# A Global Regularity Result for the 2D Generalized Magneto-Micropolar Equations

Authors: Hui Zhang
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we proved the global (in time) regularity for smooth solution to the 2D generalized magneto-micropolar equations with zero viscosity. When there is no kinematic viscosity in the momentum equation, it is difficult to examine the bounds on the any derivatives of the velocityJεuL2. In order to overcome the main obstacle, we find a new unknown quantity which is by combining the vorticity and the microrotation angular velocity; the structure of the system including the combined quantity obeys a Beale–Kato–Majda criterion. Moreover, the maximal regularity of parabolic equations together with the classic commutator estimates allows us to derive the Hs estimates for solutions of the system.


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## Body

## 1. Introduction

The standard 3D incompressible magneto-micropolar equations can be written as(1)∂tu+u⋅∇u−μ+κΔu+∇π=b⋅∇b+2κ∇×ω,∂tω+u⋅∇ω−γΔω−ν∇∇⋅ω+4κω=2κ∇×u,∂tb+u⋅∇b−ηΔb=b⋅∇u,∇⋅u=∇⋅b=0,u,ω,bx,0=u0x,ω0x,b0x.where  ux,t∈ℝ3,ωx,t∈ℝ3,bx,t∈ℝ3, and π=πx,t∈ℝ denote the velocity of the fluidthe microrotational velocity, magnetic field, and pressure, respectively. μ is the kinematic viscosity, κ is the vortex viscosity, γ and ν are spin viscosities, and η is the magnetic diffusivity. u0,ω0,b0 are the given initial data with ∇⋅u0=∇⋅b0=0.The magneto-micropolar equations have been extensively studied and applied by many engineers and physicists [1]. The mathematical study stared in the 1970s and has attracted considerable attentions. Rojas-Medar and Boldrin [2] proved the existence of weak solutions by the Galerkin method and, in 2D case, also proved the uniqueness of the weak solutions. Ortega-Torres and Rojas-Medar [3] established the local in time existence and uniqueness of strong solutions and proved global in time existence of strong solution for small initial data. However, whether the local strong solutions can exist globally or the global weak solution is regular and unique is an outstanding open problem. There are many interesting results about the theory of global well-posedness of solution to the Cauchy problem and initial-boundary-value problem for system (1) in [4–14] and the references therein.The 3D magneto-micropolar equations reduce 2D magneto-micropolar equations when(2)u=u1x1,x2,t,u2x1,x2,t,0,π=πx1,x2,t,b=b1x1,x2,t,b2x1,x2,t,0,ω=0,0,w3x1,x2,t.More explicitly, the 2D magneto-micropolar equations can be written as(3)∂tu+u⋅∇u−μ+κΔu+∇π=b⋅∇b+2κ∇×ω,∂tω+u⋅∇ω−γΔω+4κω=2κ∇×u,∂tb+u⋅∇b−ηΔb=b⋅∇u,∇⋅u=∇⋅b=0,u,ω,bx,0=u0x,ω0x,b0x,where we have written u=u1,u2 and ω is for ω3 for notational brevity. It is worth noting that, in the 2D case,(4)Ω=∇×u=∂1u2−∂2u1is a scalar function and ∇×ω=∂2w,−∂1ω is a vector function.The well-posedness problem about 2D magneto-micropolar equations and closely related equations such as the micropolar equationsb=0 has attracted considerable attention, for example, [15–18]. In [17], Dong et al. studied micropolar equations with partial dissipation case which involves no velocity dissipation. They introduced a new known quantity to obtain the existence and uniqueness of classical solutions.This paper aims at the fractional dissipation case, when (3) involves no velocity dissipation. More precisely, we study the existence and uniqueness of classical solutions to the 2D generalized magneto-micropolar equations:(5)∂tu+κu+∇π+u⋅∇u=b⋅∇b+2κ∇×ω,∂tω+u⋅∇ω−γΔω+4κω=2κ∇×u,∂tb+u⋅∇b+−Δβb=b⋅∇u,∇⋅u=∇⋅b=0,u,ω,bx,0=u0x,ω0x,b0x.We remark that the termκu does not play any significant role in the global regularity problem. It is kept in (5) simply to reflect the fact that the magneto-micropolar equations require the presence of the microrotational effect and microrotational inertia, namely, κ>0. We establish the following global regularity result for (5).Theorem 1.
Let1<β<2 and u0,ω0,b0∈Hsℝ2s>2 with ∇⋅u0=∇⋅b0=0. Then, the 2D magneto-micropolar equation (5) has a unique global solution u,ω,b satisfying, for ∀t>0,(6)u,ω,b∈L∞0,t;Hsℝ2;ω∈L20,t;Hs+1ℝ2;b∈L20,t;Hs+βℝ2.Remark 1.
It should be mentioned that, in [19], Fan and Zhao proved a similar result to MHD equations (when κ=0andω=0). However, due to the presence of microrotational term, the corresponding method is difficult to be used directly. To overcome this difficulty, we will introduce a new unknown term to obtain the desired result.Throughout this paper, we useLpℝ2 to denote the standard Lebesgue and Sobolev spaces, respectively. We use Hmℝ2 to denote the fLp norm of Lpℝ2. The letter f denotes various positive and finite constants.


## 2. Proof of Theorem 1

This section is devoted to the Proof of Theorem1. The existence and uniqueness of local smooth solutions can be done by a classical method on some fixed time interval C, see, for example, [20]. Thus, it is 0,T enough to established a priori estimates for any given t>0.Proposition 1.
Assumeu0,ω0,b0 obeys the assumptions stated in Theorem 1 with 1<β<2. Let u,ω,b be the corresponding solution; then, we have the global L2 bounds of(7)u,ω,bL22+γ∫0t∇ωτL22dτ+∫0tJβbτL22dτ≤C,for any t>0, which implies ∫0tbτL∞2dτ<+∞ due to β>1.Proof of Proposition1.
TakingL2 inner product of equations (5) and integrating by parts and using ∇⋅u=∇⋅b=0, we obtain(8)12ddtu,ω,bL22τ+κuτL22+4κωτL22+γ∇ωτL22+JβbτL22=0,where we have used the following fact:(9)∫ℝ2b⋅∇b⋅udx+∫ℝ2b⋅∇u⋅bdx=0,∫ℝ2∇×ω⋅udx+∫ℝ2∇×u⋅ω=0.
Integrating (8) in time from 0 to t implies (7).Now, we give the globalH1 bound for u,ω, and b.Proposition 2.
Under the same condition on Proposition1, then we have(10)Ω,∇ω,jL22t+γ∫0tΔωτL22dτ+∫0tJβjτL22dτ≤C,for any t>0.Proof of Proposition2.
There we denoteΩ=∇×u and j=∇×b; then, from the u-equation of (5), we have(11)∂tΩ+κΩ+u⋅∇Ω=b⋅∇j−2κΔω.
Taking theL2 inner product to equation (9) with Ω and integrating by parts yield(12)12ddtΩL22+κΩL22=∫ℝ2b⋅∇j⋅Ωdx−2κ∫ℝ2Δω⋅Ωdx=I1+I2.
Then, from the b-equation of (5), we have(13)∂tj+u⋅∇j+−Δβj=b⋅∇Ω+Q∇u,∇b,with(14)Q∇u,∇b=2∂1b1∂1u2+∂2u1+2∂2u2∂1b2+∂2b1.
Taking theL2 inner product to the equation (11) with j and integrating by parts yield(15)12ddtjL22+JβjL22=∫ℝ2b⋅∇Ω⋅jdx+∫ℝ2Q∇u,∇b⋅jdx=I3+I4.
Taking theL2 inner product to the ω−equation to (5) with −Δω and integrating by parts yield(16)12ddt∇ωL22+γΔωL22+4κ∇ωL22=∫ℝ2u⋅∇ω⋅Δωdx−2κ∫ℝ2Ω⋅Δωdx=I5+I6.
Because of the divergence-free conditions, we have(17)∫ℝ2b⋅∇j⋅Ωdx+∫ℝ2b⋅∇Ω⋅jdx=0.
We now estimateI2,I4 and I5,I6.
Through integrating by parts and applying Holder’s inequality, Young inequality, and Sobolev’s inequalities, we have(18)I2,I6≤ΩL2ΔωL2≤CΩL22+γ4ΔωL22,I4≤∫ℝ2∇u∇b2dx≤CΩL2jL22≤CΩ2bL∞2β−1/βJβjL21/β≤CΩ22β/2β−1bL∞2+12JβjL22≤CΩ22bL∞2+CbL∞2+12JβjL22,I5≤∫ℝ2∇u‖∇ω2dx≤CΩL2∇ωL42≤CΩL22∇ωL22+γ4∇ωL22,
Then, we have the following inequality:(19)12ddtΩ,∇ω,jL22+κΩL22+4κ∇ωL22+12JβjL22+γ2ΔωL22≤CΩL22∇ωL22+bL∞2+CbL∞2.
Integrating (19) in time from 0 to t and combining with Proposition 1 implies (20).Proposition 3.
Under the same condition on Proposition1, then we have(20)JβbL22+∫0tJ2βbL22dτ≤C,for any t>0.Proof of Proposition3.
Testingb− equation of (5) with −Δβb, we have(21)12ddtJβbL22+J2βbL22=∫b⋅∇u−u⋅∇bJ2βbdx≤bL∞∇uL2+uL4∇bL4J2βbL2=I1+I2.
We now estimateI1, due to Proposition 2,(22)I1≤CbL∞2+14J2βbL22.
Applying Sobolev’s inequality and Young inequality, we have(23)I2≤CJ2βbL21+3/4β≤C+14J2βbL22.
Collecting the estimates ofI1 and I2, we have(24)ddtJβbL22+J2βbL22≤C+CbL∞2.
Integrating (24) in time from 0 to t implies (20).Due toβ>1, (20) implies, for any p∈2,+∞,(25)b∈L∞0,t;L∞ℝ2;∫0t∇bτLq2dτ≤C,for∀t>0.Proposition 4.
Under the same condition on Proposition1, then corresponding solutions u,ω,b admits the following global bounds, for any q∈2,+∞ and any t>0, we have(26)Δω;−Δβb∈L20,t;Lqℝ2;Ω∈L∞0,t;Lqℝ2,for any t>0.We first recall the maximal regularity of parabolic equation [19, 21].Lemma 1.
For the parabolic equation,(27)∂tu+−Δβu=f,with the initial data ux,0=u0x. Let β≥1,2<q<∞, we have(28)∫0tJ2βuLq2dτ≤Cu0+∫0tfLq2dτ,for any t>0.From theb− equation of (5), we have(29)∂tb+−Δβb=b⋅∇u−u⋅∇b.Due toW1,pℝ2⟶L∞ℝ2,p>2, so we have(30)b⋅∇u−u⋅∇bLq≤bL∞+∇bq∇uLq.Then, from Lemma1, we have(31)∫0t−ΔβbτLq2dτ≤C∫0t‖bτ‖L∞2+∇bτq2∇uτLq2dτ.From theω−equation of (5), we have(32)∂tω−γΔω=2κΩ−4κω−u⋅∇ω.Then, from Lemma1, we have(33)∫0tΔωτp2dτ≤C∫0tΔωτp2∇uτ‖Lp2dτ+∫0tΩτp2dτ.Proof of Proposition4.
Testing (10), by Ωp−2p2<p<∞, we obtain(34)1pddtΩLpp≤CbL∞∇jLpΩLpp−1+CΔωLpΩLpp−1.
Then, we have(35)ddtΩLp2≤C∇jLp2+ΔωLp2+CΩLp2.
Thus, integrating the above inequality in time, we derive(36)ΩLp2≤C+∫0t∇jLp2+ΔωLp2+CΩLp2dτ≤C+C∫0T−ΔβbLp2+ΔωLp2+CΩLp2dτ≤C+C∫0TbL∞2+∇bLp2+∇ωLp2ΩLp2dτ.
Applying the Gronwall inequality, we have(37)ΩL∞0,t;Lp≤C.
Furthermore, combining (31), (33), and (37), we have(38)∫0t−Δβbτq2dx≤C;∫0TΔωτp2dτ≤C.Proposition 5.
Under the same condition on Proposition1, then corresponding solutions u,ω,b admits the following global bounds, for any t>0,(39)Ω∈L10,t;L∞ℝ2.Proof of Proposition5.
We setZ=Ω+2κ/γω; then, we have(40)∂tZ+u⋅∇Z+κ−4κ2γZ+8κ3γ2+6κ2γω=b⋅∇j.
Multiplying (40) by |Z|p−2Z with 2<p<∞ and integrating in ℝ2, we have(41)1pddtZtLpp≤∫ℝ2κ−4κ2γZ+8κ3γ2+6κ2γω+b⋅∇jZp−2Zdx≤CZLpp+CωLp+b⋅∇jLpZLpp−1.
Applying Gronwall inequality, one has(42)ZtLp≤CZ0Lp+C∫0TωτLp+b⋅∇jLpdτ,
We obtain, by lettingp⟶+∞,(43)ZtL∞≤CZ0L∞+C∫0tωτL∞+b⋅∇jL∞dτ≤C,and(44)ΩtL∞≤ZtL∞+ωtL∞.
Integrating (44) in time from 0 to t implies (39).Due to above estimates, we can prove (1). First, we recall commutator estimates [22] in the following. Let J=I−Δ1/2 denote the inhomogeneous differentiation operator; the proof of this lemma is based on some results due to Coifman and Meyer [23].Lemma 2.
Lets>0. Let p,p1,p3∈1,∞ and p2,p4∈1,∞ satisfy(45)1p=1p1+1p2=1p3+1p4.Then, for two constantsC1 and C2,(46)JsfgLp≤C1JsfLp1gLp2+JsgLp3fLp4,Jsfg−fJsgLp≤C2JsfLp1gLp2+Js−1gLp3∇fLp4.Proof of Theorem 1.
ApplyingJs with s>2 to system (5), taking the L2 inner product with Jsu,Jsω,Jsb, respectively, and then adding them up, we can obtain(47)12ddtJsu,Jsω,JsbL22+κJsuL22+4κJsωL22+γJsΛωL22+JsΛβbL22=2κ∫Js∇×ωJsu+Js∇×uJsωdx−∫Jsu⋅∇uJsudx+∫Jsb⋅∇bJsudx+∫Jsu⋅∇ωJsωdx−∫Jsu⋅∇bJsb+∫Jsb⋅∇uJsb=H1+⋯+H6.
Now, we estimateH1,H2,H3,H4,H5, and H6 one by one:(48)H1=2κ∫Js∇×ωJsu+Js∇×uJsωdx≤8κ2γJsuL22+γ2JsΛωL22,H2=∫Js,u⋅∇u⋅Jsudx≤C∇uL∞JsuL22.
Furthermore, we have(49)H3+H4+H5+H6≤C∇uL∞+∇ωL∞+∇bL∞Jsu,Jsω,JsbL22.
Combining these estimates yield(50)12ddtJsu,Jsω,JsbL22+κJsuL22+4κJsωL22+γ2JsΛωL22+JsΛβbL22≤C∇uL∞+∇ωL∞+∇bL∞Jsu,Jsω,JsbL22.
Combining with the above estimates and Gronwall’s inequality, we can obtain the desired globalHs bound.



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