Sadek Gala

Remarks on logarithmical regularity criteria for the Navier-Stokes equations

In this paper we establish some new logarithmically improved regularity criteria for the Navier–Stokes equations in the homogeneous Besov space B∞,∞−1.

Uniqueness criterion of weak solutions for the dissipative quasi-geostrophic equations in Orlicz–Morrey spaces

Consider the quasi-geostrophic equations with the initial data . Let and be two weak solutions with the same initial value . If where is the Orlicz–Morrey space (for a definition of this space, see Definition ), then we have . In view of the embedding with and , we see that our result improves the previous result of Dong and Chen. This is an extension of earlier regularity results in the Serrin’s type space .

A remark on two generalized Orlicz-Morrey spaces

There have been known two generalized Orlicz-Morrey spaces. One is defined earlier by Nakai and the other is by Sugano, the second and third authors. In this paper we investigate differences between these two spaces in some typical cases. The arguments rely upon property of the characteristic function of the Cantor set.

Logarithmically improved regularity criterion for the nematic liquid crystal flows in Ḃ∞,∞−1 space

Abstract In this work, we study the regularity criterion of the three-dimensional nematic liquid crystal flows. It is proved that if the vorticity satisfies ∫ 0 T ‖ ω ( t , ⋅ ) ‖ B . ∞ , ∞ − 1 2 1 + log ( e + ‖ ω ( t , ⋅ ) ‖ B . ∞ , ∞ − 1 ) d t ∞ , where B ∞ , ∞ − 1 denotes the critical Besov space, then the solution ( u , d ) becomes a regular solution on ( 0 , T ] . This result extends the recent regularity criterion obtained by Fan and Ozawa (2012) [11] .

Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices

The aim of this paper is to establish a logarithmically improved the regularity criterion in terms of the homogeneous Besov space to the Boussinesq equations. We prove the solution is smooth up to time provided that for some and .

A new regularity criterion for the nematic liquid crystal flows

In this article, we establish the Serrin-type regularity criteria for the 3D nematic liquid crystal flows in the terms of the multiplier space . Our criterion may be regarded as an extension of the recent result of Liu–Cui [Regularity of solutions to 3-D nematic liquid crystal flows, Electron. J. Diff. Equ. 2010(173) (2010), pp. 1–5].

Remarks on regularity criterion for weak solutions to the Navier–Stokes equations in terms of the gradient of the pressure

In this article, we establish a Serrin-type regularity criterion on the gradient of pressure for weak solutions to the Navier–Stokes equation in ℝ3. It is proved that if the gradient of pressure belongs to , where is the multiplier space (a definition is given in the text) for 0 ≤ r ≤ 1, then the weak solution is actually regular. Since this space is wider than , our regularity criterion covers the previous results given by Struwe [M. Struwe, On a Serrin-type regularity criterion for the Navier–Stokes equations in terms of the pressure, J. Math. Fluid Mech. 9 (2007), pp. 235–242], Berselli-Galdi [L.C. Berselli and G.P. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier–Stokes equations, Proc. Amer. Math. Soc. 130 (2002), pp. 3585–3595] and Zhou [ Y. Zhou, On regularity criteria in terms of pressure for the Navier–Stokes equations in ℝ3 , Proc. Am. Math. Soc. 134 (2006), pp. 149–156].

On the regularity criterion of strong solutions to the 3D Boussinesq equations

This note is devoted to the study of the regularity of strong solutions to the Boussinesq system in which the vorticity and to improve a result of Fan and Ozawa [J. Fan, T. Ozawa, Regularity conditions for the 3D Boussinesq equations with partial viscosity terms, Nonlinearity 22 (2009), pp. 553–568] and Gala [S. Gala, Remark on the regularity criterion on strong solutions to the 3D Boussinesq equations (Submitted)].

On the regularity criteria of the 3D Navier-Stokes equations in critical spaces

Regularity criteria of Leray-Hopf weak solutions to the three-dimensional Navier-Stokes equations in some critical spaces such as Lorentz space, Morrey space and multiplier space are derived in terms of two partial derivatives, ?1u1, ?2u2, of velocity fields.

On the uniqueness of weak solutions of the 3D MHD equations in the Orlicz–Morrey space

In this article, we show that if a Leray solution (u, b) to the 3D magneto-hydrodynamic equation belongs to with r ∈ (0, 1) and P > 1, then (u, b) is the unique Leray solution on (0, T).