Nonlinear Functional Difference Equations with ApplicationsView this Special Issue
A Regularity Criterion for the Magneto-Micropolar Fluid Equations in
The paper is dedicated to study of the Cauchy problem for the magneto-micropolar fluid equations in three-dimensional spaces. A new logarithmically improved regularity criterion for the magneto-micropolar fluid equations is established in terms of the pressure in the homogeneous Besov space .
This paper concerns with the regularity of weak solutions to the magneto-micropolar fluid equations in three dimensions as where denotes the velocity of the fluid at a point , , and denote, respectively, the microrotational velocity, the magnetic field, and the hydrostatic pressure. are positive numbers associated to properties of the material: is the kinematic viscosity, is the vortex viscosity, and are spin viscosities, and is the magnetic Reynold. are initial data for the velocity, the angular velocity, and the magnetic field with properties and . For more detailed background, we refer the readers to [1–3].
As we know, the problem of global regularity or finite time singularity for the weak solutions of the magneto-micropolar fluid equations model with large initial data still remains unsolved since (1) includes the 3D Navier-Stokes equations. It is of interest that the regularity of the weak solutions is under preassumption of certain growth conditions. There are a lot of lectures to study the regularity of weak solutions of the magneto-micropolar fluid equations (see, [4–6]). The purpose of this paper is to establish a new logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure in Besov space . Now we state the main results as follows.
Theorem 1. Let . Let and be a weak solution to the system (1). If the pressure filed satisfies the following condition: then the weak solution is regular on .
Remark 3. Since the space is wider than , hence our result extends and improves the recent results given by .
2. Preliminaries and Lemmas
Throughout this paper, we introduce some function spaces, notations, and important inequalities.
Let denote the heat semigroup defined by for and , where denotes the convolution of functions defined on .
We now recall the definition of the homogeneous Besov space with negative indices on and the homogeneous Sobolev space of exponent . It is known (p. 192 of ) that belongs to if and only if for all and . The norm of is defined, up to equivalence, by
We introduce now the homogeneous Sobolev space , which is defined by the set of functions such that . This space is endowed with the norm and when , we just let . Additionally, we have the following inclusion relations (see, e.g., ): with continuous injection.
Lemma 4 (see ). Let and . Then there exists a constant depending only on , , and such that for all ,
In particular, for , and , we get and
Lemma 5 (see ). Let , and , then there exists a positive constant independent of such that where
3. Proof of Theorem 1
For given initial data , the weak solution is the same as the local strong solution in a local interval as in the discussion of Navier-Stokes equations. For the uniqueness and existence of local strong solution, we refer to . Thus, it proves that Theorem 1 is reduced to establish a priori estimates uniformly in for strong solutions. With the use of the a priori estimates, the local strong solution can be continuously extended to by a standard process to obtain global regularity of the weak solution. Therefore, we assume that the solution is sufficiently smooth on .
Proof of Theorem 1. We show that Theorem 1 holds under condition (1). To prove the theorem, we need the -estimate. For this purpose, taking the inner product of the first equation of (1) with and integrating by parts, it can be deduced that
where we used the following relations by the divergence-free condition div:
Similarly, taking the inner product of the second equation of (1) with and integrating by parts, it can be inferred that
Using an argument similar to that used in deriving the estimate (11)–(13), it can be obtained for the third equation of (1) that
Adding up (11), (13), and (14), then we obtain
Applying the Hölder inequality and the Young inequality for , it follows that
Arguing similarly to above, it can be derived for that
Considering the term , by virtue of the Cauchy inequality, we have
Let us bound the integral . Applying the divergence operator to the first equation of (1), one formally has , where denotes the th Riesz operator. By the Calderon-Zygmund inequality, we have
With the help of (8) and (19), by the Hölder inequality and the Young inequality, we deduce that
So the term can be estimated as
Next we have the following estimate for the term :
Since and using Cauchy inequality, generalized Hölder inequality, Gagliardo-Nirenberg inequality, and Sobolev imbedding theorem, we obtain
The last term of (15) can be treated in the same way as
Inserting the estimates (15) and (21) into (14), it follows that
where is defined by
Applying Gronwall's inequality on (25) for the interval , one has
where is a positive constant depending on .
Next we will estimate the -norm of , , and . We multiply both sides of the first equation of (1) by , the second equation of (1) by , and the third equation of (1) by , by integration by parts over , we get where we have used the Gagliardo-Nirenberg inequality: Combining (29), (30), and (31) and using the definition of the weak solution, we deduce that Finally we go to the estimate for -norm of , , and . In the following calculations, we will use the following commutator estimate due to Kato and Ponce : with , and . Taking the operation on both sides of (1), then multiplying them by , , and , and integrating by parts over , we have Hence can be estimated as where we used (33) with and the following inequalities: If we use the existing estimate (31) for , (36) reduces to Using (37) again, we have For and , we have Inserting the above estimates (38)–(40) into (35), we obtain Gronwall's inequality implies the boundness of -norm of , , and provided that , which can be achieved by the absolute continuous property of integral (2). This completes the proof of Theorem 1.
The authors thank Professor Xiaohong Fan for his profitable discussion and suggestions.
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