# Regularity Criterion for Weak Solution to the 3D Micropolar Fluid Equations

Authors: Yu-Zhu Wang; Zigao Chen
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

Regularity criterion for the 3D micropolar fluid equations is investigated. We prove that, for someT>0, if ∫0T∥vx3∥Lϱρdt<∞, where 3/ϱ+2/ρ≤1 and ϱ≥3, then the solution (v,w) can be extended smoothly beyond t=T. The derivative vx3 can be substituted with any directional derivative of v.


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## Body

## 1. Introduction

In the paper, we investigate the initial value problem for the micropolar fluid equations inℝ3: (1.1)∂tv-(ν+κ)Δv+v⋅∇v+∇π-2κ∇×w=0,∂tw-γΔw-(α+β)∇∇⋅w+4κw+v⋅∇w-2κ∇×v=0,∇⋅v=0
with the initial value (1.2)t=0:v=v0(x),w=w0(x),
where v(t,x),  w(t,x), and π(t,x) stand for the divergence free velocity field, nondivergence free microrotation field (angular velocity of the rotation of the particles of the fluid), the scalar pressure, respectively ν>0 is the Newtonian kinetic viscosity, κ>0 is the dynamics microrotation viscosity, and α,β,γ>0 are the angular viscosity (see, e.g., Lukaszewicz [1]).The micropolar fluid equations was first proposed by Eringen [2]. It is a type of fluids which exhibits the microrotational effects and microrotational inertia and can be viewed as a non-Newtonian fluid. Physically, micropolar fluid may represent fluids that consists of rigid, randomly oriented (or spherical) particles suspended in a viscous medium, where the deformation of fluid particles is ignored. It can describe many phenomena appeared in a large number of complex fluids such as the suspensions, animal blood, and liquid crystals which cannot be characterized appropriately by the Navier-Stokes equations, and that is important to the scientists working with the hydrodynamic fluid problems and phenomena. For more background, we refer to [1] and references therein. Besides their physical applications, the micropolar fluid equations are also mathematically significant. The existences of weak and strong solutions for micropolar fluid equations were treated by Galdi and Rionero [3] and Yamaguchi [4], respectively. The convergence of weak solutions of the micropolar fluids in bounded domains of ℝn was investigated (see [5]). When the viscosities tend to zero, in the limit, a fluid governed by an Euler-like system was found. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research, and many interesting results have been obtained (see [6–8]). A Beale-Kato-Madja criterion (see [9]) of smooth solutions to a related model with (1.1) was established in [10].Ifκ=0 and w=0, then (1.1) reduces to be the Navier-Stokes equations. Besides its physical applications, the Navier-Stokes equations are also mathematically significant. In the last century, Leray [11] and Hopf [12] constructed weak solutions to the Navier-Stokes equations. The solution is called the Leray-Hopf weak solution. Later on, much effort has been devoted to establish the global existence and uniqueness of smooth solutions to the Navier-Stokes equations. Different criteria for regularity of the weak solutions have been proposed, and many interesting results are established (see [13–31]).The purpose of this paper is to establish the regularity criteria of weak solutions to (1.1), (1.2) via the derivative of the velocity in one direction. It is proved that if ∫0T∥vx3∥Lϱρdt<∞ with (1.3)3ϱ+2ρ≤1,ϱ≥3,
then the solution (v,w) can be extended smoothly beyond t=T.The paper is organized as follows. We first state some important inequalities in Section2, which play an important roles in the proof of our main result. Then, we give definition of weak solution and state main results in Section 3 and then prove main result in Section 4.


## 2. Preliminaries

In order to prove our main result, we need the following Lemma, which may be found in [32] (see also [33, 34]). For the convenience of the readers, the proof of the Lemmas are provided.Lemma 2.1.
Assume thatμ,λ,ι∈ℝ and satisfy
(2.1)1≤μ,λ<∞,1μ+2λ>1,1+3ι=1μ+2λ.
Assume that f∈H1(ℝ3), fx1,fx2∈Lλ(ℝ3), and fx3∈Lμ(ℝ3). Then, there exists a positive constant such that
(2.2)‖f‖Lι≤C‖fx1‖Lλ1/3‖fx2‖Lλ1/3‖fx3‖Lμ1/3.
Especially, when λ=2, there exists a positive constant C=C(μ) such that
(2.3)‖f‖L3μ≤C‖fx1‖L21/3‖fx2‖L21/3‖fx3‖Lμ1/3,
which holds for any f∈H1(ℝ3) and fx3∈Lμ(ℝ3) with 1≤μ<∞.Proof.
It is not difficult to find(2.4)|f(x1,x2,x3)|1+(1-1/λ)ι≤C∫-∞x1|f(x1,x2,x3)|(1-(1/λ))ι|∂τf(τ,x2,x3)|dτ.|f(x1,x2,x3)|1+(1-1/λ)ι≤C∫-∞x2|f(x1,x2,x3)|(1-(1/λ))ι|∂τf(x1,τ,x3)|dτ,|f(x1,x2,x3)|1+(1-1/μ)ι≤C∫-∞x3|f(x1,x2,x3)|(1-(1/μ))ι|∂τf(x1,x2,τ)|dτ.
Then, we obtain
(2.5)|f(x1,x2,x3)|ι≤C[∫-∞∞|f(x1,x2,x3)|(1-1/λ)ι|∂x1f(x1,x2,x3)|dx1]1/2×[∫-∞∞|f(x1,x2,x3)|(1-1/λ)ι|∂x2f(x1,x2,x3)|dx2]1/2×[∫-∞∞|f(x1,x2,x3)|(1-1/μ)ι|∂x3f(x1,x2,x3)|dx3]1/2.
Integrating with respect to x1 and using Hölder inequality, we have (2.6)∫-∞∞|f(x1,x2,x3)|ιdx1≤C[∫-∞∞|f(x1,x2,x3)|(1-1/λ)ι|∂x1f(x1,x2,x3)|dx1]1/2×[∫-∞∞∫-∞∞|f(x1,x2,x3)|(1-1/λ)ι|∂x2f(x1,x2,x3)|dx2dx1]1/2×[∫-∞∞∫-∞∞|f(x1,x2,x3)|(1-1/μ)ι|∂x3f(x1,x2,x3)|dx3dx1]1/2.
Integrating with respect to x2,x3 and using Hölder inequality, we obtain
(2.7)∫R3|f(x1,x2,x3)|ιdx≤C[∫-∞∞|f(x1,x2,x3)|(1-1/λ)ι|∂x1f(x1,x2,x3)|dx]1/2×[∫R3|f(x1,x2,x3)|(1-1/λ)ι|∂x2f(x1,x2,x3)|dx]1/2×[∫R3|f(x1,x2,x3)|(1-1/μ)ι|∂x3f(x1,x2,x3)|dx]1/2.
It follows from Hölder inequality that
(2.8)‖f‖Lιι≤C‖f‖Lι(1-1/λ)ι/2‖∂x1f‖Lλ1/2‖f‖Lι(1-1/λ)ι/2‖∂x2f‖Lλ1/2‖f‖Lι(1-1/μ)ι/2‖∂x3f‖Lμ1/2.
By the above inequality, we get (2.2).Lemma 2.2.
Let2≤q≤6 and assume that f∈H1(ℝ3). Then, there exists a positive constant C=C(q) such that
(2.9)‖f‖Lq≤C‖f‖L2(6-q)/2q‖∂x1f‖L2(q-2)/2q‖∂x2f‖L2(q-2)/2q‖∂x3f‖L2(q-2)/2q.Proof.
Using the interpolating inequality, we obtain(2.10)‖f‖Lq≤C‖f‖L2(6-q)/2q‖f‖L6(3q-6)/2q.
By (2.3) with μ=2, we have
(2.11)‖f‖L6≤C‖∂x1f‖L21/3‖∂x2f‖L21/3‖∂x3f‖L21/3.
Combining (2.10) and (2.11) yields (2.9).


## 3. Main Results

Before stating our main results, we introduce some function spaces. Let(3.1)C0,σ∞(R3)={φ∈(C∞(R3))3:∇⋅φ=0}⊂(C∞(R3))3.
The subspace (3.2)Lσ2=C0,σ∞(R3)¯‖⋅‖L2={φ∈L2(R3):∇⋅φ=0}
is obtained as the closure of C0,σ∞ with respect to L2-norm ∥·∥L2. Hσr is the closure of C0,σ∞ with respect to the Hr-norm
(3.3)‖φ‖Hr=‖(I-Δ)r/2φ‖L2,r≥0.
Before stating our main results, we give the definition of weak solution to (1.1), (1.2) (see [6]).Definition 3.1 (Weak solutions).
LetT>0,  v0∈Lσ2(ℝ3),  and w0∈L2(ℝ3). A measurable ℝ3-valued triple (v,w) is said to be a weak solution to (1.1), (1.2) on [0,T] if the following conditions hold the following.(1)
(3.4)v∈L∞(0,T;Lσ2(R3))⋂L2(0,T;Hσ1(R3)),w∈L∞(0,T;L2(R3))⋂L2(0,T;H1(R3)).(2)
Equations (1.1), (1.2) are satisfied in the sense of distributions; that is, for every φ∈H1((0,T);Hσ1) and ψ∈H1((0,T);H1) with φ(T)=ψ(T)=0, hold
(3.5)∫0T{-〈v,∂τφ〉+〈v⋅∇v,φ〉+(ν+κ)〈∇v,∇φ〉}dτ-∫0T{2κ〈∇×w,φ〉}dτ=〈v0,φ(0)〉,(3.6)∫0T{-〈w,∂τψ〉}+γ〈∇w,∇ψ〉+(α+β)〈∇⋅w,∇ψ〉+4κ〈w,ψ〉dτ+∫0T{〈v⋅∇w,ψ〉-2κ〈∇×v,ψ〉}dτ=〈w0,ψ(0)〉.(3)
The energy inequality, that is,(3.7)‖v(t)‖L22+‖w(t)‖L22+2∫0t(ν‖∇v(τ)‖L22+γ‖∇w(τ)‖L22)dτ+2(α+β)∫0t‖∇⋅w(τ)‖L22dτ≤‖v0‖L22+‖w0‖L22.Theorem 3.2.
Letv0∈Hσ1(ℝ3) with w0∈H1(ℝ3). Assume that (v,w) is a weak solution to (1.1), (1.2) on some interval [0,T]. If
(3.8)Θ(T)≡∫0T‖vx3‖Lϱρdt<∞,
where
(3.9)3ϱ+2ρ≤1,ϱ≥3,
then the solution (v,w) can be extended smoothly beyond t=T.


## 4. Proof of Theorem3.2

Proof.
Multiplying the first equation of (1.1) by v and integrating with respect to x on ℝ3, using integration by parts, we obtain
(4.1)12ddt‖v(t)‖L22+(ν+κ)‖∇v(t)‖L22=2κ∫R3(∇×w)⋅vdx.
Similarly, we get
(4.2)12ddt‖w(t)‖L22+γ‖∇w(t)‖L22+(α+β)‖∇⋅w‖L22+4κ‖w‖L22=2κ∫R3(∇×v)⋅wdx.
Summing up (4.1)-(4.2), we deduce that
(4.3)12ddt(‖v(t)‖L22+‖w(t)‖L22)+(ν+κ)‖∇v(t)‖L22+γ‖∇w(t)‖L22+(α+β)‖∇⋅w‖L22+4κ‖w‖L22=2κ∫R3(∇×w)⋅vdx+2κ∫R3(∇×v)⋅wdx.
By integration by parts and Cauchy inequality, we obtain
(4.4)2κ∫R3(∇×w)⋅vdx+2κ∫R3(∇×v)⋅wdx≤κ‖∇v‖L22+4κ‖w‖L22.
Combining (4.3)-(4.4) yields
(4.5)12ddt(‖v(t)‖L22+‖w(t)‖L22)+ν‖∇v(t)‖L22+γ‖∇w(t)‖L22+(α+β)‖∇⋅w‖L22≤0.
Integrating with respect to t, we have
(4.6)‖v(t)‖L22+‖w(t)‖L22+2∫0t(ν‖∇v(τ)‖L22+γ‖∇w(τ)‖L22)dτ+2(α+β)∫0t‖∇⋅w(τ)‖L22dτ≤‖v0‖L22+‖w0‖L22.
Differentiating (1.1) with respect to x3, we obtain
(4.7)∂tvx3-(ν+κ)Δvx3+vx3⋅∇v+v⋅∇vx3+∇πx3-2κ∇×wx3=0,∂twx3-γΔwx3-(α+β)∇⋅∇wx3+4κwx3+vx3⋅∇w+v⋅∇wx3-2κ∇×vx3=0.
Taking the inner product of vx3 with the first equation of (4.7) and using integration by parts yields
(4.8)12ddt‖vx3(t)‖L22+(ν+κ)‖∇vx3(t)‖L22=-∫R3vx3⋅∇v⋅vx3dx+2κ∫R3(∇×wx3)⋅vx3dx.
Similarly, we get
(4.9)12ddt‖wx3(t)‖L22+γ‖∇wx3(t)‖L22+(α+β)‖∇⋅wx3‖L22+4κ‖wx3‖L22=-∫R3vx3⋅∇w⋅wx3dx+2κ∫R3(∇×vx3)⋅wx3dx.
Combining (4.8)–(4.9) yields
(4.10)12ddt(‖vx3(t)‖L22+‖wx3(t)‖L22)+(ν+κ)‖∇vx3(t)‖L22+γ‖∇wx3(t)‖L22+(α+β)‖∇⋅wx3‖L22+4κ‖wx3‖L22=-∫R3vx3⋅∇v⋅vx3dx+2κ∫R3(∇×wx3)⋅vx3dx-∫R3vx3⋅∇w⋅wx3dx+2κ∫R3(∇×vx3)⋅wx3dx.
Using integration by parts and Cauchy inequality, we obtain
(4.11)2κ∫R3(∇×wx3)⋅vx3dx+2κ∫R3(∇×vx3)⋅wx3dx≤κ‖∇vx3‖L22+4κ‖wx3‖L22.
Combining (4.10)–(4.11) yields
(4.12)12ddt(‖vx3(t)‖L22+‖wx3(t)‖L22)+ν‖∇vx3(t)‖L22+γ‖∇wx3(t)‖L22+(α+β)‖∇⋅wx3‖L22≤-∫R3vx3⋅∇v⋅vx3dx-∫R3vx3⋅∇w⋅wx3dx≜I1+I2.
In what follows, we estimateIj(j=1,2…,5). By integration by parts and Hölder inequality, we obtain
(4.13)I1≤C‖∇vx3‖L2‖vx3‖Lσ‖v‖L3ϱ,
where
(4.14)1σ+13ϱ=12,2≤σ≤6.
It follows from the interpolating inequality that
(4.15)‖vx3‖Lσ≤C‖vx3‖L21-3(1/2-1/σ)‖∇vx3‖L23(1/2-1/σ).
From (2.3), we get
(4.16)I1≤C‖∇vx3‖L2‖vx3‖L21-3(1/2-1/σ)‖∇vx3‖L23(1/2-1/σ)‖∇v‖L22/3‖vx3‖Lϱ1/3≤C‖∇vx3‖L21+3(1/2-1/σ)‖vx3‖L21-3(1/2-1/σ)‖∇v‖L22/3‖vx3‖Lϱ1/3≤ν2‖∇vx3‖L22+C‖vx3‖L22‖∇v‖L22q‖vx3‖L2q,
where
(4.17)q=23-9(1/2-1/σ)=23(1-1/ϱ).
When ϱ≥3, we have 2q≤2 and application of Young inequality yields
(4.18)I1≤ν2‖∇vx3‖L22+C‖vx3‖L22(‖∇v‖L22+‖vx3‖Lϱδ),
where
(4.19)3ϱ+2δ=1.
From Hölder inequality, we obtain
(4.20)I2≤C‖∇w‖L2‖wx3‖L2ϱ/(ϱ-2)‖vx3‖Lϱ≤C‖∇w‖L2‖vx3‖Lϱ‖wx3‖L21-3/ϱ‖∇wx3‖L23/ϱ≤C‖∇wx3‖L22+‖∇w‖L22ϱ/(2ϱ-3)‖vx3‖Lϱ2ϱ/(2ϱ-3)‖wx3‖L2(2ϱ-6)/(2ϱ-3)≤γ2‖∇wx3‖L22+C(‖∇w‖L22+‖vx3‖Lϱδ)‖wx3‖L2(2ϱ-6)/(2ϱ-3),
where
(4.21)3ϱ+2δ=1.
Combining (4.12)–(4.20) yields
(4.22)ddt(‖vx3‖L22+‖wx3‖L22)+ν‖∇vx3‖L22+γ‖∇wx3‖L22+(α+β)‖∇⋅wx3‖L22≤C‖vx3‖L22(‖∇v‖L22+‖vx3‖Lϱδ)+C(‖∇w‖L22+‖vx3‖Lϱδ)‖wx3‖L2(2ϱ-6)/(2ϱ-3).
From Gronwall inequality, we get
(4.23)‖vx3‖L22+‖wx3‖L22+ν∫0t‖∇vx3‖L22dτ+∫0t(γ‖∇wx3‖L22+(α+β)‖∇⋅wx3‖L22)dτ≤Ce(‖v0‖L22+‖w0‖L22)eΘ(t)[‖v0‖H12+‖w0‖H12+C(‖v0‖L22+‖w0‖L22+Θ(t))2ϱ-3/ϱ].
Multiplying the first equation of (1.1) by -Δv and integrating with respect to x on ℝ3, then using integration by parts, we obtain
(4.24)12ddt‖∇v(t)‖L22+(ν+κ)‖Δv‖L22=∫R3v⋅∇v⋅Δvdx-2κ∫R3(∇×w)⋅Δvdx.
Similarly, we get
(4.25)12ddt‖∇w(t)‖L22+γ‖Δw‖L22+(α+β)‖∇∇⋅w‖L22+4κ‖∇w‖L22=∫R3v⋅∇w⋅Δwdx-2κ∫R3(∇×v)⋅Δwdx.
Collecting (4.24) and (4.25) yields
(4.26)12ddt(‖∇v(t)‖L22+‖∇w(t)‖L22)+(ν+κ)‖Δv‖L22+γ‖Δw‖L22+(α+β)‖∇∇⋅w‖L22+4κ‖∇w‖L22=∫R3v⋅∇v⋅Δvdx-2κ∫R3(∇×w)⋅Δvdx+∫R3v⋅∇w⋅Δwdx-2κ∫R3(∇×v)⋅Δwdx.
Thanks to integration by parts and Cauchy inequality, we get
(4.27)-2κ∫R3(∇×w)⋅Δvdx-2κ∫R3(∇×v)⋅Δwdx≤κ‖Δv‖L22+4κ‖∇w‖L22.
It follows from (4.26)-(4.27) and integration by parts that
(4.28)12ddt(‖∇v(t)‖L22+‖∇w(t)‖L22)+ν‖Δv‖L22+γ‖Δw‖L22+(α+β)‖∇∇⋅w‖L22≤-∫R3∇v⋅∇v⋅∇vdx-∫R3∇v⋅∇w⋅∇wdx≜J1+J2.
In what follows, we estimateJi(i=1,2).
By (2.9) and Young inequality, we deduce that
(4.29)J1≤C‖∇v‖L33≤C‖∇v‖L23/2‖∇x̃∇v‖L2‖∇vx3‖L21/2≤ν4‖∇x̃∇v‖L22+C‖∇v‖L23‖∇vx3‖L2≤ν4‖∇x̃∇v‖L22+C(‖∇v‖L22+‖∇vx3‖L22)‖∇v‖L22,
where ∇x̃=(∂x1,∂x2).
By (2.9) and Young inequality, we have
(4.30)J2≤‖∇v‖L3‖∇w‖L32≤C‖∇v‖L21/2‖∇x̃∇v‖L21/3‖∇vx3‖L21/6‖∇w‖L2‖∇x̃∇w‖L22/3‖∇wx3‖L21/3≤ν4‖∇x̃∇v‖L22+C‖∇v‖L23/5‖∇vx3‖L21/5‖∇w‖L26/5‖∇∇x̃w‖L24/5‖∇wx3‖L22/5≤ν4‖∇x̃∇v‖L22+γ2‖∇x̃∇w‖L22+C‖∇v‖L2‖∇vx3‖L21/3‖∇w‖L22‖∇wx3‖L22/3≤ν4‖∇x̃∇v‖L22+γ2‖∇x̃∇w‖L22+C‖∇w‖L22(‖∇v‖L22+‖∇vx3‖L22+‖∇wx3‖L22),
where ∇x̃=(∂x1,∂x2).
Combining (4.28)–(4.30) yields
(4.31)ddt(‖∇v(t)‖L22+‖∇w(t)‖L22)+ν‖Δv‖L22+γ‖Δv‖L22+(α+β)‖∇∇⋅w‖L22≤C(‖∇v‖L22+‖∇w‖L22)(‖∇v‖L22+‖∇vx3‖L22+‖∇wx3‖L22).
From (4.31), Gronwall inequality, (4.6), and (4.23), we know that (v,w)∈L∞(0,T;H1(ℝ3)). Thus, (v,w) can be extended smoothly beyond t=T. We have completed the proof of Theorem 3.2.



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*Source: 456547-2011-08-13.xml*