A regularity criterion for three-dimensional micropolar fluid equations in Besov spaces of negative regular indices
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A regularity criterion for three-dimensional micropolar fluid equations in Besov spaces of negative regular indices Maria Alessandra Ragusa1,2
· Fan Wu3
Received: 26 October 2019 / Revised: 20 March 2020 / Accepted: 14 May 2020 © Springer Nature Switzerland AG 2020
Abstract In this article, we study regularity criteria for the 3D micropolar fluid equations in terms of one partial derivative of the velocity. It is proved that if
T
0
2
∂3 u B1−r ˙ −r dt < ∞ with 0 < r < 1, ∞,∞
then, the solutions of the micropolar fluid equations actually are smooth on (0, T ). This improves and extends many previous results. Keywords Micropolar fluid equations · Regularity criteria · Besov spaces Mathematics Subject Classification 35Q35 · 35B65 · 76N10
1 Introduction We are interested in the regularity of weak solutions to the incompressible micropolar fluid equations in R3 :
B
Maria Alessandra Ragusa [email protected] Fan Wu [email protected]
1
Department of Mathematics, University of Catania, Viale Andrea Doria No. 6, 95128 Catania, Italy
2
RUDN University, 6 Miklukho -Maklay St, Moscow, Russia 117198
3
School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, Hunan, China 0123456789().: V,-vol
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M. A. Ragusa, F. Wu
⎧ ∂t u + (u · ∇)u + ∇π = u + ∇ × ω, ⎪ ⎪ ⎨ ∂t ω + (u · ∇)ω + 2ω = ω + ∇∇ · ω + ∇ × u, ∇ · u = 0, ⎪ ⎪ ⎩ u(x, 0) = u 0 (x), ω(x, 0) = ω0 (x),
(1.1)
where, for x ∈ R3 and t ≥ 0, u = u(x, t), ω = ω(x, t) and π = π(x, t) denote the velocity field, the micro-rotation field and the pressure, respectively. Micropolar fluids represent a class of fluids with nonsymmetric stress tensor (called polar fluids) such as fluids consisting of suspending particles, dumbbell molecules, (see, e.g., [6–8]). Mathematically, many authors [4,19,21] treated the well-posedness and large-time behaviour of solutions to system (1.1). However, the issue of global regularity of weak solutions to (1.1) remains an open problem. In the absence of microrotational effects (ω = 0), this system reduces to well-known incompressible Navier Stokes equations. Therefore, it is an interesting thing that regularity of a given weak solution of the 3D micropolar fluids or the 3D Navier–Stokes equations can be shown under some additional conditions, and over the years different criteria for regularity of the weak solutions have been proposed. Some fundamental Serrin-type regularity criteria for system (1.1) in terms of the velocity only were carried out in [3,22,23] independently. Recently, some improvement and extension were made on the basis of the present paper (see e.g. [5,14,20,25] and the references therein) were derived to guarantee the regularity of the weak solution. In this paper, we are concerned with the regularity conditions of partial components of three-dimensional micropolar fluid equations. In this respect, Jia et al. [17] showed that if 2p (1.2) ∂3 u ∈ L p−3 (0, T ; L p,∞ ) with 3 < p ≤ ∞, or
2p
∂1 u 1 , ∂2 u 2 ∈ L 2 p−3 (0, T ; L p,∞ ) with
3 < p ≤ ∞, 2
(1.3)
then the soluti
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