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In [3] L. C. Berselli, On a Regularity Criterion for the Solutions to the 3D Navier-Stokes Equations, Diff. and Integral Eq., Vol. 15, Number 9, 1129-1137 (2002). , L. Berselli showed that the additional regularity hypothesis for the velocity gradient ∇u ∈ L 2q 2q−3 (0, T;L q (Ω)), for some q ∈ (3/2, +∞], implies the strong regularity for the weak solutions of the Navier-Stokes equations. In this work, we prove that such hypothesis is also sufficient in order to obtain the strong solution for a nematic Liquid Crystal model (a coupled system of velocity u and orientation crystals vector d) when periodic boundary conditions for d are considered. For Neumann and Dirichlet boundary conditions, we obtain the same result only for the cases of q ∈ [2, 3], whereas in the cases q ∈ (3/2, 2) ∪ (3, +∞], we also need to impose an additional regularity hypothesis for d (either on ∇d or ∆d). On the other hand, when the following hypothesis for u of Serrin’s type is imposed ([18] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal. 9 (3), 187-195 (1962): u ∈ L 2p p−3 (0, T;L p (Ω)) for some p ∈ (3, +∞], we can obtain strong regularity only in the case of periodic boundary conditions for d. For Neumann or Dirichlet boundary conditions, additional regularity for d must be imposed in all cases.

In [3] L. C. Berselli, On a Regularity Criterion for the Solutions to the 3D Navier-Stokes Equations, Diff. and Integral Eq., Vol. 15, Number 9, 1129-1137 (2002). , L. Berselli showed that the additional regularity hypothesis for the velocity gradient ∇u ∈ L 2q 2q−3 (0, T;L q (Ω)), for some q ∈ (3/2, +∞], implies the strong regularity for the weak solutions of the Navier-Stokes equations. In this work, we prove that such hypothesis is also sufficient in order to obtain the strong solution for a nematic Liquid Crystal model (a coupled system of velocity u and orientation crystals vector d) when periodic boundary conditions for d are considered. For Neumann and Dirichlet boundary conditions, we obtain the same result only for the cases of q ∈ [2, 3], whereas in the cases q ∈ (3/2, 2) ∪ (3, +∞], we also need to impose an additional regularity hypothesis for d (either on ∇d or ∆d). On the other hand, when the following hypothesis for u of Serrin’s type is imposed ([18] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal. 9 (3), 187-195 (1962): u ∈ L 2p p−3 (0, T;L p (Ω)) for some p ∈ (3, +∞], we can obtain strong regularity only in the case of periodic boundary conditions for d. For Neumann or Dirichlet boundary conditions, additional regularity for d must be imposed in all cases.

In [3] L. C. Berselli, On a Regularity Criterion for the Solutions to the 3D Navier-Stokes Equations, Diff. and Integral Eq., Vol. 15, Number 9, 1129-1137 (2002). , L. Berselli showed that the additional regularity hypothesis for the velocity gradient ∇u ∈ L 2q 2q−3 (0, T;L q (Ω)), for some q ∈ (3/2, +∞], implies the strong regularity for the weak solutions of the Navier-Stokes equations. In this work, we prove that such hypothesis is also sufficient in order to obtain the strong solution for a nematic Liquid Crystal model (a coupled system of velocity u and orientation crystals vector d) when periodic boundary conditions for d are considered. For Neumann and Dirichlet boundary conditions, we obtain the same result only for the cases of q ∈ [2, 3], whereas in the cases q ∈ (3/2, 2) ∪ (3, +∞], we also need to impose an additional regularity hypothesis for d (either on ∇d or ∆d). On the other hand, when the following hypothesis for u of Serrin’s type is imposed ([18] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal. 9 (3), 187-195 (1962): u ∈ L 2p p−3 (0, T;L p (Ω)) for some p ∈ (3, +∞], we can obtain strong regularity only in the case of periodic boundary conditions for d. For Neumann or Dirichlet boundary conditions, additional regularity for d must be imposed in all cases.

In this note, a regularity criterion of weak solutions to the 3D-Boussinesq equations with respect to Serrin type condition under the framework of Besov space : B-infinity(r),(infinity). It is shown that the weak solution (u; theta) is regular on (0; T] if u satisfies integral(T)(0) parallel to u(center dot, t)parallel to(2/1+r)r(B infinity,infinity) dt < infinity, for 0 < r < 1. This result improves some previous works.

We prove some regularity criteria for smooth solutions of the three-dimensional incompressible Hall-magnetohydrodynamic equations involving only two components of the velocity and magnetic fields. © Instytut Matematyczny PAN, 2018.

The paper deals with the 3D incompressible MHD equations and aims at improving a regularity criterion in terms of the horizontal gradient of velocity and magnetic field. It is proved that the weak solution (u, b) becomes regular provided that (Formula Presented.) The result is an extension of regularity criterion for 3D Navier–Stokes equations in Besov space due to Fang and Qian (Commun Pure Appl Anal 13:585–603, 2014) [see also (Ni et al. in J Math Anal Appl 396:108–118, 2012)]. © 2017, Springer International Publishing AG, part of Springer Nature.