Regularity Criteria for the Magneto-micropolar Fluid Equations in Terms of Direction of the Velocity

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1 Applied Mathematical Sciences, Vol. 8, 01, no. 71, HIKARI td, Regulaity Citeia fo the Magneto-micopola Fluid Equations in Tems of Diection of the Velocity Chunhong Tian Depatment of Basic Sciences, Nanhang Jincheng College Nanjing, Jiangsu 11156, China Copyight c 01 Chunhong Tian. This is an open access aticle distibuted unde the Ceative Commons Attibution icense, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. Abstact This pape is dedicated to study of the Cauchy poblem fo the Magnetomicopola fluid equations. We obtain a new egulaity citeion fo the system in tems of the diection of the velocity in the Moey-Campaanto space. Keywods: Regulaity citeion, Magneto-micopola fluid equations, Moey-Campaanto space 1 Intoduction and main esults In this pape, we ae concened with following Magneto-micopola fluid equations in R + R t u μ + χ)δu + u u b b + P + b ) χ ω =0, t ω γδω κ divω +χω + u ω χ u =0, t b νδb + u b b u =0, divu = divb =0, u0,x)=u 0 x),ω0,x)=ω 0 x),b0,x)=b 0 x), 1.1) whee ut, x) =u 1 t, x),u t, x),u t, x)) R denotes the velocity of the fluid at a point x R,t [0,T), ωt, x) R,bt, x) R and P t, x) R denote, espectively, the

2 5 Chunhong Tian mico-otational velocity, the magnetic field and the hydostatic pessue. μ, χ, κ, γ, ν ae positive numbes associated to popeties of the mateial: μ is the kinematic viscosity, χ is the votex viscosity, κ and γ ae spin viscosities, and 1 is the magnetic Reynold. ν u 0,ω 0,b 0 ae initial data fo the velocity, the angula velocity and the magnetic field with popeties divu 0 = 0 and divb 0 =0. When b = 0, the equation 1.1) educes to the micopola fluid system. Micopola fluid system was fist poposed by Einge[] in Using lineaization and an almost fixed point theeom, in 1988, ukaszewicz[] established the global existence of weak solutions with sufficiently egula initial data. In 1989, using the same technique, ukaszewicz[5] poved the local and global existence and the uniqueness of the stong solutions. In 010, Yuan [9] established egulaity citeia in oentzweak p ) space. When both ω = 0 and χ = 0, then the system1.1) educes to be the magneto-hydodynamic MHD) equations, which has been studied extensively in [10-17]. To the full system, Magneto-micopola fluid equations1.1), in 1977, Galdi and Rioneo[] stated the theoem of existence and uniqueness of stong solutions, but without poof. Ahmadi and Shaninpoo[1] studied the stability of solutions fo the system in 197. By using spectal Galekin method, in 1997, Rojas-Meda[6] established local existence and uniqueness of stong solutions. In 1998, Otega-Toes and Rojas-Meda[7] poved global existence of stong solutions with small initial data. Fo the weak solution, Rojas- Meda and Boldini[8] established the local existence in two and thee dimension by using Galekin method, and also poved the uniqueness in D case. Recently, Yuan[] and Xu [] studied the egulaity of weak solutions to magneto-micopola fluid equations in diffeent spaces epecially. In[18], A. Vasseu consideed egulaity citeion fo D Navie-Stokes equations in the diection of the velocity field ux,t) and showed ux,t) div u u ) p 0, ; q R ), with p + q 1, q 6, and p, 1.) then u is smooth on 0, ) R. Vey ecently, uo[19] descibed the egulaity citeion of weak solution to the MHD equations using the diection of the velocity field u. He showed that if the inital value u u 0,b 0 H 1 R ) with div u u ) p 0,T; q R ), with p + q 1, q 6, and p, 1.) and b α 0, ; β R ), with α + 1, β, β then u, b) is smooth on 0,T) R. Motivated by the above woks, we conside the Magneto-micopola fluid equations in the Moey-Campaanto space, which is defined in Section. We obtain a egulaity

3 Regulaity citeia 5 citeion fo Magneto-micopola fluid equations in tems of the diection of the velocity. Moe pecisely, we will pove Theoem 1.1. et u, ω, b) be a weak solution to Magneto-micopola fluid equations with u 0,ω 0,b 0, R ) q R ), q>. If the solution u, ω, b) of the system 1.1) satisfies the following condition div u u ) /1 ) 0,T; Ṁ,/ R )), 0 <<1, 1.6) then u, ω, b) is smooth on 0,T) R. Remak 1.1. The definition of weak solutions to the Magneto-micopola equations 1.1) can be found in [], hee we omit it. Peliminaies and emmas Fistly, we ecall the definition and some popeties of the space we ae going to use. This kind of space play an impotant ole in studying the egulaity of solutions to patial diffeential equations see [] and efeences theein). Now we ecall the definition of Moey-Campanato spaces. Definition.1. Fo 1 <p q +, the Moey-Campanato space M p,q R ) is defined as M p,q R )= { f p loc R ): f Ṁp,q = sup R /q /p f p Bx,R)) < }, sup x R R>0 whee Bx, R) denotes the ball of cente x with adius R. It is easy to check that fλ ) Ṁp,q = 1 λ /q f Ṁ p,q, λ > 0,.1) M p, R )= R ) fo 1 p..) Additionally, fo p / and 0 </ we have the following embedding elations: / R ) /, R ) whee p, denotes the weak p space. The second elation /, R ) M p, M p,/ R ),.) R ),.)

4 5 Chunhong Tian is shown as follows: f Ṁp,/ sup E 1 1/p p fy) dy) p f /, R )) E E ) = sup E p 1/p 1 fy) p dy E E ) 1/p = sup R {x R : fy) p >R} p/ R>0 = sup R {x R : fy) >R} / R>0 = f /,. emma.1. [0] Fo 0 </, the space ŻR ) is defined as the space of fx) loc R ) such that f Ż = sup g Ḃ 1,1 fg <..5) Then f M,/ R ) if and only if f ŻR ) with equivalence of noms. emma.. [1] Fo 0 <<1, we have f Ḃ,1 C f 1 f,.6) whee C only depends on. Remak.1. Fo p / and 0 </, by the embedding.) we can see that ou esults extend and impove the known esults in [18-19]. Remak.. Compaed with the known esult [19] fo the MHD equations, ou egulaity citeion only needs velocity field u. Theefoe, ou esult impoves the known esults in [19]. Poofs of the main esults In this section, we pove Theoem 1.1. Poof of Theoem 1.1. To pove the theoem we need the a pioi estimate. Fo this pupose, we take the inne poduct of the fist equation of 1.1) with u u and integate by pats, it can be deduced that dt u +μ + χ) u u + 1 μ + χ) u.1) P u u dx +χ w u u dx b u u) b dx, R R R whee we used the following elations by the divegence fee condition divu = 0: u u u udx = 1 u u dx =0, R R

5 Regulaity citeia 55 and Δu u udx = u u dx 1 u dx, R R R ω u udx = u ω udx ω u udx, R R R u u, u u. Similaly, we take the inne poduct of the second equation of 1.1) with ω ω and integate by pats, it can be deduced that dt ω + γ ω ω + γ ω + k divω +χ ω χ u ω ω dx. R.) Using an agument simila to that used in deiving the estimate.1)-.), it can be obtained fo the thid equation of 1.1) that dt b + b b + b b b b b) u dx. R Adding up.1),.) and.), then we obtain.) dt u + ω + b )+μ + χ) u u + 1 μ + χ) u + γ ω ω + γ ω + k divω +χ ω + b b + b b P u u dx +χ w u u dx +χ u ω ω dx R R R b u u) b dx + b b b) u dx R R I 1 + I + I + I + I 5..) Applying the Hölde inequality and the Young inequality fo I, it follows that I χ + μ u u + C u + ω )..5) Aguing similaly to above it can be deived fo I that I γ ω ω + C u + ω )..6)

6 56 Chunhong Tian Applying the divegence opeato div to the fist equation of 1.1), one fomally has P = R i R j u i u j b i b j ), i,j=1 whee R j denotes the j th Riesz opeato. By the boundedness of Riesz opeato and applying the Hölde inequality again to obtain that P q C u q + b q), fo 1 <q<. Since u, b) R ) 6 R ) and Ḣ1 R ) 6 R ), by the Hölde inequality and the Young inequality, we get Pu P Pu C u + b ) P u 6 C u + b ) u 6 + b 6 ) u C u + b ) u C u + b ) + χ + μ u 8 C u + b )+ χ + μ u 8. By.6) and.7) we find using Hölde inequality and Young inequality I 1 P u u R u u dx P u u div u R u ) dx Pu u div u ) u Pu u Ḃ,1 div u ) u Ṁ. Pu u 1 u div u ) u M. Pu u 1 ) u div u ) u Ṁ. Pu ) 1 u div u ) 1 ) 1 u Ṁ. u ).7).8) C u + b )+ χ + μ u + C u div u u ) hee we use the fact u div u u )= u u u. Next we estimate the tem I. As in [5], we have I b u u dx. R 1, Ṁ.

7 Regulaity citeia 57 Since u R ) 6 R ), using Cauchy inequality, genealized Hölde inequality, Gagliado- Nienbeg inequality and Sobolev imbedding theoem, we obtain I C b u u C b u + χ + μ u C b u 6 + χ + μ u C b u u + χ + μ.9) u C b b + χ + μ u. The last tem of.) can be teated in the same way, I 5 C b u b dx C b u + 1 R 8 b C b b..10) Putting.5),.6),.8),.9) and.10) into.), we obtain dt u + ω + b )+μ + χ u u + γ ω ω + γ ω +k divω +χ ω + b b C u + b + ω )+C u + b + ω ) div u C u + b + ω ) 1+ div u which gives that by the Gonwall inequality ) ) 1 u Ṁ., T ) 1, u Ṁ. 1+ div u.11) sup u + ω + b ) u 0 + ω 0 + b 0 C ) exp 0<t<T 0 u ).1) By standad aguments of continuation of local solutions, we conclude that the solutions ut, x),ωt, x),bt, x)) can be extended beyond t = T povided that div u ) u /1 ) 0,T; Ṁ,/ R )), 0 <<1. This completes the poof of Theoem 1.1. Refeences ) 1 Ṁ. [1] G. Ahmadi and M. Shahinpoo, Univesal stability of magneto-micopola fluid motions, Intenat. J. Engg. Sci., 1 197), [] A.C. Eingen, Theoy of micopola fluids, J. Math. Mech., ), ) dt.

8 58 Chunhong Tian [] G.P. Galdi and S. Rioneo, A note on the existence and uniqueness of solutions of the micopola fluid equations, Intenat. J. Engg. Sci., ), [] G. ukaszewicz, On nonstationay flows of asymmetic fluids, Rend. Accad. Naz. Sci. X Mem. Mat., 5) ), no.1, MR 90f:5165.Zbl [5] G. ukaszewicz, On the existence, uniqueness and asymptotic popeties fo solutions of flows of asymmetic fluids, Rend. Accad. Naz. Sci. X Mem. Mat., 5) ), no.1, MR 91d:517.Zbl [6] M.A. Rojas-Meda, Magneto-micopola fluid motion: Existence and uniqueness of stong solution, Math. Nach., ), [7] E.E. Otega-Toes and M.A. Rojas-Meda, Magneto-micopola fluid motion: Global existence of stong solutions, Abstact and Applied Analysis, 1999), [8] M.A. Rojas-Meda and J.. Boldini, Magneto-micopola fluid motion: Existence of weak solutions, Revista Mathmática Complutense, ), -60. [9] B. Yuan, On the egulaity citeia of weak solutions to the micopola fluid equations in oentz spaces Po. Ame. Math. Soc., ), [10] J. Wu, Bounds and new appoaches fo the D MHD equations, J. Nonlinea Sci., 1 00), [11] C. He and Z. Xin, On the egulaity of weak solutions to the magnetohydodynamic equations, J. Diff. Equations, 1 005), 5-5. [1] J. Wu, Regulaity esults fo weak solutions of the D MHD equations, Discete Cont. Dyn. S., 10 00), [1] Q. Chen and C. Miao, Existence theoem and blow-up citeion of stong solutions to the two-fluid MHD equation in R J. Diff. Equations, 9 007), [1] Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda Citeion fo the D Magneto-Hydodynamics Equations Commu. Math. Phys., ), [15] Q. Chen, C. Miao and Z. Zhang, On the Regulaity Citeion of Weak Solution fo the D Viscous Magneto-Hydodynamics Equations Commu. Math. Phys., 8 008), [16] M. Cannon, Q. Chen and C. Miao, A losing estimate fo the Ideal MHD equations with application to Blow-up citeion, SIAM J. Math. Anal., 8 007), [17] M. Cannon, C. Miao, N. Pioux and B. Yuan, The cauchy poblem fo the magnetohydodynamic system, Self-simila solutions of nonlinea PDE, Banach Cente Publications, Institute of mathematics, Polish Academy of Sciences, Waszawa 7 006), 59-9.

9 Regulaity citeia 59 [18] A. Vasseu, Regulaity citeion fo D Navie-Stokes equations in tems of the diection of the velocity, Appl. Math., 5 009), 7-5. [19] Y. uo, Regulaity of weak solutions to the magneto-hydodynamic equations in tems of the diection of the velocity, Electon. J. Diffe. Equ., 1 009), 1-7. [0] P.G. emaié-rieusset, The Navie-Stokes equations in the citical Moey- Campanato space, Rev. Mat. Ibeoam., 007), [1] S. Machihaa, T. Ozawa, Intepolation inequalities in Besov spaces, Poc. Ame. Math. Soc ), [] S. Gala, A note on the uniqueness of mild solutions to the Navie-Stokes equations, Ach. Math.Basel), 8 007), 8-5. [] B. Yuan The egulaity of weak solutions to magneto-micopola fluid equations, Acta. Math. Sci., 0 010), [] F. Xu, Regulaity citeion of weak solution fo the D Magneto-micopola fluid equations in Besov spaces, Commun. Nonlinea Sci. Nume. Simulat, 17 01), 6-. [5] Y. Zhou, Regulaity citeia fo the MHD equations in tems of the pessue, J. Non- inea. Mech., 1 006), Received: Apil 7, 01

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