A logarithmically improved blow-up criterion for smooth solutions to the micropolar fluid equations in weak multiplier spaces
© Zhao; licensee Springer 2013
Received: 11 March 2013
Accepted: 26 April 2013
Published: 10 May 2013
In this paper, we study the initial value problem for the three-dimensional micropolar fluid equations. A new logarithmically improved blow-up criterion for the three-dimensional micropolar fluid equations in a weak multiplier space is established.
where , and represent the divergence free velocity field, non-divergence free micro-rotation field and the scalar pressure, respectively. is the Newtonian kinetic viscosity and is the dynamics micro-rotation viscosity, are the angular viscosity (see ).
The micropolar fluid equations were first proposed by Eringen . The micropolar fluid equations are a generalization of the Navier-Stokes model. It takes into account the microstructure of the fluid, by which we mean the geometry and microrotation of particles. It is a type of fluids which exhibit the micro-rotational effects and micro-rotational inertia, and can be viewed as a non-Newtonian fluid. Physically, it may represent adequately the fluids consisting of bar-like elements. Certain anisotropic fluids, e.g., liquid crystals that are made up of dumbbell molecules, are of the type. For more background, we refer to  and references therein.
Due to its importance in mathematics and physics, there is lots of literature devoted to the mathematical theory of the 3D micropolar fluid equations. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research and many interesting results have been established (see [3–12]). The regularity of weak solutions is examined by imposing some critical growth conditions only on the pressure field in the Lebesgue space, Morrey space, multiplier space, BMO space and Besov space, respectively (see ). A new logarithmically improved blow-up criterion for the 3D micropolar fluid equations in an appropriate homogeneous Besov space was obtained by Wang and Yuan . A Serrin-type regularity criterion for the weak solutions to the micropolar fluid equations in in the critical Morrey-Campanato space was built . Wang and Zhao  established logarithmically improved blow-up criteria of a smooth solution to (1.1), (1.2) in the Morrey-Campanto space.
If and , then equations (1.1) reduce to be the Navier-Stokes equations. The Leray-Hopf weak solution was constructed by Leray  and Hopf , respectively. Later on, much effort has been devoted to establishing the global existence and uniqueness of smooth solutions to the Navier-Stokes equations. Different criteria for regularity of the weak solutions have been proposed and many interesting results were established (see [15–24]).
Without loss of generality, we set , in the rest of the paper. The purpose of this paper is to establish a new logarithmically improved blow-up criterion to (1.1), (1.2) in a weak multiplier space. Now we state our results as follows.
then the solution can be extended beyond .
We have the following corollary immediately.
The paper is organized as follows. We first state some preliminaries on functional settings and some important inequalities in Section 2, which play an important role in the proof of our main result. Then we prove the main result in Section 3.
Definition 2.1 
where we denote by the completion of the space with respect to the norm .
The following lemma comes from .
where and .
We also need the following interpolation inequalities in three space dimensions.
3 Proof of Theorem 1.1
In what follows, for simplicity, we set .
for all .
where C depends on . (3.31) and (3.5) imply . Thus, can be extended smoothly beyond . We have completed the proof of Theorem 1.1.
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