Nonlinear Analysis 74 (11) 5 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na A regularity criterion for the 3D magneto-micropolar fluid equations in Triebel Lizorkin spaces Zujin Zhang a,, Zheng-an Yao a, Xiaofeng Wang b a Department of Mathematics, Sun Yat-sen University, Guangzhou 5175, China b College of Mathematics, Guangzhou University, Guangzhou 516, China a r t i c l e i n f o a b s t r a c t Article history: Received 6 July 1 Accepted 14 November 1 MSC: 35Q35 76W5 35B65 Keywords: Magneto-micropolar fluid equations Conditional regularity Triebel Lizorkin spaces Beal Kato Majda criterion We consider the regularity criterion for the 3D magneto-micropolar fluid equations in Triebel Lizorkin spaces. It is proved that if u L, p T; Ḟ q,q/3 with p + 3 =, 3/ < q, q then the solution remains smooth in (, T). As a corollary, we obtain the classical Beal Kato Majda criterion, that is, the condition u L 1 (, T; Ḃ, ), ensures the smoothness of the solution. Crown Copyright 1 Published by Elsevier Ltd. All rights reserved. 1. Introduction We consider the magneto-micropolar fluid equations in : t u + (u ) u (µ + χ) u (b ) b + p + b χ ω =, t ω γ ω κ div ω + χω + u ω χ u =, t b ν b + (u ) b (b ) u =, u = b =, u(x, ) = u (x), ω(, x) = ω (x), b(, x) = b (x). Here u = u(x, t) represents the velocity field, b = b(x, t) represents the magnetic field, ω = ω(x, t) represents the microrotational velocity; p denotes the hydrodynamic pressure; µ > is the kinematic viscosity, χ > is the vortex viscosity, κ > and γ > are spin viscosities, 1/ν (with ν > ) is the magnetic Reynolds; while u, b, ω are the corresponding initial data with div u = div b =. This system is of interest for various reasons. For example, it includes some well-known equations, say the Navier Stokes equations (ω = b = ) and the MHD equations (ω = ). Moreover, it has similar scaling properties and energy estimates as the Navier Stokes and MHD equations. We believe that the regularity theory of system (1.1) can improve the understanding of the Navier Stokes and MHD equations. System (1.1) was first proposed by Galdi and Rionero [1]. The existence of global-in-time weak solutions were then established by Rojas-Medar and Boldrini [], while the local strong solutions and global strong solutions for the small initial (1.1) Corresponding author. E-mail addresses: uia.china@gmail.com (Z. Zhang), mcsyao@sysu.edu.cn (Z.-a. Yao), wangxiaofeng514@tom.com (X. Wang). 36-546X/$ see front matter Crown Copyright 1 Published by Elsevier Ltd. All rights reserved. doi:1.116/j.na.1.11.6
Z. Zhang et al. / Nonlinear Analysis 74 (11) 5 1 data were considered, respectively, by Rojas-Medar [3] and Ortega-Torres and Rojas-Medar [4]. However, whether the local strong solutions can exist globally or the global weak solution is regular and unique is an outstanding open problem. Hence there are many regularity criteria to ensure the smoothness of solutions; see [5 15] for the Navier Stokes equations, [16 7] for the MHD equations, [8] for the magneto-micropolar fluid equations. Motivated by Gala [8], we consider the regularity criteria for system (1.1). Before stating the precise result, let us recall the definition of weak solutions (see []). Definition 1.1. Let T >, (u, ω, b ) L ( ) with div u = div b =. A measurable -valued triple (u, ω, b) is said to be a weak solution of system (1.1) on [, T] if the following conditions hold: 1. (u, ω, b) L (, T; L ( )) L (, T; H 1 ( ));. system (1.1) is satisfied in the sense of distributions; 3. the energy inequality, that is (u, ω, b) L (t) + (µ + χ) + χ u L (s)ds + γ ω L (s)ds + ν b L (s)ds ω L (s)ds (u, ω, b ) L. (1.) Now our regularity criterion for system (1.1) reads Theorem 1.1. Let (u, ω, b ) L (R 3 ) with div u = div b =. Assume that the triple (u, ω, b) is a weak solution to system (1.1). If u L, p T; Ḟ q,q/3 with p + 3 =, 3/ < q, q then the solution is smooth on (, T). (1.3) Remark 1.1. If we take p = 1 in Theorem 1.1, and notice that the classical Riesz transformation is bounded in Ḃ,, then the classical Beal Kato Majda criterion for the magneto-micropolar fluid equations is obtained, that is, the condition u L 1 (, T; Ḃ, ) ensures the smoothness of the solution. The function spaces that appeared in Theorem 1.1 will be introduced in Section. And in Section 3, we shall prove Theorem 1.1. Throughout this whole paper, the usual Lebesgue space L p ( ) (1 p ) is endowed with the norm L p. For a Banach space X, we do not distinguish it with its vector analogues X 3 ; however, all vector- or tensor-valued functions are printed in boldface. A constant C which may depend on the initial data may change from line to line.. Preliminaries We first introduce the Littlewood Paley decomposition. Let S( ) be the Schwartz class of rapidly decreasing functions. For f S( ), its Fourier transform F f = ˆf is defined by ˆf (ξ) = f (x)e ix ξ dx. Let us choose a nonnegative radial function ϕ S( ) such that 1, if ξ 1, ˆϕ(ξ) 1, ˆϕ(ξ) =, if ξ, and let ψ(x) = ϕ(x) 3 ϕ(x/), ϕ j (x) = 3j ϕ( j x), ψ j (x) = 3j ψ( j x), j Z. For j Z, the Littlewood Paley projection operators S j and j are, respectively, defined by S j f = ϕ j f, j f = ψ j f. (.4) (.5)
Z. Zhang et al. / Nonlinear Analysis 74 (11) 5 Observe that j = S j S j1. Also it is easy to check that if f L, then S j f, as j, S j f f, as j +, in the L sense. By telescoping the series, we thus have the Littlewood Paley decomposition + f = j f, j= (.6) for all f L, where the summation is in the L sense. Notice that j f = j+ l=j l j f = j+ l=j ψ l ψ j f, then from Young s inequality, it readily follows that j f q C 3j(1/p1/q) j f p, (.7) where 1 p q, C is a constant independent of f, j. Let s R, p, q [1, ], the homogeneous Besov space Ḃ s p,q is defined by the full-dyadic decomposition such as where Ḃ s p,q = {f Z ( ); f Ḃs p,q < }, f Ḃs p,q = { js j f L p} j= l q, and Z ( ) denotes the dual space of Z( ) = {f S( ); D α ˆf () =, α N 3 }. On the other hand, for s R, p [1, ), q [1, ] and for s R, p =, q =, the homogeneous Triebel Lizorkin space is defined as where Ḟ s p,q = {f Z ( ); f Ḟ s p,q < }, f Ḟ s p,q = js j f l q L p. For p = and q [1, ), the space Ḟ s p,q is defined by means of Carleson measures which is not treated in this paper. Notice that by Minkowski inequality, we have the following inclusions: Ḃ s p,q Ḟ s p,q, if q p; Ḟ s p,q Ḃs p,q, if q p. Also it is well-known that Ḣ s = Ḃ s, = Ḟ s,, L Ḟ, = Ḃ,. (.8) (.9) We refer to [9] for more detailed properties. Throughout the proof of Theorem 1.1 in Section 3, we shall use the following interpolation inequality frequently, f L q C f 3/q1/ f 3/3/q, q 6, (.1) L L for f L ( ) Ḣ 1 ( ). 3. Proof of Theorem 1.1 Applying i to both sides of system (1.1), and then multiplying both sides by i u, i ω, i b, respectively, integrating over, after suitable integration by parts, one obtains
1 d dt ( iu, i ω, i b) + L Z. Zhang et al. / Nonlinear Analysis 74 (11) 5 3 3 {(µ + χ) ij u + γ L ij ω + ν L ij b } + χ L i ω + L κ div ω L j=1 i u, i ω + i u u, i u + i u ω, i ω + i b b, i u + i u b, i b + i b u, i b 4 I k, (3.11) k=1 where we use the following facts: div u = div b =, b i b, i u + b i u, i b =, i u, i ω = i ω, i u, and, denotes the inner product in L ( ). Now we shall show that (u, ω, b) L (, T; H 1 ) L (, T; H ), (3.1) under condition (1.3) in Theorem 1.1. Thus a standard bootstrap argument concludes the proof. By the Hölder inequality, the first term I 1 is easily estimated as I 1 = i u, i ω χ iu L + χ ω L. The other three terms are bounded similarly. For simplicity, we detail the last one, I 4. Invoking the Littlewood Paley decomposition (.6), we have u = l ( u) = l ( u) + l= N l=n l ( u) + l ( u), where N is a positive integer to be determined. Substituting this into I 4, we obtain N I 4 C l ( u) b dx + C l ( u) b dx + C l=n I 1 4 + I 4 + I3 4. l ( u) b dx (3.13) (3.14) Using the Hölder inequality, (.7), (.8) and Young s inequality, I 1 4 is dominated as = C l ( u) b dx I 1 4 C l ( u) L b L C 3l/ l ( u) L b L C 3N/ u L b L C 3N/ u L + b L 3/. (3.15) For I 4, from the Hölder inequality and (.7), it follows that N = C l ( u) b dx I 4 Here q = C l=n 3/q N l ( u) q/3 N 13/q b dx l=n CN 13/q u Ḟ b q. q,3q/ q q1 is the Hölder conjugate of q. Since 3/ < q, we have q < 6, (.1) then implies that
4 Z. Zhang et al. / Nonlinear Analysis 74 (11) 5 I 4 CN 13/q u Ḟ b 3/q b 3/q q,q/3 L L ν b L + CN u ṗ F q,q/3 b L. (3.16) Finally, I 3 4 can be estimated as = C l ( u) b dx I 3 4 C b L 3 l ( u) L 3 C b L 3 l/ l ( u) L 1/ 1/ C b L b L l l l ( u) L C N/ b L b L u L C N/ b L u + b L L. (3.17) In the above calculations, we utilize the Hölder inequality, (.7) and (.1). Now combining the estimates of I i (1 i 4), and substituting into (3.11), we are led to 1 d µ + χ dt (u, ω, b) + u + γ L L ω + ν L b L {C N (u, ω, b) L } 3/ + CN u ṗ F q,q/3 (u, ω, b) L + {C N (u, ω, b) L } 1/ (u, ω, b) L. (3.18) Now we take N in (3.18) so that i.e. C N (u, ω, b) L 1 16 min{µ + χ, γ, ν}, log e + (u, ω, b) L N C + 4, log where the constant C may depend on µ, χ, γ, ν. Then (3.18) implies that d dt (u, ω, b) L C + C u ṗ F q,q/3 log e + (u, ω, b) L (u, ω, b) L. Applying Gronwall s inequality twice, we gather that T (u, ω, b) (t) L C exp exp C u ṗ, F q,q/3(s)ds for all t (, T). The proof is complete. Acknowledgements Yao is partially supported by the National Nature Science Foundation of China under Grant no. 197134, and the Fundamental Research Funds for the Central Universities under Grant no. 13431651. References [1] G.P. Galdi, S. Rionero, A note on the existence and uniqueness of solutions of the microploar fluid equations, Internat. J. Engrg. Sci. 15 (1997) 15 18. [] M.A. Rojas-Medar, J.L. Boldrini, Magneto-microploar fluid motion: existence of weak solutions, Internet. Rev. Mat. Complut. 11 (1998) 443 46. [3] M.A. Rojas-Medar, Magneto-microploar fluid motion: existence and uniqueness of strong solutions, Math. Nachr. 188 (1997) 31 319. [4] E.E. Ortega-Torres, M.A. Rojas-Medar, Magneto-microploar fluid motion: global existence of strong solutions, Abstr. Appl. Anal. 4 (1999) 19 15. [5] J.T. Beal, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equations, Comm. Math. Phys. 94 (1984) 61 66. [6] H. Beirão da Veiga, A new regularity class for the Navier Stokes equations in R n, Chinese Ann. Math. 16 (1995) 47 41. [7] L. Escauriaza, G. Seregin, V. Šverák, Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal. 169 (3) 147 157. [8] J.S. Fan, S. Jiang, G.X. Ni, On regularity criteria for the n-dimensional Navier Stokes equations in terms of the pressure, J. Differential Equations 44 (8) 963 979. [9] I. Kukavica, M. Ziane, One component regularity for the Navier Stokes equations, Nonlinearity 19 (6) 453 469.
Z. Zhang et al. / Nonlinear Analysis 74 (11) 5 5 [1] I. Kukavica, M. Ziane, Navier Stokes equations with regularity in one direction, J. Math. Phys. 48 (7) 1 pp. [11] F. Planchon, An extension of the Beale Kato Majda criterion for the Euler equations, Comm. Math. Phys. 3 (3) 319 36. [1] G. Prodi, Un teorema di unicità per el equazioni di Navier Stokes, Ann. Mat. Pura Appl. IV (1959) 173 18. [13] J. Serrin, On the interior regularity of weak solutions of the Navier Stokes equations, Arch. Ration. Mech. Anal. 9 (196) 187 195. [14] Z.F. Zhang, Q.L. Chen, Regularity criterion via two components of vorticity on weak solutions to the Navier Stokes equations in, J. Differential Equations 16 (5) 47 481. [15] Y. Zhou, S. Gala, Logrithmically improved regularity criteria for the Navier Stokes equations in multiplier spaces, J. Math. Anal. Appl. 356 (9) 498 51. [16] Q.L. Chen, C.X. Miao, Z.F. Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys. 84 (8) 919 93. [17] Y. Du, Y. Liu, Z. Yao, Remarks on the blow-up criteria for 3D ideal magnetohydrodynamic equations, J. Math. Phys. 5 (9) 8 pp.. [18] C. He, Z.P. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations 3 (5) 35 54. [19] J.H. Wu, Regularity results for the weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst. 1 (4) 543 556. [] J.H. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations 33 (8) 385-36. [1] Z.J. Zhang, Remarks on the regularity criteria for generalized MHD equations, J. Math. Anal. Appl. 375 (11) 799 8. [] Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst. 1 (5) 881 886. [3] Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Nonlinear Mech. 41 (6) 1174 118. [4] Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (7) 491 55. [5] Y. Zhou, J.S. Fan, On regularity criteria in terms of pressure for the 3D viscous MHD equations (9). Preprint. [6] Y. Zhou, S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys. 61 (1) 193 199. [7] Y. Zhou, S. Gala, A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field, Nonlinear Anal. 7 (1) 3643 3648. [8] S. Gala, Regularity criteria for the 3D magneto-microploar fluid equations in the Morrey Campanato space, Nonlinear Differential Equations Appl. (9) doi:1.17/s3-9-47-4. [9] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam, New-York, Oxford, 1978.