A new regularity criterion of weak solutions to the 3D micropolar fluid flows in terms of the pressure
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A new regularity criterion of weak solutions to the 3D micropolar fluid flows in terms of the pressure Sadek Gala1,2 · Maria Alessandra Ragusa2,3
· Michel Théra4,5
Received: 16 April 2020 / Accepted: 8 September 2020 © Unione Matematica Italiana 2020
Abstract In this study, we establish a new regularity criterion of weak solutions to the three-dimensional micropolar fluid flows by imposing a critical growth condition on the pressure field. Keywords Micropolar fluid flows · Weak solutions · Pressure criterion · Besov spaces Mathematics Subject Classification 35Q35 · 35B65 · 76D05
1 Introduction and the main result In this paper we consider the following Cauchy problem for the incompressible micropolar fluid equations in R3 : ⎧ ∂t u − u + (u · ∇) u + ∇π − ∇ × ω = 0, ⎪ ⎪ ⎨ ∂t ω − ω − ∇∇ · ω + 2ω + (u · ∇)ω − ∇ × u = 0, (1.1) ∇ · u = 0, ⎪ ⎪ ⎩ u(x, 0) = u 0 (x), ω(x, 0) = ω0 (x), where u = u(x, t) ∈ R3 , ω = ω(x, t) ∈ R3 and π = π (x, t) denote the unknown velocity of the fluid, the micro-rotational velocity of the fluid particles and the unknown scalar pressure of the fluid at the point (x, t) ∈ R3 × (0, T ), respectively, while u 0 , ω0 are given initial data satisfying ∇ · u = 0 in the sense of distributions. This model for micropolar fluid flows proposed by Eringen [6] enables to consider some physical phenomena that cannot be treated by the classical Navier–Stokes equations for the
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Maria Alessandra Ragusa [email protected]
1
University of Mostaganem, P. O. Box 270, 27000 Mostaganem, Algeria
2
Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6, 95125 Catania, Italy
3
RUDN University, 6 Miklukho-Maklay St, Moscow 117198, Russia
4
XLIM UMR-CNRS 7252 Université de Limoges, Limoges, France
5
Centre for Informatics and Applied Optimisation, Federation University Australia, Ballarat, Australia
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viscous incompressible fluids, such as for example, the motion of animal blood, muddy fluids, liquid crystals and dilute aqueous polymer solutions, colloidal suspensions, etc. When the micro-rotation effects are neglected or ω = 0, (1.1) reduces to the incompressible Navier–Stokes equations, and it is well known that the regularity criteria for weak solution for the fluid dynamical models attracts more and more attention. There are many important results on the velocity or vorticity or pressure blow-up criteria for Navier–Stokes equations, micropolar fluid equations and MHD equations and so on (see e.g., [1–4,7–10,14,16,17] and the references therein). As for the pressure criterion, let us first recall some interesting results on pressure regularity of Navier–Stokes equations. In Ref. [13], He and Gala proved regularity of weak solutions under the condition T
0
π(·, t)2˙ −1 dt < ∞. B ∞,∞
(1.2)
−1 stands for the homogeneous Besov space, (for the definition see Here and thereafter, B˙ ∞,∞ e.g. [12,13]). Later on, Guo and Gala [12] refined the condition (1.2) to
T 0
π(·, t)2˙ −1 B∞,∞ dt < ∞. 1 + log e + π(·, t) B˙ ∞,∞ −1
(1.3)
Motiva
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