Blow-up criterion of smooth solutions for magneto-micropolar fluid equations with partial viscosity
Boundary Value Problems volume 2011, Article number: 11 (2011)
In this paper, we investigate the Cauchy problem for the incompressible magneto-micropolar fluid equations with partial viscosity in ℝn(n = 2, 3). We obtain a Beale-Kato-Majda type blow-up criterion of smooth solutions.
MSC (2010): 76D03; 35Q35.
The incompressible magneto-micropolar fluid equations in ℝn(n = 2, 3) takes the following form
where u(t, x), v(t, x), b(t, x) and p(t, x) denote the velocity of the fluid, the micro-rotational velocity, magnetic field and hydrostatic pressure, respectively. μ, χ, γ, κ and ν are constants associated with properties of the material: μ is the kinematic viscosity, χ is the vortex viscosity, γ and κ are spin viscosities, and is the magnetic Reynold. The incompressible magneto-micropolar fluid equations (1.1) has been studied extensively (see [1–8]). Rojas-Medar  established the local in time existence and uniqueness of strong solutions by the spectral Galerkin method. Global existence of strong solution for small initial data was obtained in . Rojas-Medar and Boldrini  proved the existence of weak solutions by the Galerkin method, and in 2D case, also proved the uniqueness of the weak solutions. Wang et al.  obtained a Beale-Kato-Majda type blow-up criterion for smooth solution (u, v, b) to the magneto-micropolar fluid equations with partial viscosity that relies on the vorticity of velocity ∇ × u only (see also ). For regularity results, refer to Yuan  and Gala .
If b = 0, (1.1) reduces to micropolar fluid equations. The micropolar fluid equations was first proposed by Eringen . It is a type of fluids which exhibits the micro-rotational effects and micro-rotational inertia, and can be viewed as a non-Newtonian fluid. Physically, micropolar fluid may represent fluids that consisting of rigid, randomly oriented (or spherical particles) suspended in a viscous medium, where the deformation of fluid particles is ignored. It can describe many phenomena appeared in a large number of complex fluids such as the suspensions, animal blood, liquid crystals which cannot be characterized appropriately by the Navier-Stokes equations, and that it is important to the scientists working with the hydrodynamic-fluid problems and phenomena. For more background, we refer to  and references therein. The existences of weak and strong solutions for micropolar fluid equations were treated by Galdi and Rionero  and Yamaguchi , respectively. The global regularity issue has been thoroughly investigated for the 3D micropolar fluid equations and many important regularity criteria have been established (see [13–19]). The convergence of weak solutions of the micropolar fluids in bounded domains of ℝn was investigated (see ). When the viscosities tend to zero, in the limit, a fluid governed by an Euler-like system was found.
If both v = 0 and χ = 0, then Equations 1.1 reduces to be the magneto-hydrodynamic (MHD) equations. The local well-posedness of the Cauchy problem for the incompressible MHD equations in the usual Sobolev spaces Hs(ℝ3) is established in  for any given initial data that belongs to Hs(ℝ3), s ≥ 3. But whether this unique local solution can exist globally is a challenge open problem in the mathematical fluid mechanics. There are numerous important progresses on the fundamental issue of the regularity for the weak solution to (1.1), (1.2) (see [22–34]). In this paper, we consider the magneto-micropolar fluid equations (1.1) with partial viscosity, i.e., μ = χ = 0. Without loss of generality, we take γ = κ = ν = 1. The corresponding magneto-micropolar fluid equations thus reads
We obtain a blow-up criterion of smooth solutions to (1.2), which improves our previous result (see ).
In the absence of global well-posedness, the development of blow-up/non-blow-up theory is of major importance for both theoretical and practical purposes. For incompressible Euler and Navier-Stokes equations, the well-known Beale-Kato-Majda's criterion  says that any solution u is smooth up to time T under the assumption that . Beale-Kato-Majda's criterion is slightly improved by Kozono et al.  under the assumption . In this paper, we obtain a Beale-Kato-Majda type blow-up criterion of smooth solutions to Cauchy problem for the magneto-micropolar fluid equations (1.2).
Now, we state our results as follows.
Theorem 1.1 Assume that u0, v0, b0 ∈ Hm(ℝn)(n = 2, 3), m ≥ 3 with ∇ · u0 = 0, ∇ · b0 = 0. Let (u, v, b) be a smooth solution to Equations 1.2 with initial data u(0, x) = u0(x), v(0, x) = v0(x), b(0, x) = b0(x) for 0 ≤ t < T . If u satisfies
then the solution (u, v, b) can be extended beyond t = T.
We have the following corollary immediately.
Corollary 1.1 Assume that u0, v0, b0 ∈ Hm(ℝn)(n = 2, 3), m ≥ 3 with ∇ · u0 = 0, ∇ · b0 = 0. Let (u, v, b) be a smooth solution to Equations 1.2 with initial data u(0, x) = u0(x), v(0, x) = v0(x), b(0, x) = b0(x) for 0 ≤ t < T . Suppose that T is the maximal existence time, then
The plan of the paper is arranged as follows. We first state some preliminary on functional settings and some important inequalities in Section 2 and then prove the blow-up criterion of smooth solutions to the magneto-micropolar fluid equations (1.2) in Section 3.
Let be the Schwartz class of rapidly decreasing functions. Given , its Fourier transform is defined by
and for any given , its inverse Fourier transform is defined by
In what follows, we recall the Littlewood-Paley decomposition. Choose a non-negative radial functions , supported in such that
The frequency localization operator is defined by
Next, we recall the definition of homogeneous function spaces (see ). For (p, q) ∈ [1, ∞]2 and s ∈ ℝ, the homogeneous Besov space is defined as the set of f up to polynomials such that
In what follows, we shall make continuous use of Bernstein inequalities, which comes from .
Lemma 2.1 For any s ∈ ℕ, 1 ≤ p ≤ q ≤ ∞ and f ∈ Lp(ℝn), then the following inequalities
hold, where c and C are positive constants independent of f and k.
The following inequality is well-known Gagliardo-Nirenberg inequality.
Lemma 2.2 Let j, m be any integers satisfying 0 ≤ j < m, and let 1 ≤ q, r ≤ ∞, and such that
Then for all f ∈ Lq(ℝn) ∩Wm,r(ℝn), there is a positive constant C depending only on n, m, j, q, r, θ such that the following inequality holds:
with the following exception: if 1 < r < 1 and is a nonnegative integer, then (2.3) holds only for a satisfying .
The following lemma comes from .
Lemma 2.3 Assume that 1 < p < ∞. For f, g ∈ Wm,p, and 1 < q1, q2 ≤ ∞, 1 < r1, r2 < 1, we have
where 1 ≤ α ≤ m and .
Lemma 2.4 There exists a uniform positive constant C, such that
holds for all vectors f ∈ H3(ℝn)(n = 2, 3) with ∇ · f = 0.
Proof. The proof can be founded in . For the convenience of the readers, the proof will be also sketched here. It follows from Littlewood-Paley composition that
Using (2.1), ( 2.2) and (2.6), we obtain
It follows from (2.7), (2.8) and Calderon-Zygmand theory that (2.5) holds. Thus, we have completed the proof of lemma. □
In order to prove Theorem 1.1, we need the following interpolation inequalities in two and three space dimensions.
Lemma 2.5 In three space dimensions, the following inequalities
hold, and in two space dimensions, the following inequalities
Proof. (2.9) and (2.10) are of course well known. In fact, we can obtain them by Sobolev embedding and the scaling techniques. In what follows, we only prove the last inequality in (2.9) and (2.10). Sobolev embedding implies that H3(ℝn), ↪ L4(ℝn) for n = 2, 3. Consequently, we get
For any given 0 ≠ f ∈ H3(ℝn) and δ > 0, let
By (2.11) and (2.12), we obtain
which is equivalent to
Taking and n = 3 and n = 2, respectively. From (2.14), we immediately get the last inequality in (2.9) and (2.10). Thus, we have completed the proof of Lemma 2.5. □
3 Proof of main results
Proof of Theorem 1.1. Adding the inner product of u with the first equation of (1.2), of v with the second equation of (1.2) and of b the third equation of (1.2), then using integration by parts, we get
where we have used ∇ ·· u = 0 and ∇ · b = 0.
Integrating with respect to t, we have
Applying ∇ to (1.2) and taking the L2 inner product of the resulting equation with (∇u, ∇v, ∇b), with help of integration by parts, we have
By (3.3) and ∇ · u = 0, ∇ · b = 0, we deduce that
Using Gronwall inequality, we get
Owing to (1.3), we know that for any small constant ε > 0, there exists T* < T such that
It follows from (3.5), (3.6), (3.7) and Lemma 2.4 that
where C1 depends on , while C0 is an absolute positive constant.
Applying ∇m to the first equation of (1.2), then taking L2 inner product of the resulting equation with ∇mu and using integration by parts, we have
Likewise, we obtain
It follows (3.9), (3.10), (3.11), ∇ · u = 0, ∇ · b = 0 and integration by parts that
In what follows, for simplicity, we will set m = 3.
With help of Hölder inequality and Lemma 2.3, we derive
Using integration by parts and Hölder inequality, we get
Thanks to Lemma 2.5, Young inequality and (3.8), we get
in 3D and
It follows from Lemmas 2.2, 2.5, Young inequality and (3.8) that
in 3D and
Consequently, we get
It follows from (3.14), (3.15) and (3.16) that
Likewise, we have
Collecting (3.12), (3.13), (3.17), (3.18), (3.19) and (3.20) yields
for all T* ≤ t < T.
Integrating (3.21) with respect to time from T* to τ and using Lemma 2.4, we have
Owing to (3.22), we get
For all T* ≤ t < T, with help of Gronwall inequality and (3.23), we have
where C depends on .
Noting that (3.2) and the right-hand side of (3.24) is independent of t for T* ≤ t < T , we know that (u(T, ·), v(T, ·), b(T, ·)) ∈ H3(ℝn). Thus, Theorem 1.1 is proved.
Gala S: Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey-Campanato space. Nonlinear Differ Equ Appl 2010, 17: 181-194. 10.1007/s00030-009-0047-4
Wang Y, Hu L, Wang Y: A Beale-Kato Majda criterion for magneto-micropolar fluid equations with partial viscosity. Bound Value Prob 2011, 2011: 14. Article ID 128614 10.1186/1687-2770-2011-14
Ortega-Torres E, Rojas-Medar M: On the uniqueness and regularity of the weak solution for magneto-micropolar fluid equations. Revista de Matemáticas Aplicadas 1996, 17: 75-90.
Ortega-Torres E, Rojas-Medar M: Magneto-micropolar fluid motion: global existence of strong solutions. Abstract Appl Anal 1999, 4: 109-125. 10.1155/S1085337599000287
Rojas-Medar M: Magneto-micropolar fluid motion: existence and uniqueness of strong solutions. Mathematische Nachrichten 1997, 188: 301-319. 10.1002/mana.19971880116
Rojas-Medar M, Boldrini J: Magneto-micropolar fluid motion: existence of weak solutions. Rev Mat Complut 1998, 11: 443-460.
Yuan B: regularity of weak solutions to magneto-micropolar fluid equations. Acta Mathematica Scientia 2010, 30: 1469-1480. 10.1016/S0252-9602(10)60139-7
Yuan J: Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations. Math Methods Appl Sci 2008, 31: 1113-1130. 10.1002/mma.967
Eringen A: Theory of micropolar fluids. J Math Mech 1966, 16: 1-18.
Lukaszewicz G: Micropolar fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Baston 1999.
Galdi G, Rionero S: A note on the existence and uniqueness of solutions of the micropolar fluid equations. Int J Eng Sci 1977, 15: 105-108. 10.1016/0020-7225(77)90025-8
Yamaguchi N: Existence of global strong solution to the micropolar fluid system in a bounded domain. Math Methods Appl Sci 2005, 28: 1507-1526. 10.1002/mma.617
Yuan B: On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space. Proc Am Math Soc 2010, 138: 2025-2036. 10.1090/S0002-9939-10-10232-9
Fan J, Zhou Y, Zhu M: A regularity criterion for the 3D micropolar fluid flows with zero angular viscosity. 2010, in press.
Fan J, He X: A regularity criterion of the 3D micropolar fluid flows. 2011, in press.
Fan J, Jin L: A regularity criterion of the micropolar fluid flows. 2011, in press.
Dong B, Chen Z: Regularity criteria of weak solutions to the three-dimensional micropolar flows. J Math Phys 2009, 50: 103525-1-103525-13.
Szopa P: Gevrey class regularity for solutions of micropolar fluid equations. J Math Anal Appl 2009, 351: 340-349. 10.1016/j.jmaa.2008.10.026
Ortega-Torres E, Rojas-Medar M: On the regularity for solutions of the micropolar fluid equations. Rendiconti del Seminario Matematico della Università de Padova 2009, 122: 27-37.
Ortega-Torres E, Rojas-Medar M, Villamizar-Roa EJ: Micropolar fluids with vanishing viscosity. Abstract Appl Anal 2010, 2010: 18. Article ID 843692
Sermange M, Temam R: Some mathematical questions related to the MHD equations. Commun Pure Appl Math 1983, 36: 635-666. 10.1002/cpa.3160360506
Caisch R, Klapper I, Steele G: Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Commun Math Phys 1997, 184: 443-455. 10.1007/s002200050067
Cannone M, Chen Q, Miao C: A losing estimate for the ideal MHD equations with application to blow-up criterion. SIAM J Math Anal 2007, 38: 1847-1859. 10.1137/060652002
Cao C, Wu J: Two regularity criteria for the 3D MHD equations. J Diff Equ 2010, 248: 2263-2274. 10.1016/j.jde.2009.09.020
He C, Xin Z: Partial regularity of suitable weak sokutions to the incompressible magnetohydrodynamics equations. J Funct Anal 2005, 227: 113-152. 10.1016/j.jfa.2005.06.009
Lei Z, Zhou Y: BKM criterion and global weak solutions for Magnetohydrodynamics with zero viscosity. Discrete Contin Dyn Syst A 2009, 25: 575-583.
Wu J: Regularity results for weak solutions of the 3D MHD equations. Discrete Contin Dyn Syst 2004, 10: 543-556.
Wu J: Regularity criteria for the generalized MHD equations. Commun Partial Differ Equ 2008, 33: 285-306. 10.1080/03605300701382530
Zhou Y: Remarks on regularities for the 3D MHD equations. Discrete Contin Dyn Syst 2005, 12: 881-886.
Zhou Y: Regularity criteria for the 3D MHD equations in term of the pressure. Int J Nonlinear Mech 2006, 41: 1174-1180. 10.1016/j.ijnonlinmec.2006.12.001
Zhou Y: Regularity criteria for the generalized viscous MHD equations. Ann Inst H Poincaré Anal Non Linéaire 2007, 24: 491-505. 10.1016/j.anihpc.2006.03.014
Zhou Y, Gala S: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Z Angew Math Phys 2010, 61: 193-199. 10.1007/s00033-009-0023-1
Zhou Y, Gala S: A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field. Nonlinear Anal 2010, 72: 3643-3648. 10.1016/j.na.2009.12.045
Zhou Y, Fan J: Logarithmically improved regularity criteria for the 3D viscous MHD equations. Forum Math 2010, in press.
Beale J, Kato T, Majda A: Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun Math Phys 1984, 94: 61-66. 10.1007/BF01212349
Kozono H, Ogawa T, Taniuchi Y: The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math Z 2002, 242: 251-278. 10.1007/s002090100332
Triebel H: Theory of Function Spaces. Monograph in Mathematics. Birkhauser, Basel 1983., 78:
Chemin J: Perfect Incompressible Fluids. In Oxford Lecture Ser Math Appl. Volume 14. The Clarendon Press/Oxford University Press, New York; 1998.
Majda A, Bertozzi A: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge; 2002.
The authors would like to thank the referee for his/her pertinent comments and advice. This work was supported in part by Research Initiation Project for High-level Talents (201031) of North China University of Water Resources and Electric Power.
The authors declare that they have no competing interests.
YZW completed the main part of theorem in this paper, YL and YXW revised the part proof. All authors read and approve the final manuscript.
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Wang, YZ., Li, Y. & Wang, YX. Blow-up criterion of smooth solutions for magneto-micropolar fluid equations with partial viscosity. Bound Value Probl 2011, 11 (2011). https://doi.org/10.1186/1687-2770-2011-11