# Global Regularity Criterion for the Magneto-Micropolar Fluid

Authors: Fanhui Meng; Gang Wang
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

We are concerned with  the magneto-micropolar  fluid equations inℝ3. Using Littlewood-Paley decomposition, we obtain an Osgood type global regularity criterion for the system.


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## Body

## 1. Introduction

In this paper, we consider the following magneto-micropolar fluid equations inℝ3:
(1)∂tu-(μ+χ)Δu+u·∇u-b·∇b+∇(p+b2)-χ∇×ω=0,∂tω-γΔω-κ∇divω+2χω+u·∇ω-χ∇×u=0,∂tb-νΔb+u·∇b-b·∇u=0,divu=divb=0,u(0,x)=u0(x),ω(0,x)=ω0(x),b(0,x)=b0(x),
where u(t,x)=(u1(t,x),u2(t,x),u3(t,x))∈ℝ3 denotes the velocity of the fluid at a point x∈ℝ3,t∈[0,T), ω(t,x)∈ℝ3andb(t,x)∈ℝ3, and p(t,x)∈ℝ denote, respectively, the microrotational velocity, the magnetic field, and the hydrostatic pressure. μ,χ,κ,γ,andν are positive numbers associated with 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,andb0 are initial data for the velocity, the angular velocity, and the magnetic field with properties divu0=0 and divb0=0.It is well known that the question of global existence or finite time blowup 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 magneto-micropolar fluid equations. This challenging problem has attracted significant attention. Therefore, it is interesting to study the global regularity criterion of the solutions for system (1). But there are few theories about regularity and blow-up criteria of magneto-micropolar fluid equations. Some blow-up criterion are obtained by Yuan [1] in 2010. His paper implies that most classical blow-up criteria of smooth solutions to Navier-Stokes or magneto-hydrodynamic equations also hold for magneto-micropolar fluid equations. In particular, using Fourier frequency localization, Yuan proved the Beale-Kato-Majda criterion only relying on ∇u; that is, if
(2)∫0T‍∥∇u(t)∥B˙∞,∞0dt<∞,
then the solution (u,ω,b) can be extended past time t=T. In 2008, Yuan [2] obtain the following blow-up criteria: if
(3)limε→0supj∈Z∫T-εT‍∥Δj(∇×u)(t)∥∞dt<∞,
then the solution (u,ω,b) can be extended t=T. Recently,  Xu [3] also studied the regularity of weak solutions to magneto-micropolar fluid equations in Besov spaces.In this paper, we establish a refined global regularity criterion by means of Osgood norm which improves the results (2) and (3). As we know, Osgood condition plays an important role in solving uniqueness of solutions to the incompressible fluids equations. This induces us to apply it to global regularity criterion problems of smooth solution. To achieve this goal, taking full advantage of Fourier frequency localization method and using the low-high decomposition technique, we show the following main results.Theorem 1.
Suppose that fors>1/2, (u,ω,b)∈C([0,T);Hs)∩C1((0,T);Hs)∩C((0,T);Hs+2) is the smooth solution to (1). If the following Osgood type condition:
(4)sup2≤q<∞∫0T‍∥S-q∇u(t)∥∞qlogqdt<∞,
then the solution (u,ω,b) can be extended past time t=T. Here one denotes that S-q:=∑k=-qqΔ˙k.Remark 2.
The Osgood type condition (4) is weaker than (2) and (3). Note that, for q∈[2,∞), we have
(5)∥S-q∇u∥∞qlogq≤∑k=-qq∥Δ˙k(∇u)∥∞qlogq≤C∥∇u∥B˙∞,∞0.


## 2. Preliminaries

The proof of the results presented in this paper is based on a dyadic partition of unity in Fourier variables, the so-called homogeneous Littlewood-Paley decomposition. So, we first introduce the Littlewood-Paley decomposition and review the so-called Beinstein estimate and commutator estimate, which are to be used in the proof of our theorem.Let𝒮(ℝ3) be the Schwartz class of rapidly decreasing functions. Given f∈𝒮(ℝ3), the Fourier transform of f is defined by
(6)f^(ξ)=(2π)-(3/2)∫ℝ3e-ix·ξf(x)dx.We consider thatχ,φ∈𝒮(ℝ3), respectively, support in B={ξ∈ℝ3,|ξ|≤4/3} and 𝒞={ξ∈ℝ3,3/4≤|ξ|≤8/3}, such that
(7)χ(ξ)+∑j≥0‍φ(2-jξ)=1,∀ξ∈ℝ3,∑j∈ℤ‍φ(2-jξ)=1,∀ξ∈ℝ3∖{0}.
Setting φj=φ(2-jξ), then suppφj∩suppφj′=∅ if |j-j′|≥2 and suppχ∩suppφj′=∅ if |j-j′|≥1. Let h=F-1φ and h~=F-1χ; the dyadic blocks are defined as follows:
(8)Δjf=φ(2-jD)f=23j∫ℝ3h(2jy)f(x-y)dy,Sjf=∑k≤j-1‍Δkf=23j∫ℝ3h~(2jy)f(x-y)dy,j∈ℤ.
Informally, Δj=Sj+1-Sj is a frequency projection to the annulus |ξ|≈2j, while Sj is frequency projection to the ball |ξ|≲2j.  The details of Littlewood-Paley decomposition can be found in Triebel [4] and Chemin [5]. Now Besov spaces in ℝ3 can be defined as follows:
(9)B˙p,qs={f∈𝒵′(ℝ3)∣∥f∥B˙p,qs=(∑j∈ℤ2jsq∥Δjf∥pq)1/q<∞},q≠∞,B˙p,∞s={f∈𝒵′(ℝ3)∣∥f∥B˙p,qs=supj∈ℤ2js∥Δjf∥p<∞},
where 𝒵' denotes the dual space of 𝒵={f∈𝒮;Dαf^(0)=0;∀α∈ℕnmulti-index}.Now we introduce well-known Bernstetin's Lemma and commutator estimate, the proof is omitted here, and we can find the details in Chemin [5], Chemin and Lerner [6], and Kato and Ponce [7].Lemma 3 (Bernstein's lemma).
Let1≤p≤q≤∞. Assume that f∈Lp, then there exist constants C,C1,andC2 independent of f,j, such that
(10)sup|α|=k∥∂αf∥q≤C2jk+3j(1/p-1/q)∥f∥psuppf^⊂{|ξ|≲2j},C12jk∥f∥p≤sup|α|=k∥∂αf∥p≤C22jk∥f∥psuppf^⊂{|ξ|≈2j}.Remark 4.
From the above Beinstein estimate, we easily know that inℝ3, for the Reisz transform Rk(k=1,2,3), it has for ∀1≤p≤q≤∞(11)∥RkΔju∥q≤C23j(1/p-1/q)∥u∥p.
If suppose vector valued funtion u be divergence free, by Biot Savard law ∇u=(-Δ)-1∇∇×v with v=∇×u and the boundedness of Reisz transform on Lp(1<p<∞), we have, there exist constants C independent u such that
(12)∥∇u∥p≤C∥v∥p,∀1<p<∞.
If the frequency of u is restricted to annulus |ξ|≈2j, then (11) implies that
(13)∥∇u∥p≤C∥v∥p,∀1≤p≤∞.Now we denote thatΛ=(I-Δ)1/2, which satisfies
(14)Λf^(ξ)=(1+|ξ|2)1/2f^(ξ).Λs(s∈ℝ) can be defined in the same way as follows:
(15)Λsf^(ξ)=(1+|ξ|2)s/2f^(ξ).
Using the perivious notation, we define the norm of Sobolev space Ws,p(16)∥f∥Ws,p≜∥Λsf∥Lp,
especially by Fourier transform, and  Hs≜Ws,2 can be defined as
(17)Hs≜{f∈𝒮′(ℝ3)∣∥f∥Hs<∞},
where
(18)∥f∥Hs≜∥Λsf∥L2(∫ℝ3(1+|ξ|2)s|f^(ξ)|2dξ)1/2.Lemma 5 (Commutator estimate).
Let1<p<∞,s>0; assume that f,g∈Ws,p, then there exists a constant C independent of f,g, such that
(19)∥[Λs,f]g∥Lp≤C(∥∇f∥Lp1∥g∥Ws-1,p2+∥f∥Ws,p3∥g∥Lp4)
with 1<p2,p3<∞, such that
(20)1p=1p1+1p2=1p3+1p4.
Here [Λs,f]g=Λs(fg)-fΛsg.


## 3. Proof of the Theorem1

First we go on with theH1 estimates of the solution (u,ω,b) under the condition (3). Denote  that H=∇×u,I=∇×ω,  and J=∇×b we take curl on both sides of (1); we get the following equation:
(21)∂tH-(μ+χ)ΔH+u·∇H-H·∇u-b·∇J+J·∇b-χ∇×I=0,∂tI-γΔI+2χI+u·∇I-H·∇ω-χ∇×H=0,∂tJ-νΔJ+u·∇J-H·∇b-b·∇H+J·∇u=2T(b,u)
with
(22)T(b,u)=(∂2b·∂3u-∂3b·∂2u∂3b·∂1u-∂1b·∂3u∂1b·∂2u-∂2b·∂1u)
which uses the fact ∇×∇divω=0.Multiplying the three equations with(H,I,J), respectively, integrating by parts over ℝ3 about the variable x, then adding the resulting equations yields that
(23)12ddt(∥H∥22+∥I∥22+∥J∥22)+(μ+χ)∥∇H∥22+γ∥∇I∥22+ν∥∇J∥22+2χ∥I∥22=∫ℝ3(H·∇)u·Hdx+∫ℝ3(H·∇)ω·Idx+∫ℝ3(J·∇)u·Jdx-∫ℝ3(J·∇)b·Hdx+∫ℝ3(H·∇)b·Jdx+2χ∫ℝ3(∇×H)·Idx+2∫ℝ3T(b,u)·Jdx=II1+II2+II3+II4+II5+II6+II7,
where we use the following fact due to the divergence free condition of u,b:
(24)∫ℝ3(u·∇)H·Hdx=∫ℝ3(u·∇)I·Idx=∫ℝ3(u·∇)J·Jdx=0,∫ℝ3(b·∇)J·Hdx+∫ℝ3(b·∇)H·Jdx=0,∫ℝ3(∇×H)·Idx=∫ℝ3(∇×I)·Hdx.Let us begin with estimatingII1 and II3. Using Littlewood-Paley decomposition to ∇u, we have
(25)II1=∑j<-N∫ℝ3(H·∇)Δju·Hdx+∑-N≤j≤N∫ℝ3(H·∇)Δju·Hdx+∑j>N∫ℝ3(H·∇)Δju·Hdx,=II11+II12+II13.
For the terms II11 and II12, using Hölder's inequality, Beinstein's inequality, and (12), (13), we obtain
(26)|II11|≤∥H∥22∑j<-N‍∥∇Δju∥∞≤C∥H∥22∑j<-N‍2(3/2)j∥ΔjH∥2≤C2-(3/2)N∥H∥23,|II12|≤∥H∥22∑-N≤j≤N‍∥∇Δju∥∞≤C∥H∥22∥S-N∇u∥∞,
for II13, we get
(27)|II13|≤∥H∥32∑j>N‍∥∇Δju∥3≤C∥H∥32∑j>N‍2j/2∥ΔjH∥2≤C∥H∥32(∑j>N‍2-(j/2)·2)1/2(∑j>N‍2j·2∥H∥22)1/2≤C2-(N/2)∥H∥2∥∇H∥22,
where we use the interpolation inequality
(28)∥H∥3≤C∥H∥21/2∥∇H∥21/2.
Summing up (26)-(27), we have
(29)|II1|≤C(2-(3/2)N∥H∥23+∥H∥22∥S-N∇u∥∞+2-(N/2)∥H∥2∥∇H∥22).II3 can be treated in the same way, and we decompose it as
(30)II3=∑j<-N∫ℝ3(J·∇)Δju·Jdx+∑-N≤j≤N∫ℝ3(J·∇)Δju·Jdx+∑j>N∫ℝ3(J·∇)Δju·Jdx,
then we obtain
(31)|II3|≤C(2-(3/2)N∥J∥22∥H∥2+∥J∥22∥S-N∇u∥∞+2-(N/2)∥J∥2∥∇J∥2∥∇H∥2).
Now we study II2, we decompose H by using Littlewood-Paley theory; that is,
(32)II2=∑j<-N∫ℝ3(ΔjH·∇)ω·Idx+∑-N≤j≤N∫ℝ3(ΔjH·∇)ω·Idx+∑j>N∫ℝ3(ΔjH·∇)ω·Idx,
then
(33)|II2|≤C(2-(3/2)N∥I∥22∥H∥2+∥I∥22∥S-N∇u∥∞+2-(N/2)∥I∥2∥∇I∥2∥∇H∥2).
Similarly for II4,II5, and II7, we have
(34)|II4|+|II5|+|II7|≤C(2-(3/2)N∥J∥22∥H∥2+∥J∥22∥S-N∇u∥∞+2-(N/2)∥J∥2∥∇J∥2∥∇H∥2).
Using Young's inequality, the term II6 can be written as
(35)|II6|≤2χ∥∇×H∥2∥I∥2≤χ2∥∇H∥22+2χ∥I∥22.
Summing up (29)–(35) and taking the sum into (23), by Young's inequality, we get
(36)ddt(∥H∥22+∥I∥22+∥J∥22)+(2μ+χ)∥∇H∥22+2γ∇I22+2ν∥∇J∥22≤C(2-(3/2)N(∥H∥23+∥I∥23+∥J∥23))+∥S-N∇u∥∞(∥H∥22+∥I∥22+∥J∥22)+2-(N/2)(∥H∥2+∥I∥2+∥J∥2)×(∥∇H∥22+∥∇I∥22+∥∇J∥22).
If we let 2-(N/2)(∥H∥2+∥I∥2+∥J∥2)≤min(μ,γ,ν), that is, if we choose
(37)N≥[2log2log+(Cmin(μ,γ,ν)(∥H∥2+∥I∥2+∥J∥2))]+1,
where [a] stands for the integral parts of a∈ℝ, log+(x)=log(x+e), then we have
(38)ddt(∥H∥22+∥I∥22+∥J∥22)+(μ+χ)∥∇H∥22+γ∥∇I∥22+ν∥∇J∥22≤C∥S-N∇u∥∞(∥H∥22+∥I∥22+∥J∥22)+C≤CfNNlogN(∥H∥22+∥I∥22+∥J∥22)+C≤CfNlog+(∥H∥2+∥I∥2+∥J∥2)×log+log+(∥H∥2+∥I∥2+∥J∥2)×(∥H∥22+∥I∥22+∥J∥22)+C,
where fN=∥S-N∇u∥∞/NlogN.Using Gronwall's inequality, we have(39)(∥H∥22+∥I∥22+∥J∥22)+∫0t‍[(μ+χ)∥∇H∥22+γ∥∇I∥22+ν∥∇J∥22]dt≤expexpexp(C∫0t‍fN(t′)dt′).
On the other hand, by multiplying (u,ω,b), it can be easily derived from magneto-micropolar fluid equation (1) that
(40)∥u∥22+∥ω∥22+∥b∥22+2μ∫0t‍∥∇u∥22t′+2γ∫0t‍∥∇ω∥22dt′+2ν∫0t‍∥∇b∥22dt′+2κ∫0t‍∥divω∥22dt′+2χ∫0t‍∥ω∥22dt′≤∥u0∥22+∥ω0∥22+∥b0∥22.
Equation (39) along with (40) implies that the H1 estimate of solution (u,ω,b).Next, we will show how to deduceHs estimates based on the H1 estimates. We apply operator Λs on the two sides of (1), multiply (Λsu,Λsω,Λsb) by the resulting equations and integrate the final form over ℝ3, and ge
(41)12ddt(∥Λsu∥22+∥Λsω∥22+∥Λsb∥22)+(μ+χ)∥∇Λsu∥22+γ∥∇Λsω∥22+ν∥∇Λsb∥22+κ∥divΛsω∥22+2χ∥Λsb∥22=-∫ℝ3Λs(u·∇u)Λsudx-∫ℝ3Λs(u·∇ω)Λsωdx-∫ℝ3Λs(u·∇b)Λsbdx+∫ℝ3Λs(b·∇b)Λsudx+∫ℝ3Λs(b·∇u)Λsbdx-2χ∫ℝ3Λs(∇×u)Λsωdx,
where we use the fact
(42)∫ℝ3Λs(∇×ω)Λsudx=∫ℝ3Λs(∇×u)Λsωdx.
Furthermore, the divergence free conditions of (u,b) imply that
(43)∫ℝ3(u·∇Λsu)Λsudx=∫ℝ3(u·∇Λsω)Λsωdx=∫ℝ3(u·∇Λsb)Λsbdx=0,∫ℝ3(b·∇Λsb)Λsudx+∫ℝ3(b·∇Λsu)Λsbdx=0,
then
(44)12ddt(∥Λsu∥22+∥Λsω∥22+∥Λsb∥22)+(μ+χ)∥∇Λsu∥22+γ∥∇Λsω∥22+ν∥∇Λsb∥22+κ∥divΛsω∥22+2χ∥Λsω∥22=-∫ℝ3[Λs,u]∇uΛsudx-∫ℝ3[Λs,u]∇ωΛsωdx-∫ℝ3[Λs,u]∇bΛsbdx+∫ℝ3([Λs,b]∇bΛsu+[Λs,b]∇uΛsb)dx+-2χ∫ℝ3Λs(∇×u)Λsωdx.=III1+III2+III3+III4+III5.
By Lemma 5, Hölder's inequality, Gagliardo-Nirenberg's inequality
(45)∥f∥Ws,4≤∥f∥Ws,21/4∥∇f∥Ws,23/4,
and Young's inequality
(46)ab≤1pap+1qbq,1p+1q=1,
we deduce that
(47)|III1|≤∥[Λs,u]∇u∥4/3∥Λsu∥4≤C(∥∇u∥2∥∇u∥Ws-1,4+∥u∥Ws,4∥∇u∥2)∥u∥Ws,4≤C∥∇u∥2∥u∥Hs1/2∥∇u∥Hs3/2≤C∥∇u∥24∥u∥Hs2+μ4∥∇u∥Hs2.
Similarly, we estimate III2,III3,andIII4 as follows:
(48)|III2+III3+III4|≤C(∥∇u∥24+∥∇ω∥24+∥∇b∥24)×(∥u∥Hs2+∥ω∥Hs2+∥b∥Hs2)+μ4∥∇u∥Hs2+γ2∥∇ω∥Hs2+ν2∥∇b∥Hs2.
Finally, we estimate the last term
(49)|III5|≤2χ∥Λs(∇×u)∥2∥Λsω∥2≤χ2∥∇u∥Hs2+2χ∥ω∥Hs2.
Summing up (47)–(49) with (44), we obtain
(50)ddt(∥u∥Hs2+∥ω∥Hs2+∥b∥Hs2)+(μ+χ)∥∇u∥Hs2+γ∥∇ω∥Hs2+ν∥∇b∥Hs2+κ∥divω∥Hs2≤C(∥u∥H14+∥ω∥H14+∥b∥H14)×(∥u∥Hs2+∥ω∥Hs2+∥b∥Hs2).
Using Gronwall's inequality we obtain
(51)(∥u∥Hs2+∥ω∥Hs2+∥b∥Hs2)+∫0t‍((μ+χ)∥∇u∥Hs2+γ∥∇ω∥Hs2+ν∥∇b∥Hs2+κ∥divω∥Hs2)(t′)dt′≤C(∥u0∥Hs2+∥ω0∥Hs2+∥b0∥Hs2)×exp(tsupt′∈[0,t)(∥u∥H14+∥ω∥H14+∥b∥H14)).
Hence by (39) and (51), we can get the Hs regularity at time t=T; that is, the smooth solution (u,ω,b) can be extended past time T.



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