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 velocity . 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 estimates for solutions of the system.
1. Introduction
The standard 3D incompressible magneto-micropolar equations can be written aswhere , and denote the velocity of the fluidthe microrotational velocity, magnetic field, and pressure, respectively. is the kinematic viscosity, is the vortex viscosity, and are spin viscosities, and is the magnetic diffusivity. are the given initial data with .
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
More explicitly, the 2D magneto-micropolar equations can be written aswhere we have written and is for for notational brevity. It is worth noting that, in the 2D case,is a scalar function and is a vector function.
The well-posedness problem about 2D magneto-micropolar equations and closely related equations such as the micropolar equations 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:
We remark that the term 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, . We establish the following global regularity result for (5).
Theorem 1. Let and with . Then, the 2D magneto-micropolar equation (5) has a unique global solution satisfying, for ,
Remark 1. It should be mentioned that, in [19], Fan and Zhao proved a similar result to MHD equations (when ). 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 use to denote the standard Lebesgue and Sobolev spaces, respectively. We use to denote the norm of . The letter denotes various positive and finite constants.
2. Proof of Theorem 1
This section is devoted to the Proof of Theorem 1. The existence and uniqueness of local smooth solutions can be done by a classical method on some fixed time interval , see, for example, [20]. Thus, it is enough to established a priori estimates for any given .
Proposition 1. Assume obeys the assumptions stated in Theorem 1 with . Let be the corresponding solution; then, we have the global bounds offor any , which implies due to .
Proof of Proposition 1. Taking inner product of equations (5) and integrating by parts and using , we obtainwhere we have used the following fact:Integrating (8) in time from 0 to implies (7).
Now, we give the global bound for , and .
Proposition 2. Under the same condition on Proposition 1, then we havefor any .
Proof of Proposition 2. There we denote and ; then, from the u-equation of (5), we haveTaking the inner product to equation (9) with and integrating by parts yieldThen, from the b-equation of (5), we havewithTaking the inner product to the equation (11) with and integrating by parts yieldTaking the inner product to the equation to (5) with and integrating by parts yieldBecause of the divergence-free conditions, we haveWe now estimate and .
Through integrating by parts and applying Holder’s inequality, Young inequality, and Sobolev’s inequalities, we haveThen, we have the following inequality:Integrating (19) in time from 0 to and combining with Proposition 1 implies (20).
Proposition 3. Under the same condition on Proposition 1, then we havefor any .
Proof of Proposition 3. Testing equation of (5) with , we haveWe now estimate , due to Proposition 2,Applying Sobolev’s inequality and Young inequality, we haveCollecting the estimates of and , we haveIntegrating (24) in time from 0 to implies (20).
Due to , (20) implies, for any ,
Proposition 4. Under the same condition on Proposition 1, then corresponding solutions admits the following global bounds, for any and any , we havefor any .
We first recall the maximal regularity of parabolic equation [19, 21].
Lemma 1. For the parabolic equation,with the initial data . Let , we havefor any .
From the equation of (5), we have
Due to , so we have
Then, from Lemma 1, we have
From the equation of (5), we have
Then, from Lemma 1, we have
Proof of Proposition 4. Testing (10), by , we obtainThen, we haveThus, integrating the above inequality in time, we deriveApplying the Gronwall inequality, we haveFurthermore, combining (31), (33), and (37), we have
Proposition 5. Under the same condition on Proposition 1, then corresponding solutions admits the following global bounds, for any ,
Proof of Proposition 5. We set ; then, we haveMultiplying (40) by with and integrating in , we haveApplying Gronwall inequality, one hasWe obtain, by letting ,andIntegrating (44) in time from 0 to implies (39).
Due to above estimates, we can prove (1). First, we recall commutator estimates [22] in the following. Let denote the inhomogeneous differentiation operator; the proof of this lemma is based on some results due to Coifman and Meyer [23].
Lemma 2. Let . Let and satisfy
Then, for two constants and ,
Proof of Theorem 1. Applying with to system (5), taking the inner product with , respectively, and then adding them up, we can obtainNow, we estimate , and one by one:Furthermore, we haveCombining these estimates yieldCombining with the above estimates and Gronwall’s inequality, we can obtain the desired global bound.
Data Availability
All data used to support the findings of the study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
H. Zhang was partially supported by NSFC (no. 31971185) and Anhui Education Bureau (no. KJ2019A0556).