects, and microrotational inertia. When the microrotation effects are neglected or 𝑤=0, the micropolar fluid equations (1.1) reduce to the incompressible Navier-Stokes flows (see, e.g., [7, 8]): 𝜕_𝑡()𝑢()𝑢−Δ𝑢+𝑢⋅∇𝑢+∇𝜋=0,∇⋅𝑢=0,𝑥,0=𝑢₀. (1.3) That is to say, Navier-Stokes equations are viewed as a subclass of the micropolar fluid equations.

Mathematically, there is a large literature on the existence, uniqueness and large time behaviors of solutions of micropolar fluid equations (see [9–15] and references therein); however, the global regularity of the weak solution in the three-dimensional case is still a big open problem. Therefore it is interesting and important to consider the regularity criterion of the weak solutions under some assumptions of certain growth conditions on the velocity or on the pressure.

On one hand, as for the velocity regularity criteria, by means of the Littlewood-Paley decomposition methods, Dong and Chen [16] proved the regularity of weak solutions under the velocity condition: ∇𝑢∈𝐿^𝑞0,𝑇;𝐵̇⁰_𝑝,𝑟ℝ³ (1.4) with 2𝑞+3𝑝3=2,2<𝑝≤∞,𝑟≤2𝑝3. (1.5) Moreover, the result is further improved by Dong and Zhang [17] in the margin case: 𝑢∈𝐿^2/(1+𝑟)0,𝑇;𝐵^𝑟_∞,∞ℝ³,−1<𝑟<1. (1.6)

On the other hand, as for the pressure regularity criteria, Yuan [18] investigated the regularity criterion of weak solutions of the micropolar fluid equations in Lebesgue spaces and Lorentz spaces: 𝜋∈𝐿^𝑞0,𝑇;𝐿^𝑝ℝ³2,for𝑞+3𝑝3=2,2<𝑝<∞,𝜋∈𝐿^𝑞0,𝑇;𝐿^𝑝,∞ℝ³2,for𝑞+3𝑝3=2,2<𝑝<∞, (1.7) where 𝐿^𝑝,∞(ℝ³) is the Lorents space (see the definitions in the next section).

Recently, Dong et al. [19] improved the pressure regularity of the micropolar fluid equations in Morrey spaces: 𝜋∈𝐿^𝑞0,𝑇;𝑀̇_𝑝,𝑟ℝ³, (1.8) where 2𝑞+3𝑝3=2,2<𝑝≤∞,𝑟≤𝑝. (1.9) Furthermore, Jia et al. [20] refined the regularity from Morrey spaces to Besov spaces: 𝜋∈𝐿^𝑞0,𝑇;𝐵^𝑟_𝑝,∞ℝ³ (1.10) with 2𝑞+3𝑝3=2+𝑟,2+𝑟<𝑝<∞,−1<𝑟≤1. (1.11) One may also refer to some interesting results on the regularity criteria of Newtonian and non-Newtonian fluid equations (see [21–27] and references therein).

The aim of the present study is to investigate the pressure regularity criterion of the three-dimensional micropolar fluid equations in the multiplier spaces which are larger than the Lebesgue spaces, Lorentz spaces, and Morrey spaces.

2. Preliminaries and Main Result

Throughout this paper, we use 𝑐 to denote the constants which may change from line to line. 𝐿^𝑝(ℝ³),𝑊^𝑘,𝑝(ℝ³) with 𝑘∈𝐙,1≤𝑝≤∞ denote the usual Lebesgue space and Sobolev space. 𝐻̇^𝑠(ℝ³),𝑠∈ℝ denote the fractional Sobolev space with 𝑓_̇𝐻𝑠=_ℝ3𝜉^2𝑠𝑓²𝑑𝜉^1/2. (2.1)

Consider a measurable function 𝑓 and define for 𝑡≥0 the Lebesgue measure 𝑚𝑓,𝑡∶=𝑚𝑥∈ℝ³:𝑓(𝑥)>𝑡 (2.2) of the set {𝑥∈ℝ³∶|𝑓(𝑥)|>𝑡}. The Lorentz space is defined by 𝑓∈𝐿^𝑝,𝑞(ℝ³) if and only if 𝑓_{𝐿𝑝,𝑞}=^∞₀𝑡^𝑞𝑚𝑓,𝑡^𝑞/𝑝𝑑𝑡𝑡^1/𝑞𝑓<∞,for1≤𝑞<∞,_𝐿𝑝,∞=sup_𝑡≥0𝑡𝑚𝑓,𝑡^1/𝑝<∞,for𝑞=∞. (2.3)

We defined 𝑀̇_𝑝,𝑞(ℝ³), 1≤𝑝,𝑞≤∞ the homogeneous Morrey space associated with norm 𝑓_{̇𝑀𝑝,𝑞}=sup_𝑟>0sup_𝑥∈ℝ3𝑟^{3/𝑝−3/𝑞}_{|𝑥−𝑦|<𝑟}𝑓𝑦^𝑞𝑑𝑦^1/𝑞<∞. (2.4)

We now recall the definition and some properties of the multiplier space 𝑋̇^𝑟.

Definition 2.1 (see Lemarié-Rieusset [28]). For 0≤𝑟<3/2, the space 𝑋̇^𝑟 is defined as the space of 𝑓(𝑥)∈𝐿²_loc(ℝ³) such that 𝑓_̇𝑋𝑟=sup_{‖𝑔‖̇𝐻𝑟≤1ℝ3}𝑓𝑔²𝑑𝑥^1/2. (2.5)

According the above definition of the multiplier space, it is not difficult to verify the homogeneity properties. For all 𝑥₀∈ℝ³𝑓(⋅+𝑥₀)_̇𝑋𝑟=𝑓_̇𝑋𝑟,𝑓(𝜆)_̇𝑋𝑟=1𝜆^𝑟𝑓_̇𝑋𝑟,𝜆>0. (2.6)

When 𝑟=0, it is clear that (see Lemarié-Rieusset [28]) 𝑋̇⁰≅BMO, (2.7) where BMO denotes the homogenous space of bounded mean oscillations associated with the norm 𝑓_BMO=sup_{𝑥∈ℝ3,𝑟>0}1𝐵_𝑟(𝑥)_{𝐵𝑟(𝑥)}𝑓𝑦−1𝐵_𝑟𝑦_{𝐵𝑟(𝑦)}𝑓(𝑧)𝑑𝑧𝑑𝑦. (2.8)

In particular, the following imbedding (see Lemarié-Rieusset [28]) 𝐿^𝑝ℝ³⊂𝐿^𝑝,∞ℝ³⊂𝑀̇_𝑝,𝑞ℝ³⊂𝑋̇^𝑟ℝ³⊂𝐵̇^−𝑟_∞,∞ℝ³3,𝑝=𝑟>𝑞>2 (2.9) holds true.

In order to state our main results, we recall the definition of the weak solution of micropolar flows (see, e.g., Łukaszewicz [9]).

Definition 2.2. Let 𝑇>0, (𝑢₀,𝑤₀)∈𝐿²(ℝ³), and ∇⋅𝑢₀=0. (𝑢,𝑤) is termed as a weak solution to the 3D micropolar flows (1.1) and (1.2) on (0,𝑇), if (𝑢,𝑤) satisfies the following properties:(i)(𝑢,𝑤)∈𝐿^∞(0,𝑇;𝐿²(ℝ³))∩𝐿²(0,𝑇;𝐻¹(ℝ³));(ii)equations (1.1) and (1.2) are valid in the sense of distributions.

Our main results are now read as follows.

Theorem 2.3. Suppose 𝑇>0, (𝑢₀,𝑤₀)∈𝐿²(ℝ³)∩𝐿⁴(ℝ³), and ∇⋅𝑢₀=0 in the sense of distributions. Assume that (𝑢,𝑤) is a weak solution of the 3D micropolar fluid flows (1.1) and (1.2) on (0,𝑇). If the pressure 𝜋(𝑥,𝑡) satisfies the logarithmically growth condition: ^𝑇₀‖𝜋(𝑠,⋅)‖_̇𝑋^2/(2−𝑟)_𝑟1+ln𝑒+‖𝜋(𝑠,⋅)‖_𝐿²𝑑𝑠<∞,0≤𝑟≤1, (2.10) then the weak solution (𝑢,𝑤) is regular on (0,𝑇].

Thanks to ^𝑇₀‖𝜋(𝑠,⋅)‖_̇𝑋^2/(2−𝑟)_𝑟1+ln𝑒+‖𝜋(𝑠,⋅)‖_𝐿²𝑑𝑠≤^𝑇₀‖𝜋(𝑠,⋅)‖_̇𝑋^2/(2−𝑟)_𝑟𝑑𝑠, (2.11) it is easy to deduce the following pressure regularity criterion of the three-dimensional micropolar equations (1.1) and (1.2).

Corollary 2.4. On the substitution of the pressure condition (2.10) by the following conditions: ^𝑇₀‖𝜋(𝑠,⋅)‖_̇𝑋^2/(2−𝑟)_𝑟𝑑𝑠<∞,0≤𝑟≤1, (2.12) the conclusion of Theorem 2.3 holds true.

Remark 2.5. According to the embedding relation (2.9), our results obviously largely improve the previous results (1.7) and (1.8). Moreover, it seems incomparable with the Besov space (1.10).

Remark 2.6. Furthermore, since we have no additional growth condition on the microrotation vector field 𝑤(𝑥,𝑡), Theorem 2.3 is also valid for the pressure regularity problem of the three-dimensional Navier-Stokes equations (see, e.g., Zhou [29, 30]).

3. Proof of Theorem 2.3

In order to prove our main results, we first recall the following local existence theorem of the three-dimensional micropolar fluid equations (1.1) and (1.2).

Lemma 3.1 (see Dong et al. [19]). Assume 3<𝑝<∞ and (𝑢₀,𝑤₀)∈𝐿^𝑝(ℝ³) with ∇⋅𝑢₀=0 in the sense of distributions. Then there exist a constant 𝑇>0 and a unique strong solution (𝑢,𝑤) of the 3D micropolar fluid equations (1.1) and (1.2) such that 𝑢∈𝐵𝐶[0,𝑇);𝐿^𝑝ℝ³,𝑡^1/2∇𝑢∈𝐵𝐶[0,𝑇);𝐿^𝑝ℝ³. (3.1)

By means of the local existence result, (1.1) and (1.2) with (𝑢₀,𝑤₀)∈𝐿²(ℝ³)∩𝐿⁴(ℝ³) admit a unique 𝐿⁴-strong solution (𝑢,𝑤) on a maximal time interval. For the notation simplicity, we may suppose that the maximal time interval is [0,𝑇). Thus, to prove Theorem 2.3, it remains to show that lim_𝑡→𝑇‖𝑢(𝑡)‖₄+‖𝑤(𝑡)‖₄<∞. (3.2) This will lead to a contradiction to the estimates to be derived below. We now begin to follow these arguments.

Taking the inner product of the second equation of (1.1) with 𝑢|𝑢|² and the third equation of (1.1) with 𝑤|𝑤|², respectively, and integrating by parts, it follows that 14𝑑‖𝑢‖𝑑𝑡^4𝐿4+‖|𝑢|∇𝑢‖²_𝐿2+12∇|𝑢|²²_𝐿2=_ℝ3(∇×𝑤)⋅𝑢|𝑢|²𝑑𝑥−_ℝ3𝑢⋅∇𝜋|𝑢|²1𝑑𝑥,4𝑑‖𝑤‖𝑑𝑡^4𝐿4+‖|𝑤|∇𝑤‖²_𝐿2+12‖|𝑤|∇⋅𝑤‖²_𝐿2+∇|𝑤|²²_𝐿2=_ℝ3(∇×𝑢)⋅𝑤|𝑤|²𝑑𝑥−2‖𝑤‖⁴_𝐿4, (3.3) where we have used the following identities due to the divergence free property of the velocity field 𝑢: _ℝ3(𝑢⋅∇𝑢)⋅𝑢|𝑢|²𝑑𝑥=0,_ℝ3(𝑢⋅∇𝑤)⋅𝑤|𝑤|²()⋅∇𝑢|𝑢|𝑑𝑥=0,∇𝑢²=|∇𝑢|²|𝑢|²+12∇|𝑢|²²,(∇𝑤)⋅∇𝑤|𝑤|²=|∇𝑤|²|𝑤|²+12∇|𝑤|²²,_ℝ3(∇⋅𝑤)∇⋅𝑤|𝑤|²𝑑𝑥=_ℝ3|∇⋅𝑤|²|𝑤|²𝑑𝑥+_ℝ3(∇⋅𝑤)𝑤⋅∇|𝑤|²≥1𝑑𝑥2_ℝ³|∇⋅𝑤|²|𝑤|²1𝑑𝑥−2_ℝ³∇|𝑤|²²𝑑𝑥. (3.4)

Furthermore, applying Young inequality, Hölder inequality, and integration by parts, we have _ℝ3(∇×𝑤)⋅𝑢|𝑢|²𝑑𝑥+_ℝ3(∇×𝑢)⋅𝑤|𝑤|²𝑑𝑥−2‖𝑤‖⁴_𝐿4≤‖𝑤‖_𝐿⁴‖|𝑢|∇𝑢‖_𝐿²‖𝑢‖_𝐿⁴+‖𝑢‖_𝐿⁴‖|𝑤|∇𝑤‖_𝐿²‖𝑤‖_𝐿⁴−2‖𝑤‖⁴₄≤‖𝑢‖⁴_𝐿4+‖|𝑤|∇𝑤‖²_𝐿2+‖|𝑢|∇𝑢‖²_𝐿2. (3.5)

Combining the above inequalities, it follows that 𝑑‖𝑢‖𝑑𝑡^4𝐿4+‖𝑤‖⁴_𝐿4+2‖|𝑢|∇𝑢‖²_𝐿2+2∇|𝑢|²²_𝐿2≤𝐶‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4+𝐶_ℝ3𝑢⋅∇𝜋|𝑢|²𝑑𝑥. (3.6)

In order to estimate the last term of the right-hand side of (3.6), taking the divergence operator ∇⋅ to the first equation of (1.1) produces the expression of the pressure: 𝜋=(−Δ)⁻¹∇⋅(𝑢⋅∇𝑢). (3.7) Employing Calderón-Zygmund inequality and the divergence free condition of the velocity derives the estimate of the pressure: ‖𝜋‖_𝐿^𝑟≤𝑐‖𝑢‖²_𝐿2𝑟,‖∇𝜋‖_𝐿^𝑟≤𝑐‖𝑢⋅∇𝑢‖_𝐿^𝑟,1<𝑟<∞. (3.8)

Therefore, we estimate the pressure term as −_ℝ3𝑢⋅∇𝜋|𝑢|²𝑑𝑥≤_ℝ3|𝜋||𝑢|∇|𝑢|²1𝑑𝑥≤2_ℝ³∇|𝑢|²²𝑑𝑥+_ℝ3|𝜋|²|𝑢|²𝑑𝑥. (3.9) Now we estimate the integral 𝐼=∶_ℝ3|𝜋|²|𝑢|²𝑑𝑥 (3.10) on the right-hand side of (3.9). By the Hölder inequality and the Young inequality we have 𝐼≤𝜋|𝑢|²_𝐿2‖𝜋‖_𝐿²≤‖𝜋‖_̇𝑋^𝑟|𝑢|²_̇𝐻𝑟‖𝜋‖_𝐿²≤‖𝜋‖_̇𝑋^𝑟|𝑢|²_𝐿^1−𝑟₂∇|𝑢|^2𝑟_𝐿2‖𝑢‖²_𝐿4≤‖𝜋‖_̇𝑋^𝑟‖𝑢‖_𝐿^2(2−𝑟)₄∇|𝑢|^2𝑟_𝐿2≤𝜖∇|𝑢|²²_𝐿2+𝐶(𝜖)‖𝜋‖_̇𝑋^2/(2−𝑟)_𝑟‖𝑢‖⁴_𝐿4, (3.11) where we have used the following interpolation inequality: 𝑓_̇𝐻𝑟≤𝑓_𝐿^1−𝑟₂∇𝑓^𝑟_𝐿2. (3.12)

Hence, combining the above inequalities, we derive 𝑑‖𝑢‖𝑑𝑡^4𝐿4+‖𝑤‖⁴_𝐿4+‖|𝑢|∇𝑢‖²_𝐿2+‖|𝑤|∇𝑤‖²_𝐿2≤𝐶‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4+𝐶(𝜖)‖𝜋‖_̇𝑋^2/(2−𝑟)_𝑟‖𝑢‖⁴_𝐿4. (3.13)

Furthermore, we have the second term of the right-hand side of (3.13) rewritten as 𝐶(𝜖)‖𝜋‖_̇𝑋^2/(2−𝑟)_𝑟‖𝑢‖⁴_𝐿4≤𝐶‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4‖𝜋‖_̇𝑋^2/(2−𝑟)_𝑟‖𝜋(≤𝐶𝑡,⋅)‖_̇𝑋^2/(2−𝑟)_𝑟1+ln𝑒+‖𝜋(𝑡,⋅)‖_𝐿²‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿41+ln𝑒+‖𝜋(𝑡,⋅)‖_𝐿²‖𝜋(≤𝐶𝑡,⋅)‖_̇𝑋^2/(2−𝑟)_𝑟1+ln𝑒+‖𝜋(𝑡,⋅)‖_𝐿²‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿41+ln𝑒+‖𝑢(𝑡,⋅)‖²_𝐿4‖𝜋(≤𝐶𝑡,⋅)‖_̇𝑋^2/(2−𝑟)_𝑟1+ln𝑒+‖𝜋(𝑡,⋅)‖_𝐿²‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿41+ln𝑒+‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4. (3.14) Inserting (3.14) into (3.13) and applying the Gronwall inequality, one shows that 1+ln‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4≤1+ln𝑢₀⁴_𝐿4+𝑤₀⁴_𝐿4×exp𝐶𝑇+^𝑇₀‖𝜋(𝑠,⋅)‖_̇𝑋^2/(2−𝑟)_𝑟1+ln𝑒+‖𝜋(𝑠,⋅)‖_𝐿²𝑑𝑠 (3.15) which implies _ℝ3|𝑢(𝑡)|⁴+|𝑤(𝑡)|⁴𝑑𝑥<∞,0<𝑡≤𝑇. (3.16)

Hence we complete the proof of Theorem 2.3.