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.
MSC: 35K15, 35K45.
Keywords:micropolar fluid equations; smooth solution; blow-up criterion
In the paper, we consider the initial value problem for the micropolar fluid equations in
with the initial value
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.
Theorem 1.1Assume that , with . Let be a smooth solution to equations (1.1), (1.2) for . Ifvsatisfies
then the solution can be extended beyond .
We have the following corollary immediately.
Corollary 1.1Assume that , with . Let be a smooth solution to equations (1.1), (1.2) for . Suppose thatTis the maximal existence time, then
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.
For , is a Banach space of all distributions f on such that there exists a constant C such that for all , we have and
where we denote by the completion of the space with respect to the norm .
The norm of is given by the operator norm of pointwise multiplication
The following lemma comes from .
Lemma 2.2Assume that . For , and , , we have
where and .
We also need the following interpolation inequalities in three space dimensions.
Lemma 2.3In three space dimensions, the following inequalities hold:
3 Proof of Theorem 1.1
Multiplying the first equation of (1.1) by v and integrating with x respect to on , using integration by parts, we obtain
Similarly, we get
Summing up (3.1)-(3.2), we deduce that
We apply integration by parts and the Cauchy inequality. This yields
Substituting (3.3) into (3.4) yields
Integrating with respect to t, we have
Multiplying the first equation of (1.1) by , then integrating the resulting equation with respect to x over and using integrating by parts, we obtain
We multiply the second equation of (1.1) by , then integrate the resulting equation with respect to x over and use integrating by parts. This yields
where we have used
Equations (3.6) and (3.7) give
Making use of the Young inequality, we have
Applying the divergence operator ∇⋅ to the first equation of (1.1) produces the expression of the pressure
It follows from (3.10) that
By integration by parts and (3.11), we obtain
Combining (3.8), (3.9), (3.12) and (3.5) yields
where we have used
By (1.3), we know that for any small constant , there exists such that
The Gronwall inequality and (3.13)-(3.15) give
Applying ∇ to the first equation, then multiplying the resulting equation by ∇v and using integration by parts, the Hölder inequality, (2.2) and the Young inequality, we obtain
Similarly, we have
Adding (3.17) and (3.18), we arrive at
Equations (3.5), (3.16), (3.19) and the Gronwall inequality give
Applying to the first equation in (1.1), then taking inner product of the resulting equation with and using integration by parts, we have
Similarly, we obtain
Summing (3.21), (3.22) and using , we get
In what follows, for simplicity, we set .
By the Hölder inequality, (2.1), (2.2) and the Young inequality, we obtain
From the Young inequality and (2.2), we deduce that
Similarly, we have
Inserting (3.24)-(3.27) into (3.23) and taking ε small enough such that , we obtain
for all .
Integrating (3.28) with respect to time from to τ, we have
We get from (3.29)
For all , with the help of Gronwall inequality and (3.30), we have
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.
The author declares that she has no competing interests.
The author completed the paper herself. The author read and approved the final manuscript.
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