Regularity criterion via two components of velocity on weak solutions to the shear thinning fluids in $${{\mathbb {R}}}^
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Regularity criterion via two components of velocity on weak solutions to the shear thinning fluids in R3 Ahmad M. Alghamdi1 · Sadek Gala2,3
· Maria Alessandra Ragusa3,4 · J. Q. Yang5
Received: 3 March 2020 / Revised: 20 July 2020 / Accepted: 29 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract We consider the regularity of weak solutions to the shear thinning fluids in R3 . Let u be a weak solution in R3 × (0, T ) and u = (u 1 , u 2 , 0). It is proved that u becomes a strong 5 p−6
solution if u ∈ L 5 p−8 (0, T ; B M O(R3 )) , for 85 < p ≤ 2. This is an improvement of the result given by Yang (Comput Math Appl 77:2854–2858). Keywords Shear thinning fluids · Regularity criteria · Two velocity components · BMO space Mathematics Subject Classification 35B65, 35K92, 76D03
Communicated by Cassio Oishi.
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Sadek Gala [email protected] Ahmad M. Alghamdi [email protected] Maria Alessandra Ragusa [email protected] J. Q. Yang [email protected]
1
Department of Mathematical Science, Faculty of Applied Science, Umm Alqura University, P. O. Box 14035, Mecca 21955, Saudi Arabia
2
Department of Sciences Exactes, ENS of Mostaganem, University of Mostaganem, Box 227, Mostaganem 27000, Algeria
3
Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6 95125 Catania, Italy
4
RUDN University, 6 Miklukho-Maklay St, Moscow 117198, Russia
5
School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129, China 0123456789().: V,-vol
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A. M. Alghamdi et al.
1 Introduction
In this paper, we consider the regularity of weak solutions to the non-Newtonian incompressible fluids which is governed by the following system: ⎧ ⎨ ∂t u + (u · ∇)u − div(|D(u)| p−2 D(u)) + ∇π = 0, (1.1) ∇ · u = 0, ⎩ u(x, 0) = u 0 (x), where u = (u 1 , u 2 , u 3 ) is the velocity field and π is the scalar pressure, while u 0 is the corresponding initial data satisfying ∇ · u 0 = 0 in the sense of distribution. The symmetric matrix D(u) is defined by D(u) = 21 (∇u + (∇u)T ). When p = 2, then (1.1) represents the well-known classical Navier–Stokes system, which has been an object of many studies (see, for example, Cao and Titi 2008; Chemin and Zhang 2016; Kukavica and Ziane 2006; Yamazaki 2016a; Zhou and Pokorný 2010 and references therein). For the case of threedimensional incompressible magnetohydrodynamic (MHD) equations, Ji and Lee (2010) got some regularity criteria for the weak solutions in terms of partial components of the velocity and magnetic fields (see also Cao and Wu 2010; Jia and Zhou 2012; Yamazaki 2016b). If 1 < p < 2, the fluid is said to be shear thinning; in case p > 2, the fluid is called shear thickening (see Wolf 2007, for instance). The system (1.1) was first considered by Ladyzhenskaya (1969) and the global regularity for p > 95 is shown there (see also Pokorny 1996). When p > 85 , existence of weak solutions with Dirichlet boundary condition for (1.1) has been studied by Wolf (2007) . For 75
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