# A Regularity Criterion for the Magneto-Micropolar Fluid Equations inB˙∞,∞−1

Authors: Zhihao Tang; Gang Wang; Haiwa Guan
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

The paper is dedicated to study of the Cauchy problem for the magneto-micropolar fluid equations in three-dimensional spaces. A new logarithmically improved regularity criterion for the magneto-micropolar fluid equations is established in terms of the pressure in the homogeneous Besov spaceB˙∞,∞−1.


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## Body

## 1. Introduction

This paper concerns with the regularity of weak solutions to the magneto-micropolar fluid equations in three dimensions as(1)∂tv-(μ+χ)Δv+v·∇v-b·∇b+∇(p+b2)-χ∇×ω=0,∂tω-γΔω-κ∇divω+2χω+v·∇ω-χ∇×v=0,∂tb-νΔb+v·∇b-b·∇v=0,divv=divb=0,v(0,x)=v0(x),ω(0,x)=ω0(x),b(0,x)=b0(x),
where v(t,x)=(v1(t,x),v2(t,x),v3(t,x))∈ℝ3 denotes the velocity of the fluid at a point x∈ℝ3,t∈[0,T), ω(t,x)∈ℝ3,b(t,x)∈ℝ3, and p(t,x)∈ℝ denote, respectively, the microrotational velocity, the magnetic field, and the hydrostatic pressure. μ,χ,κ,γ,ν are positive numbers associated to properties of the material: μ is the kinematic viscosity, χ is the vortex viscosity, κ and γ are spin viscosities, and 1/ν is the magnetic Reynold. u0,ω0,b0 are initial data for the velocity, the angular velocity, and the magnetic field with properties divu0=0 and divb0=0. For more detailed background, we refer the readers to [1–3].As we know, the problem of global regularity or finite time singularity for the weak solutions of the magneto-micropolar fluid equations model with large initial data still remains unsolved since (1) includes the 3D Navier-Stokes equations. It is of interest that the regularity of the weak solutions is under preassumption of certain growth conditions. There are a lot of lectures to study the regularity of weak solutions of the magneto-micropolar fluid equations (see, [4–6]). The purpose of this paper is to establish a new logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure in Besov space B˙∞,∞-1. Now we state the main results as follows.Theorem 1.
Let(v0(x),ω0(x),b0(x))∈H1(ℝ3). Let T>0 and (v,ω,b) be a weak solution to the system (1). If the pressure filed P satisfies the following condition:
(2)∫0T‍∥P(t,·)∥B˙∞,∞-121+ln(e+∥P(t,·)∥B˙∞,∞-1)dt<∞,
then the weak solution (v,ω,b) is regular on [0,T].Remark 2.
Since the spaceB˙∞,∞-1 is wider than ℳ˙2,3, so our result resolves the limit case r=1 in [7], which greatly improves the result in [7].Remark 3.
Since the spaceB˙∞,∞-1 is wider than L3/r,∞, hence our result extends and improves the recent results given by [4].


## 2. Preliminaries and Lemmas

Throughout this paper, we introduce some function spaces, notations, and important inequalities.LetetΔ denote the heat semigroup defined by
(3)etΔf=Kt*f,Kt=(4πt)-3/2exp(-|x|24t)
for t>0 and x∈ℝ3, where * denotes the convolution of functions defined on ℝ3.We now recall the definition of the homogeneous Besov space with negative indicesB˙∞,∞-α on ℝn and the homogeneous Sobolev space H˙qα of exponent α>0. It is known (p. 192 of [8]) that f∈𝒮′(ℝ3) belongs to B˙∞,∞-α if and only if etΔ∈L∞ for all t>0 and tα/2∥etΔ∥∞∈L∞(0,∞;L∞). The norm of B˙∞,∞-α is defined, up to equivalence, by
(4)∥f∥B˙∞,∞-α=supt>0(tα/2∥etΔ∥∞).We introduce now the homogeneous Sobolev spaceH˙qα(ℝ3), which is defined by the set of functions f∈Lr(ℝ3),1/r=(1/q)-(s/3) such that (-Δ)s/2f∈Lq(ℝ3). This space is endowed with the norm
(5)∥f∥H˙qα=∥(-Δ)s/2f∥Lq,
and when q=2, we just let H˙2α(ℝ3)=H˙α(ℝ3). Additionally, we have the following inclusion relations (see, e.g., [9]):
(6)H˙1/2(ℝ3)⊂L3(ℝ3)⊂L3,∞(ℝ3)⊂B˙∞,∞-1(ℝ3),H˙1/2(ℝ3)⊂L3(ℝ3)⊂ℳ˙2,3(ℝ3)⊂B˙∞,∞-1(ℝ3)
with continuous injection.Lemma 4 (see [10]).
Let1<p<q<∞ and s=α((q/p)-1)>0. Then there exists a constant C depending only on α, p, and q such that for all f∈H˙pα(ℝ3)∩B˙∞,∞-α(ℝ3),
(7)∥f∥Lq≤C∥(-Δ)s/2f∥Lpp/q∥f∥B˙∞,∞-α1-(p/q).In particular, fors=1,p=2, and q=4, we get α=1 and
(8)∥f∥L4≤C∥f∥H˙11/2∥f∥B˙∞,∞-11/2.Lemma 5 (see [11]).
Letf∈W1,s(ℝ3)(s≥1), and r≥1, then there exists a positive constant C independent of f such that
(9)∥f∥Lγ≤C∥f∥L21-α∥∇f∥L2α,
where
(10)α=(1/r)-(1/γ)(1/3)-(1/s)-(1/r).


## 3. Proof of Theorem1

For given initial data(v0,ω0,b0)∈H1(ℝ3), the weak solution is the same as the local strong solution (v,ω,b) in a local interval (0,T) as in the discussion of Navier-Stokes equations. For the uniqueness and existence of local strong solution, we refer to [1]. Thus, it proves that Theorem 1 is reduced to establish a priori estimates uniformly in (0,T) for strong solutions. With the use of the a priori estimates, the local strong solution (v,ω,b) can be continuously extended to t=T by a standard process to obtain global regularity of the weak solution. Therefore, we assume that the solution (v,ω,b) is sufficiently smooth on (0,T).Proof of Theorem1.
We show that Theorem1 holds under condition (1). To prove the theorem, we need the L4-estimate. For this purpose, taking the inner product of the first equation of (1) with |u|2u and integrating by parts, it can be deduced that
(11)14ddt∥u∥L44+(μ+χ)∥|∇u||u|∥L22+12(μ+χ)∥∇|u|2∥L22≤2∫ℝ3‍|P||u|2|∇u|dx+3χ∫ℝ3‍|w||u|2|∇u|dx-∫ℝ3‍|b||∇(|u|2u)||b|dx,
where we used the following relations by the divergence-free condition divu=0:
(12)∫ℝ3‍u·∇u·|u|2udx=12∫ℝ3‍u·∇|u|4dx=0,∫ℝ3‍Δu·|u|2udx=-∫ℝ3|∇u|2‍|u|2dx-12∫ℝ3|∇|u|2|2‍dx,∫ℝ3‍∇×ω·|u|2udx=-∫ℝ3|u|‍2ω·∇×udx-∫ℝ3‍ω·∇|u|2×udx,|∇×u|≤|∇u|,|∇|u||≤|∇u|.
Similarly, taking the inner product of the second equation of (1) with |ω|2ω and integrating by parts, it can be inferred that
(13)14ddt∥ω∥L44+γ∥|∇ω||ω|∥L22+γ2∥∇|ω|2∥L22+k∥divω∥L22+2χ∥ω∥L44≤3χ∫ℝ3‍|u||ω|2|∇ω|dx.
Using an argument similar to that used in deriving the estimate (11)–(13), it can be obtained for the third equation of (1) that
(14)14ddt∥b∥L44+∥|∇b||b|∥L22+2∥∇|b||b|∥L22≤∫ℝ3‍|b||∇(|b|2b)||u|dx.
Adding up (11), (13), and (14), then we obtain
(15)14ddt(∥u∥L44+∥ω∥L44+∥b∥L44)+(μ+χ)∥|∇u||u|∥L22+12(μ+χ)∥∇|u|2∥L22+γ∥|∇ω||ω|∥L22+γ2∥∇|ω|2∥L22+k∥divω∥L22+2χ∥ω∥L44+∥|∇b||b|∥L22+2∥∇|b||b|∥L22≤2∫ℝ3‍|P||u|2|∇u|dx+3χ∫ℝ3‍|w||u|2|∇u|dx+3χ∫ℝ3‍|u||ω|2|∇ω|dx-∫ℝ3‍|b||∇(|u|2u)||b|dx+∫ℝ3‍|b||∇(|b|2b)||u|dx≜I1+I2+I3+I4+I5.
Applying the Hölder inequality and the Young inequality for I2, it follows that
(16)I2≤χ+μ2∥|∇u||u|∥L22+C(∥u∥L44+∥ω∥L44).
Arguing similarly to above, it can be derived for I3 that
(17)I3≤γ2∥|∇ω||ω|∥L22+C(∥u∥L44+∥ω∥L44).
Considering the term I1, by virtue of the Cauchy inequality, we have
(18)I1≤12∫ℝ3‍|∇|v|2|2dx+12∫ℝ3|P|2‍|v|2dx.
Let us bound the integral (1/2)∫ℝ3|P|2‍|v|2dx. Applying the divergence operator div to the first equation of (1), one formally has P=∑i,j=13‍RiRj(uiuj-bibj), where Rj denotes the jth Riesz operator. By the Calderon-Zygmund inequality, we have
(19)∥∇P∥L2≤C(∥|v||∇v|∥L2+∥|b||∇b|∥L2).
With the help of (8) and (19), by the Hölder inequality and the Young inequality, we deduce that
(20)12∫ℝ3‍|P|2|v|2dx≤12∥P∥L42∥v∥L42≤C∥∇P∥L2∥P∥B˙∞,∞-1∥v∥L42≤C(∥|v||∇v|∥L2+∥|b||∇b|∥L2)∥P∥B˙∞,∞-1∥v∥L42=(∥|v||∇v|∥L2+∥|b||∇b|∥L2)(C∥P∥B˙∞,∞-12∥v∥L44)1/2≤14(∥|v||∇v|∥L22+∥|b||∇b|∥L22)+C∥P∥B˙∞,∞-12∥v∥L44.
So the term I1 can be estimated as
(21)I1≤12∫ℝ3|∇|v|2|2dx+14(∥|v||∇v|∥L22+∥|b||∇b|∥L22)+C∥P∥B˙∞,∞-12∥v∥L44.
Next we have the following estimate for the term I4:
(22)I4≤∫ℝ3|b|2|u||∇|u|2|dx.
Since u∈L2(ℝ3)∩L6(ℝ3) and using Cauchy inequality, generalized Hölder inequality, Gagliardo-Nirenberg inequality, and Sobolev imbedding theorem, we obtain
(23)I4≤C∥|b|2|u|∥L2∥∇|u|2∥L2≤C∥|b|2|u|∥L22+χ+μ4∥∇|u|2∥L22≤C∥|b|2∥L62∥u∥L32+χ+μ4∥∇|u|2∥L22≤C∥∇|b|2∥L22∥u∥L2∥∇u∥L2+χ+μ4∥∇|u|2∥L22≤C∥|b|∇|b|∥L22+χ+μ4∥∇|u|2∥L22.
The last term of (15) can be treated in the same way as
(24)I5≤C∫ℝ3‍|b|2|u||∇|b|2|dx≤C∥|b|2|u|∥L22+18∥∇|b|2∥L2≤C∥|b|∇|b|∥L22.
Inserting the estimates (15) and (21) into (14), it follows that
(25)ddt(∥v∥44+∥ω∥44+∥b∥44)≤C∥P∥B˙∞,∞-12∥v∥L44+C(∥v∥44+∥ω∥44+∥b∥44)≤C∥P∥B˙∞,∞-12(∥v∥44+∥ω∥44)+C(∥v∥44+∥ω∥44+∥b∥44)≤C(1+∥P(t,·)∥B˙∞,∞-121+ln(e+∥P(t,·)∥B˙∞,∞-1))×[1+ln(e+∥P(t,·)∥B˙∞,∞-1)](∥v∥44+∥ω∥44+∥b∥44)≤C(1+∥P(t,·)∥B˙∞,∞-121+ln(e+∥P(t,·)∥B˙∞,∞-1))×[1+ln(e+∥P(t,·)∥L3)](∥v∥44+∥ω∥44+∥b∥44)≤C(1+∥P(t,·)∥B˙∞,∞-121+ln(e+∥P(t,·)∥B˙∞,∞-1))×[1+ln(e+∥v(t,·)∥L62)](∥v∥44+∥ω∥44+∥b∥44)≤C(1+∥P(t,·)∥B˙∞,∞-121+ln(e+∥P(t,·)∥B˙∞,∞-1))×[1+ln(e+y(t))](∥v∥44+∥ω∥44+∥b∥44),
where y(t) is defined by
(26)y(t)=:supT0≤s≤t(∥Λ3v∥L22+∥Λ3ω∥L22+∥Λ3b∥L22).
Applying Gronwall's inequality on (25) for the interval [T0,t], one has
(27)supT0≤s≤t(∥v∥44+∥ω∥44+∥b∥44)≤C0exp(Cε(1+ln(e+y(t))))≤C0exp(2Cεln(e+y(t)))≤C0(e+y(t))2Cε
provided that
(28)∫T0t‍∥P(t,·)∥B˙∞,∞-121+ln(e+∥P(t,·)∥B˙∞,∞-1)ds<ε≪1,
where C0 is a positive constant depending on T0.
Next we will estimate theL2-norm of ∇v, ∇ω, and ∇b. We multiply both sides of the first equation of (1) by (-Δv), the second equation of (1) by (-Δω), and the third equation of (1) by (-Δb), by integration by parts over ℝ3, we get
(29)12ddt∥∇v∥L22+∥Δv∥L22=∫ℝ3‍(v·∇)v·Δvdx+∫ℝ3‍(b·∇)b·Δvdx-∫ℝ3‍curlωΔvdx≤∥v∥L4∥∇v∥L4∥Δv∥L2+∥b∥L4∥∇b∥L4∥Δv∥L2+∥∇ω∥L2∥Δv∥L2≤∥v∥L4∥v∥L21/8∥Δv∥L27/8∥Δv∥L2+∥b∥L4∥b∥L21/8∥Δb∥L27/8∥Δv∥L2+116∥Δv∥L22+C∥∇ω∥L22≤18∥Δv∥L22+18∥Δω∥L22+18∥Δb∥L22+C∥b∥L416∥b∥L22+C∥v∥L416∥v∥L22+C∥ω∥L22,(30)12ddt∥∇ω∥L22+∥Δω∥L22+∥∇divω∥L22+2∥∇ω∥L22=∫‍(v·∇)ω·Δωdx-∫‍curlvΔωdx≤∥v∥L4∥∇ω∥L4∥Δω∥L2+∥∇v∥L2∥Δω∥L2≤∥v∥L4∥ω∥L21/8∥Δω∥L27/8∥Δω∥L2+C∥v∥L21/2∥Δv∥L21/2∥Δω∥L2≤18∥Δv∥L22+18∥Δω∥L22+C∥v∥L416∥ω∥L22+C∥v∥L22,(31)12ddt∥∇b∥L22+∥Δb∥L22=∫ℝ3‍(v·∇b)Δbdx-∫ℝ3‍(b·∇v)Δbdx≤∥v∥L4∥∇b∥L4∥Δb∥L2+∥b∥L4∥∇v∥L4∥Δb∥L2≤∥v∥L4∥b∥L21/8∥Δb∥L27/8∥Δv∥L2+∥b∥L4∥v∥L21/8∥Δv∥L27/8∥Δb∥L2≤18∥Δv∥L22+18∥Δb∥L22+C∥v∥L416∥b∥L22+C∥b∥L416∥v∥L22,
where we have used the Gagliardo-Nirenberg inequality:
(32)∥∇f∥L4≤C∥f∥L21/8∥Δf∥L27/8.
Combining (29), (30), and (31) and using the definition of the weak solution, we deduce that
(33)∥∇v∥L22+∥∇ω∥L22+∥∇b∥L22≤CC0(e+y(t))6Cε(t-T0)+∥∇v(·,T0)∥L22+∥∇ω(·,T0)∥L22.
Finally we go to the estimate for H3-norm of v, ω, and b. In the following calculations, we will use the following commutator estimate due to Kato and Ponce [12]:
(34)∥Λs(fg)-fΛsg∥Lp≤(∥∇f∥Lp1∥Λs-1g∥Lq1+∥Λsf∥Lp2∥g∥Lq2),
with s>0, Λs=(-Δ)s/2 and (1/p)=(1/p1)+(1/q1)=(1/p2)+(1/q2). Taking the operation Λ3 on both sides of (1), then multiplying them by Λ3v, Λ3ω, and Λ3b, and integrating by parts over ℝ3, we have
(35)12ddt∫ℝ3‍|Λ3v|2+|Λ3ω|2+|Λ3b|2dx+∫ℝ3‍|Λ4v|2dx+∫ℝ3‍|Λ4ω|2dx+∫ℝ3‍|Λ4b|2dx+∫ℝ3‍|Λ3divv|2dx+2∫ℝ3‍|Λ3ω|2dx=-∫ℝ3‍[Λ3(v·∇v)-v·∇Λ3v]Λ3vdx+∫ℝ3‍Λ3curlω·Λ3vdx-∫ℝ3‍[Λ3(v·∇ω)-v·∇Λ3ω]Λ3ωdx+∫ℝ3‍Λ3curlv·Λ3ωdx+∫ℝ3‍[Λ3(b·∇b)-b·∇Λ3b]Λ3vdx-∫ℝ3‍[Λ3(v·∇b)-v·∇Λ3b]Λ3bdx+∫ℝ3‍[Λ3(b·∇v)-b·∇Λ3v]Λ3bdx≡A1+A2+A3+A4+A5+A6+A7.
Hence A1 can be estimated as
(36)A1≤C∥∇v∥L3∥Λ3v∥L32≤C∥∇v∥L213/2∥Λ3v∥L21/4∥Λ4v∥L25/3≤16∥Λ4v∥L22+C∥∇v∥L213/2∥Λ3v∥L23/2,
where we used (33) with s=3,p=3/2,p1=q1=p2=q2=3 and the following inequalities:
(37)∥∇v∥L3≤C∥∇v∥L23/4∥Λ3v∥L21/4,∥Λ3v∥L3≤C∥∇v∥L21/6∥Λ4v∥L25/6.
If we use the existing estimate (31) for T0<t<T, (36) reduces to
(38)A1≤16∥Λ4v∥L22+CC0(e+y(t))(3/4)+(39/2)Cε.
Using (37) again, we have
(39)A3+A5+A6+A7≤16(∥Λ4v∥L22+∥Λ4ω∥L22+∥Λ4b∥L22)+CC0(e+y(t))(3/4)+(39/2)Cε.
For A2 and A4, we have
(40)A2+A4≤16(∥Λ4v∥L22+∥Λ4ω∥L22)+C(∥Λ3v∥L22+∥Λ3ω∥L22)≤16(∥Λ4v∥L22+∥Λ4ω∥L22)+CC0(e+y(t)).
Inserting the above estimates (38)–(40) into (35), we obtain
(41)ddt∫ℝ3‍|Λ3v|2+|Λ3ω|2dx≤CC0(e+y(t))(3/4)+(39/2)Cε+CC0(e+y(t)).
Gronwall's inequality implies the boundness of H3-norm of v, ω, and b provided that 39Cε<(1/2), which can be achieved by the absolute continuous property of integral (2). This completes the proof of Theorem 1.



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