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.
Keywordsmicropolar fluid equations smooth solution blow-up criterion
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.
- Lukaszewicz G: Micropolar Fluids. Theory and Applications. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Boston; 1999.Google Scholar
- Eringen A: Theory of micropolar fluids. J. Math. Mech. 1966, 16: 1–18.MathSciNetGoogle Scholar
- Dong B, Chen Z: Regularity criteria of weak solutions to the three-dimensional micropolar flows. J. Math. Phys. 2009., 50: Article ID 103525Google Scholar
- Dong B, Jia Y, Chen Z: Pressure regularity criteria of the three-dimensional micropolar fluid flows. Math. Methods Appl. Sci. 2011, 34: 595–606. 10.1002/mma.1383MathSciNetView ArticleGoogle Scholar
- He X, Fan J: A regularity criterion for 3D micropolar fluid flows. Appl. Math. Lett. 2012, 25: 47–51. 10.1016/j.aml.2011.07.007MathSciNetView ArticleGoogle Scholar
- Jia Y, Zhang W, Dong B: Remarks on the regularity criterion of the 3D micropolar fluid flows in terms of the pressure. Appl. Math. Lett. 2011, 24: 199–203. 10.1016/j.aml.2010.09.003MathSciNetView ArticleGoogle Scholar
- Ortega-Torres E, Rojas-Medar M: On the regularity for solutions of the micropolar fluid equations. Rend. Semin. Mat. Univ. Padova 2009, 122: 27–37.MathSciNetView ArticleGoogle Scholar
- Wang Y, Chen Z: Regularity criterion for weak solution to the 3D micropolar fluid equations. J. Appl. Math. 2011., 2011: Article ID 456547Google Scholar
- Wang Y, Yuan H: A logarithmically improved blow up criterion of smooth solutions to the 3D micropolar fluid equations. Nonlinear Anal., Real World Appl. 2012, 13: 1904–1912. 10.1016/j.nonrwa.2011.12.018MathSciNetView ArticleGoogle Scholar
- Gala S: On regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey-Campanato space. Nonlinear Anal., Real World Appl. 2011, 12: 2142–2150. 10.1016/j.nonrwa.2010.12.028MathSciNetView ArticleGoogle Scholar
- Gala S, Yan J: Two regularity criteria via the logarithmic of the weak solutions to the micropolar fluid equations. J. Partial Differ. Equ. 2012, 25: 32–40.MathSciNetGoogle Scholar
- Wang Y, Zhao H: Logarithmically improved blow up criterion of smooth solutions to the 3D micropolar fluid equations. J. Appl. Math. 2012., 2012: Article ID 541203Google Scholar
- Leray J: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 1934, 63: 183–248.MathSciNetView ArticleGoogle Scholar
- Hopf E: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 1951, 4: 213–231.MathSciNetView ArticleGoogle Scholar
- Fan J, Ozawa T: Regularity criterion for weak solutions to the Navier-Stokes equations in terms of the gradient of the pressure. J. Inequal. Appl. 2008., 2008: Article ID 412678Google Scholar
- Fan J, Jiang S, Nakamura G, Zhou Y: Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations. J. Math. Fluid Mech. (Printed Ed.) 2011, 13: 557–571. 10.1007/s00021-010-0039-5MathSciNetView ArticleGoogle Scholar
- He C: New sufficient conditions for regularity of solutions to the Navier-Stokes equations. Adv. Math. Sci. Appl. 2002, 12: 535–548.MathSciNetGoogle Scholar
- Serrin J: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 1962, 9: 187–195.MathSciNetView ArticleGoogle Scholar
- Zhang Z, Chen Q:Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in . J. Differ. Equ. 2005, 216: 470–481. 10.1016/j.jde.2005.06.001View ArticleGoogle Scholar
- Zhou Y, Gala S: Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces. J. Math. Anal. Appl. 2009, 356: 498–501. 10.1016/j.jmaa.2009.03.038MathSciNetView ArticleGoogle Scholar
- Zhou Y, Pokorný M: On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component. J. Math. Phys. 2009., 50: Article ID 123514Google Scholar
- Zhou Y, Pokorný M: On the regularity of the solutions of the Navier-Stokes equations via one velocity component. Nonlinearity 2010, 23: 1097–1107. 10.1088/0951-7715/23/5/004MathSciNetView ArticleGoogle Scholar
- Guo Z, Gala S: Remarks on logarithmical regularity criteria for the Navier-Stokes equations. J. Math. Phys. 2011., 52: Article ID 063503Google Scholar
- He X, Gala S:Regularity criterion for weak solutions to the Navier-Stokes equations in terms of the pressure in the class . Nonlinear Anal., Real World Appl. 2011, 12: 3602–3607. 10.1016/j.nonrwa.2011.06.018MathSciNetView ArticleGoogle Scholar
- Jiang Z, Gala S, Ni L: On the regularity criterion for the solutions of 3D Navier-Stokes equations in weak multiplier spaces. Math. Methods Appl. Sci. 2011, 34: 2060–2064. 10.1002/mma.1506MathSciNetView ArticleGoogle Scholar
- Kato T, Ponce G: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 1988, 41: 891–907. 10.1002/cpa.3160410704MathSciNetView ArticleGoogle Scholar
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