GLOBAL REGULARITY OF THE TWO-DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM WITH ZERO ANGULAR VISCOSITY

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1 GLOBAL REGULARITY OF THE TWO-DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM WITH ZERO ANGULAR VISCOSITY KAZUO YAMAZAKI Abstract. We study the two-dimensional magneto-micropolar fluid system. Making use of the structure of the system, we show that with zero angular viscosity the solution triple remains smooth for all time. Keywords: Magneto-micropolar fluid, micropolar fluid, global regularity, regularity criteria, Besov spaces. Introduction The magneto-micropolar fluid (MMPF) system consists of the following euations: u t + (u )u r(b )b + (π + r b ) + (µ + χ)λ r u = χ( w), (a) ρ w t + ρ(u )w + w = χw + (α + β) divw + χ( u), γλr (b) b t + (u )b (b )u + νλr3 b =, (c) u = b =, (u, w, b)(x, ) = (u, w, b )(x), (d) where we denoted u, w, b, π the velocity, micro-rotational velocity, the magnetic and the hydrostatic pressure fields respectively. We also denoted physically meaningful uantities: r = M ReRm where M is the Hartmann number, Re the Reynolds number, Rm the magnetic Reynolds number, χ the vortex viscosity, µ the kinematic viscosity, ρ the microinertia, α, β, γ the angular viscosities, ν = Rm all of which we assume to be positive taking into account of conditions such as Clausius-Duhem ineuality. Finally, we denoted fractional Laplacians defined through Fourier transform by Λ ri f(ξ) = ξ ri ˆf(ξ), i =,, 3. Hereafter let us write t = t, i = x i and also assume ρ =. Let us discuss the rich history of mathematical and physical study concerning the MMPF system and its related systems. Firstly, the MMPF system at w b The author expresses gratitude to Professor Jiahong Wu and Professor David Ullrich for their teaching. MSC : 35B65, 35Q35, 35Q86 Department of Mathematics, Oklahoma State University, 4 Mathematical Sciences, Stillwater, OK 7478, USA

2 KAZUO YAMAZAKI reduces to the Navier-Stokes euations (NSE). The NSE has ample engineering applications in fluid mechanics and has been investigated mathematically with much intensity. Secondly, the MMPF system at χ =, w reduces to the magnetohydrodynamics (MHD) system which describes the motion of electrically conducting fluids and has broad applications in applied sciences such as astrophysics, geophysics and plasma physics (cf. [3]). The mathematical analysis of the MHD system has attracted much attention in particular concerning the global regularity issue in two-dimensional case (cf. [3, 4, 7, 4, 7, 3, 3, 33, 38]). Thirdly, the MMPF system at b reduces to the micropolar fluid (MPF) system: t u + (u )u + π + (µ + χ)λ r u = χ( w), t w + (u )w + γλ r w = χw + (α + β) divw + χ( u), (a) (b) u =, (u, w)(x, ) = (u, w )(x). (c) The microfluids and micropolar fluids were introduced in [3, 4] respectively. In particular, the micropolar fluids represent the fluids consisting of bar-like elements, e.g. anisotropic fluids, such as liuid crystals made up of dumbbell molecules and animal blood. The study of this system was continued by many (e.g. [5, 6, 8, 9,,, 8, 9, 9, 3, 37]). In particular, the authors in [] obtained global regularity result of the two-dimensional MPF system with zero angular viscosity and r =. Finally, the MMPF system (a)-(d) was considered in [] in which the authors obtained Serrin-type stability criteria. This system has also found much applications and attraction by mathematicians (cf. [6,,, 5, 8, 3, 35, 36, 39]). In particular, the authors in [5] obtained the global existence of weak solution triple (u, w, b), global existence of strong solution triple (u, w, b) in case initial data is small if dimension is three, while the uniue weak solution triple (u, w, b) if dimension is two, both with r = r = r 3 =. We now consider, following the work of [, ], the two-dimensional problem by setting and denote by u = (u, u, ), w = (,, w 3 ) Ω := u = u u, j =: b = b b, w = ( w, w). We present our results concerning the following special case of () with x R : t u + (u )u (b )b + (π + b ) = (µ + χ) u + χ( w), t w + (u )w = χw + χ( u), t b + (u )b (b )u = ν b. (3a) (3b) Theorem.. For every (u, w, b ) H s (R ), s >, there exists a uniue solution triple (u, w, b) to (3a)-(3c) such that u, b C([, ); H s (R )) L ([, ); H s+ (R )), w C([, ); H s (R )). (3c)

3 TWO DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM 3 Remark.. () Theorem. improves previous global regularity results with angular viscosity (cf. []) and extend those in [] from MPF system to the MMPF system. Furthermore, it is also an extension of the global wellposedness result of the two-dimensional classical MHD system (cf. [3]). () In three-dimension, employing standard energy method on (a)-(d), one can show that the global regularity is attained as long as r + N, min{r + r, r + r 3 } + N, N = 3, 4 (cf. [6]). Hence, the result of Theorem. at r = r 3 =, r = is a significant improvement, attainable due to the advantage of the structure of the system upon taking curls, very similarly to the recent developments in the two-dimensional generalized MHD system. (3) Local existence proof can be performed in many ways, for example using mollifiers following the work on the NSE. Because the authors in [] proved the local existence in the case of the MPF system in detail and one can follow their work line by line, just adding the magnetic field euation, we omit this proof. Let us use the notation A a,b B, A a,b B to imply that there exists a nonnegative constant c that depends on a, b such that A cb, A = cb and briefly discuss the difficulty of this problem before we present our proof. Firstly, we take L -inner products with (u, w, b) on (3a)-(3c) respectively to obtain due to the incompressibility of u, t u L + (µ + χ) u L = (b )b u + χ ( w) u t w L + χ w L = χ ( u)w t b L + ν b L = (b )u b In sum, using the fact that ( w) u = ( u)w, we obtain t( u L + w L + b L ) + (µ + χ) u L + χ w L + ν b L χ w L u L χ w L + χ u L by the incompressibility of b, Hölder s and Young s ineualities. Therefore, after absorbing the right hand side, integrating in time we obtain sup t [,T ] ( u L + w L + ) T b L (t) + u L + b L dτ u,w,b. (4) We will obtain via a commutator estimate a regularity criteria that in order to prove Theorem., it suffices to show the time integrability of Ω L, w L, j L (See Proposition 3.). Now taking a curl on (3a) leads to t Ω (µ + χ) Ω = (u )Ω + (b )j χ w. (5)

4 4 KAZUO YAMAZAKI Due to the lack of angular viscosity, we have no obvious way to handle χ w. The key observation in [] was that by defining ( ) χ Z := Ω w, (6) µ + χ we can take advantage of its evolution in time governed by t Z = (µ + χ) Z (u )Z c Z + c w + (b )j (7) where we denoted c := χ µ + χ, c := χ µ + χ χ 3 (µ + χ). This leads to the following L p -estimate of Z after multiplying (7) by Z p Z, integrating in space, using the incompressibility of u and Hölder s ineuality: p t Z p L w p L p Z p L + (b )j p L p Z p L. p Dividing by Z p Lp, we obtain taking limit p, t Z L w L + (b )j L w L + b L j L. (8) On the other hand, from (3b) we can estimate similarly p t w p L + p χ w p L χ Ω p L p w p L Z p L p w p L + p w p L p. by the incompressibility of u, Hölder s ineuality and (6). Dividing by w p L p taking limit p lead to t w L Z L + w L. (9) In sum of (8) and (9) we obtain t ( Z L + w L ) Z L + w L + b L j L. Thus, in the absence of (b )j, the authors in [] were able to show the global regularity for the two-dimensional MPF system with zero angular viscosity. If we replace the diffusive term b by Λ r3 b so that this becomes familiar to the recent developments in the two-dimensional generalized MHD system (cf. [3]), then this problem in our case reuires, according to (4), r 3 > 3 considering a Sobolev embedding H r3 (R ) L (R ). A natural idea to reduce the lower bound on r 3 is to improve the bound on b. To create some cancellations, it is optimal to estimate Ω and j together. Because taking a curl on (3c) gives t j ν j = (u )j + (b )Ω + [ b ( u + u ) u ( b + b )], () we have the L -estimates of Ω and j as follows: t Ω L + (µ + χ) Ω L = (b )jω χ wω, t j L + ν j L = (b )Ωj + [ b ( u + u ) u ( b + b )]j. and

5 TWO DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM 5 Considering the term χ wω, we have no choice but to try the estimate on w simultaneously. However, taking L -inner products of (3b) with w leads to t w L + χ w L = u w w + χ Ω w, after integration by parts and the first nonlinear term becomes problematic. The novelty of this manuscript is the observation that by defining Z in (6), a certain bound of j will lead to a bound of Z, which in turn leads to the bound of w, and then Ω. In the next section, we set up notations and state key lemmas. Thereafter we prove our result. The proof consists of two parts: a regularity criteria for the solution triple to (3a)-(3c) using a commutator estimate and an a priori estimate making use of Z in (6).. Preliminaries We state some key lemmas and facts that will be crucial in our proofs. Lemma.. (cf. [7]) Let f be divergence-free vector field such that f L p, p (, ). Then f L p c p p curl f L p. The next lemma consists of variations of the classical result from []; for the proofs of these specific ineualities we refer readers to the Appendices of [3, 34]: Lemma.. Suppose f L (R ) H s (R ), s >. Then f L c ( f L + curl f L log ( + f H s) + ), () f L c ( f L + f H log ( + f H s) + ). () Let us recall the notion of Besov spaces (cf. [7]). We denote by S(R ) the Schwartz class functions and S (R ), its dual. We define S to be the subspace of S in the following sense: S = {φ S, φ(x)x l dx =, l =,,,...}. R Its dual S is given by S = S/S = S /P where P is the space of polynomials. For k Z we define A k = {ξ R : k < ξ < k+ }. It is well-known that there exists a seuence {Φ k } S(R ) such that supp ˆΦ k A k, ˆΦk (ξ) = ˆΦ ( k ξ) or Φ k (x) = k Φ ( k x) and k= Conseuently, for any f S, ˆΦ k (ξ) = { if ξ R \ {}, if ξ =.

6 6 KAZUO YAMAZAKI k= Φ k f = f. To define the homogeneous Besov spaces, we set k f = Φ k f, k =, ±, ±,... With such we can define for s R, p, [, ], the homogeneous Besov spaces where f Ḃs p, = Ḃp, s = {f S : f Ḃs < }, p, { ( k (ks k f L p) ) if <, sup k ks k f L p if =. To define the inhomogeneous Besov spaces, we let Ψ C (R ) be such that = ˆΨ(ξ) + ˆΦ k (ξ), Ψ f + Φ k f = f, k= for any f S. With that, we set k= if k, k f = Ψ f if k =, Φ k f if k =,,,..., and define for any s R, p, [, ], the inhomogeneous Besov spaces where f B s p, = For any s R, < p <, we have Bp, s = {f S : f B s p, < }, { ( k= (ks k f L p) ), if <, sup k< ks k f L p if =. B s p,min{p,} W s,p B s p,max{p,}, Ḃs p,min{p,} Ẇ s,p Ḃs p,max{p,}. Moreover, Bony s paraproduct decomposition will be used freuently (cf. [7]) fg = T f g + T g f + R(f, g) where T f g = k S k f k g, R(f, g) = k f k g, S k = m. k,k : k k m:m k 3. Commutator estimate and regularity criteria In this section, we obtain a regularity criteria for the solution (u, w, b) to (3a)-(3c) using a commutator estimate, namely the following:

7 TWO DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM 7 Proposition 3.. Let s >, f H s (R ) L (R ), g H s (R ) L (R ), g =. Then for any k 3, [ k, g i i ]f L c k ks ( g L f H s + g H s f L ). (3) Moreover, for s > if additionally f L (R ), then for any k, [ k, g i i ]f L c k ks ( g L f H s + g H s f L ) (4) where {c k } l k. Proof. The ineuality (3) is shown in []; we focus on (4). We first write [ k, g i i ]f = k (T g i i f + T if g i + R(g i, i f)) (T g i i k f + T i k f g i + R(g i, i k f)) =[ k, T g i i ]f + k (T if g i ) + k (R(g i, i f)) T i k f g i R(g i, i k f) where we used the Bony s paraproduct decomposition. Furthermore, we write [ k, T g i i ]f = k We rewrite this as k = = = = k : k k k : k k k : k k k : k k k : k k k : k k S k g i k i f S k g i k i f k : k k k : k k (S k g i ) k ( k i f) [(S k g i ) k ( k i f) k (S k g i k i f)] R k Φ ( k y)[ ( ) k Φ ( k y) y i R 3k R (5) (S k g i ) k ( k i f). τ S k g i (x τy)dτ] k f(x y)dy y i [ τ S k g i (x τy)dτ] k f(x y)dy y S k g i (x τy)dτ Φ ( k y) ( k y) i k f(x y)dy. by definition of k and integration by parts using divergence-free property of g. Thus, [ k, T g i i ]f L k : k k S k g i L 3k x Φ ( k x) i ( k x) by Young s ineuality for convolution. Hence, because Φ S(R ), L k f L [ k, T g i i ]f L g L k f L c k ks g L f H s (6) where we used that under restriction of k k, we may replace k by k modifying constants. Next,

8 8 KAZUO YAMAZAKI k (T if g i ) L S k f L k g L k : k k (7) f L k g L c k ks f L g H s by Hölder s ineuality. Next, k R(g i, i f) L k g i k i f L f L ks k :k k +3 k,k k +3 (k k )s k s k g L c k ks f L g H s (8) by Hölder s ineuality. Next, T i k f g i L = ( k ( i k f)) k g i L k k :k k k :k k k i f k g i L c k ks f L g H s (9) where we used that k k f = if k k > and Hölder s ineuality. Finally, R(g i, i k f) L k g i k i k f L k,k : k k () f L k g L c k ks f L g H s by Hölder s ineualities. obtain Taking L -norm on (5) and considering (6)-(), we [ k, g i i ]f L c k ks ( f L g H s + g L f H s). This completes the proof of Proposition 3.. Proposition 3.. Suppose (u, w, b ) H s (R ), s > and its corresponding solution triple (u, w, b) to (3a)-(3c) in time interval [, T ] satisfies T Ω L + w L + j Ldτ where Ω = u, j = b. Then (u, w, b) H s (R ) for all time t [, T ]. Proof. We apply k, k and take L -inner products of (3a)-(3c) with ( k u, k w, k b) respectively and sum to obtain

9 TWO DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM 9 t( k u L + kw L + kb L ) + (µ + χ) k u L + χ kw L + ν kb L = k ((u )u) k u (u ) k u k u + k ((b )b) k u (b ) k b k u k ((u )w) k w (u ) k w k w k ((u )b) k b (u ) k b k + k ((b )u) k b (b ) k u k b + χ k w k ( u) () where we used the incompressibility of u and b. For k 3, we now bound t( k u L + kw L + kb L ) + (µ + χ) k u L + χ kw L + ν kb L = [ k, u ]u k u + [ k, b ]b k u [ k, u ]w k w [ k, u ]b k b + [ k, b ]u k b + χ k w k ( u) [ k, u ]u L k u L + [ k, b ]b L k u L + [ k, u ]w L k w L + [ k, u ]b L k b L + [ k, b ]u L k b L + χ k w L k u L c k ks (( u L u H s + b L b H s) k u L + ( u L w H s + u H s w L ) k w L + ( u L b H s + b L u H s) k b L ) + χ k w L + χ ku L by Hölder s ineualities, (4) on the first, second, fourth and fifth commutators and (3) on the third and Young s ineualities. Absorbing the last two terms, multiplying by ks, k 3, s >, we obtain t ks ( k u L + kw L + kb L ) + (µ + χ) ks k u L + νks k b L c k ks ( k u L + k w L + k b L ) ( u H s + w H s + b H s + )( u L + b L + ) + c k ks u H s w L k w L.

10 KAZUO YAMAZAKI For the case k =,,,, we use the fact that k is a convolution operator, Ψ, Φ k S(R ), Hölder s and Young s ineualities for convolution and (4) to obtain k= k= ks k ((u )u) k u + k ((u )b) k b + k ((b )b) k u k ((b )u) k b + χ ks ( u u L u L + b b L u L + u w L w L + u b L b L + b u L b L + w L u L ). k ((u )w) k w k w k ( u) Thus, we can multiply () by ks, sum over k and apply Hölder s ineualities to obtain t ( u H s + w H s + b H s) + (µ + χ) u H s + ν b H s ( + c k l )( u H s + w H s + b H s + )( u L + b L + ) + ( + c k l ) u H s w L w H s. () For the last term on the right hand side, we take Young s ineualities to obtain c( + c k l ) u H s w L w H s µ + χ u H + s c w L w Hs. (3) We apply (3) to (), absorb the first term on the right hand side of (3) to obtain t ( u H s + w H s + b H s) ( u H s + w H s + b H s + )( u L + w L + b L + ). By () of Lemma. applied to u L, () of Lemma. applied to b L and (4) we obtain t log ( + u H s + w H s + b H s) log ( + u H s + w H s + b H s)( + Ω L + w L + j L ). Gronwall s ineuality completes the proof of Proposition A priori estimate We consider Z from (6) and first obtain L -estimates. Proposition 4.. Suppose (u, w, b) solves (3a)-(3c) in time interval [, T ]. Then sup t [,T ] ( Z L + Ω L + j L )(t) + T Z L + j Ldτ.

11 TWO DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM Proof. We take L -inner products (7) and () with (Z, j) respectively to obtain t Z L +(µ + χ) Z L + c Z L c w L Z L + (b )j ( ( ) ) χ Ω w, µ + χ t j L +ν j L = (b )Ωj + [ b ( u + u ) u ( b + b )]j. (4) (5) where we used (6). We sum (4) and (5) and bound t( Z L + j L ) + (µ + χ) Z L + ν j L + c Z L Z L + b L j L w L + b L 4 u L j L 4 Z L + b L j 3 L + j L j L u L ν j L + c( Z L + + j L Ω L ) ν j L + c( Z L + + j L ( Z L + w L )) ν j L + c( Z L + j L )( + j L ) by Hölder s and Gagliardo-Nirenberg ineualities, (4), Lemma., Young s ineualities and (6). By Gronwall s ineuality and (4), this implies sup t [,T ] ( Z L + j L ) (t) + T Z L + j Ldτ. Because we have the bound on w(t) L from (4), by (6) we obtain the bound on Ω(t) L. Thus, the proof of Proposition 4. is complete. We obtain slightly higher integrability now: Proposition 4.. Suppose (u, w, b) solves (3a)-(3c) in time interval [, T ]. Then for any 4, sup ( Z L + Ω L + j L )(t). (6) t [,T ] Proof. We assume < 4 below as the case = is already done in the Proposition 4.. We multiply (7) by Z Z, integrate in space to obtain t Z L (µ + χ) Z Z Z + c Z L =c w Z Z + (b )j Z Z. For the dissipative term, we integrate by parts to obtain (7)

12 KAZUO YAMAZAKI (µ + χ) On the other hand, Z Z Z = 4(µ + χ)( ) Z L. (8) Z L ( ) = Z L 4( ) Z ( L 4 ) Z ( ) L Z ( L ( ) ) where we used the Gagliardo-Nirenberg ineuality and Proposition 4.. This implies Now Z L ( ) Z ( ) L. (9) c w Z Z + (b )j Z Z c w L Z ( ) b ( Z) Z j L ( ) c Z ( ) ( ) ( ) L b Z Z j ( ) (µ + χ)( ) Z L + c + b Z j by Hölder s ineuality, (4), (9), integration by parts and Young s ineuality. We can absorb the dissipative term whereas for the second term we estimate b Z j b L Z L j L b L j L Z L j ( ) L ( j ) L (3) +( j L ( Z L + ) (3) by Hölder s and Gagliardo-Nirenberg ineualities, (4) and Proposition 4.. Thus, we have after absorbing, from (7), (8), (3) and (3), t Z (µ + χ)( ) L + Z L ( j L + )( Z L + ) (3) by Young s ineuality. Thus, by Proposition 4. and Gronwall s ineuality applied to (3) we obtain sup Z(t) L. (33) t [,T ] Next, we multiply (3b) by w w, integrate in space to obtain t w L + χ w L Ω L w L Z L w L + w L ( + w L ) by Hölder s ineualities, (6) and (33). This implies by Gronwall s ineuality and hence by (6) and (33), sup w(t) L t [,T ]

13 TWO DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM 3 sup Ω(t) L. (34) t [,T ] Finally, we multiply () by j j and integrate in space to estimate t j L ν j j j = (b )Ω j j + [ b ( u + u ) u ( b + b )] j j. On the diffusive term, we apply integration by parts, ν j j j = ν( ) j j j = ν( ) j j (35) (36) and estimate the first non-linear term of (35) by integration by parts and Young s ineuality as follows: (b )Ω j j = ( ) ν( ) 4 b j j Ω + c b j Ω. j j We can absorb the diffusive term whereas for the second term of (37), c b j Ω b L j L Ω L (37) b L j L j L ( + j L )( j L + ) (38) by Hölder s and Gagliardo-Nirenberg ineualities, (34) and (4). Moreover, we can also write On the other hand, ν( ) j j ( j L = j L j 4 = L 4 ν( )4 j L. (39) ) j L j ( ) L by the Gagliardo-Nirenberg ineualities and Proposition 4. and therefore j L j 4( ) L. (4) Now we estimate the second integral in (35): [ b ( u + u ) u ( b + b )] j j b L u L j L j L j L j L Ω L j 4( ) L j ν( ) L j L + c( j L + ) (4)

14 4 KAZUO YAMAZAKI by Hölder s ineualities, Lemma., (34), (4) and Young s ineualities. Therefore, from (35)-(39), (4), after absorbing we obtain t j ν( ) j j L + ν( ) + 4 j L (+ j L )( j L +). By Proposition 4., Gronwall s ineuality completes the proof of Proposition 4.. We now show that the criteria according to Proposition 3. is satisfied Bound of w(t) L. Firstly, we take L -inner products on (7) with Z to obtain t Z L + (µ + χ) Z L + c Z L = (u )Z Z c w Z (b )j Z u L Z L Z L + w L Z L + b L j L Z L u 4 L u 3 4 L 3 Z L Z L + Z L + b 4 L b 3 4 L 3 j L Z L µ + χ Z L + c( + Z L )( + j L ) by Hölder s ineualities, (4), the Gagliardo-Nirenberg and Young s ineualities and Proposition 4.. After absorbing, by Gronwall s ineuality we obtain sup t [,T ] Z(t) L + T Z Ldτ (4) due to Proposition 4.. Next, we multiply (3b) by w p w, integrate in space to obtain p t w p L + p χ w p L Ω p L p w p L Z p L p w p L + p w p L p by Hölder s ineualities and (6). Dividing by w p, we obtain w(t) L p w L pe ct + Taking limit p, we obtain t L p t e c(t τ) Z L pdτ t + e T Z L pdτ. T w(t) L T + e T Z L dτ. Therefore, by Proposition 4., (4) and Sobolev embedding, we obtain sup w(t) L. (43) t [,T ] 4... Bound of Ω(t) L. By (4), (43) and (6), we immediately obtain T Ω(t) L dτ T Z(t) L dt + T w(t) L dt. (44)

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