Blow-up criterion of smooth solutions for magneto-micropolar fluid equations with partial viscosity
© Wang et al; licensee Springer. 2011
Received: 4 April 2011
Accepted: 15 August 2011
Published: 15 August 2011
In this paper, we investigate the Cauchy problem for the incompressible magneto-micropolar fluid equations with partial viscosity in ℝ n (n = 2, 3). We obtain a Beale-Kato-Majda type blow-up criterion of smooth solutions.
MSC (2010): 76D03; 35Q35.
Keywordsmagneto-micropolar fluid equations smooth solutions; blow-up criterion
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 ℝ n was investigated (see ). When the viscosities tend to zero, in the limit, a fluid governed by an Euler-like system was found.
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.
then the solution (u, v, b) can be extended beyond t = T.
We have the following corollary immediately.
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.
In what follows, we shall make continuous use of Bernstein inequalities, which comes from .
hold, where c and C are positive constants independent of f and k.
The following inequality is well-known Gagliardo-Nirenberg inequality.
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 .
where 1 ≤ α ≤ m and .
holds for all vectors f ∈ H3(ℝ n )(n = 2, 3) with ∇ · f = 0.
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.
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
where we have used ∇ ·· u = 0 and ∇ · b = 0.
where C1 depends on , while C0 is an absolute positive constant.
In what follows, for simplicity, we will set m = 3.
for all T* ≤ t < T.
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(ℝ n ). Thus, Theorem 1.1 is proved.
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.
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