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We investigate regularity criteria for weak solutions to the Cauchy problem of the 3d co-rotational Beris-Edwards system for nematic liquid crystals, which couples the Navier–Stokes equations for the fluid velocity u with an evolution-diffusion equations for the Q -tenser. Our results yield that for any positive constant γ > 0 , if either the negative part of the associated pressure Π satisfies Π − [ ln ⁡ (1 + Π −) ] 1 + γ ∈ L ∞ (R + ; L 3 2 , ∞ (R 3)) , or the quantity 2 Π + | u | 2 + | ∇ Q | 2 satisfies (2 Π + + | u | 2 + | ∇ Q | 2) [ ln ⁡ (1 + 2 Π + + | u | 2 + | ∇ Q | 2) ] 1 + γ ∈ L ∞ (R + ; L 3 2 , ∞ (R 3)) , then the weak solution (u , Q) , to the 3d co-rotational Beris-Edwards system, is global-in-time smooth. Here, the subscript "−" and "+" denote the negative and the nonnegative part, respectively. L 3 2 , ∞ (R 3) denotes the standard weak Lebesgue space. If Q ≡ 0 , then our results extend some previous known results from the theory of the 3d Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

We study the partial regularity problem for a three dimensional simplified Ericksen–Leslie system, which consists of the Navier–Stokes equations for the fluid velocity coupled with a convective Ginzburg-Landau type equations for the molecule orientation, modelling the incompressible nematic liquid crystal flows. Base on the recent studies on the Navier–Stokes equations, we first prove some new local energy bounds and an εε-regularity criterion for suitable weak solutions to the simplified Ericksen-Leslie system, i.e., for σ ∈[0,1], there exists a ε > 0 such that if (u,d,P) is a suitable weak solution in Qr(z0) with 0 < r ≤ 1 and z0 = (x0,t0), and satisfies r −3/2−σ ∫t0t0−r2(∥|u|2∥2/2−σ/H−σ(Br(x0)) + ∥|∇d|2∥2/2−σ/H−σ(Br(x0)) + ∥P∥2/2−σ/H−σ(Br(x0))) dt ≤ ε, then (u,d) is regular at z0. Here, H−σ(Br(x)) is the dual space of Hσ0(Br(x)), the space of functions ƒ in the homogeneous Sobolev space Hσ(R3) such that supp ƒ ⊂ Br(x). Inspired by this ε-regularity criterion, we then improve the known upper Minkowski dimension of the possible interior singular points for suitable weak solutions from 95/63(≈1.50794) given by [24] (Nonlinear Anal. RWA, 44 (2018), 246–259.) to 835/613(≈1.36215). [ABSTRACT FROM AUTHOR]

This paper is concerned with the global well-posedness of the n-dimensional generalized incompressible Navier–Stokes equations with initial data in the critical Besov space. By fully using the advantage of suitable weighted functions and the Fourier localization technique, the global well-posedness is obtained without the hypothesis of smallness on initial data, but for some nonlinear smallness of the first iterate. We also give an example of initial data satisfying the nonlinear smallness condition, but whose norm is arbitrarily large. [ABSTRACT FROM AUTHOR]

This work establishes a new regularity criterion for the 3D incompressible MHD equations in term of one directional derivative of the pressure (i.e., ∂ 3 P ) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that for T > 0 , if ∂ 3 P ∈ L β (0 , T ; L α (R x 1 x 2 2 ; L γ (R x 3))) with 2 β + 1 γ + 2 α = k ∈ [ 2 , 3) and 3 k ≤ γ ≤ α ≤ 1 k - 2 , then the corresponding solution (u, b) to the 3D MHD equations is regular on [0, T]. [ABSTRACT FROM AUTHOR]

In this paper, we consider partial regularity of the Leray–Hopf type weak solutions to the Cauchy problem of the 3d co-rotational Beris–Edwards system for the nematic liquid crystal flows with Landau–De Gennes potential. The system under investigation consists of the Navier–Stokes equations for the velocity u coupled with an evolution equations for the Q -tensor. It is proved that a Leray–Hopf type weak solution (u , Q) ∈ L ∞ (0 , T ; L 2 (R 3 ; R 3)) ∩ L 2 (0 , T ; H 1 (R 3 ; R 3)) × L ∞ (0 , T ; H 1 (R 3 ; S 0 (3) )) ∩ L 2 (0 , T ; H 1 (R 3 ; S 0 (3) )) , to the 3d co-rotational Beris–Edwards system, is locally bounded if there exists a absolute constant ɛ so that the scaled local L p (1 ≤ p ≤ 5) norm of (u , ∇ Q) is less than ɛ. This implies that the one-dimensional parabolic Hausdorff measure for the possible singular point set is zero, which extends the corresponding result of Du et al. (2020) to the Leray–Hopf type weak solution. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for the 3D incompressible nematic liquid crystal flows is investigated. A new logarithmically improved blow-up criterion for the local in time smooth solution to the 3D incompressible nematic liquid crystal flows in an appropriate homogeneous Besov space is established. [ABSTRACT FROM AUTHOR]

In Guo and Wang (2012) [10] , Y. Guo and Y. Wang developed a general new energy method for proving the optimal time decay rates of the solutions to dissipative equations. In this paper, we generalize this method in the framework of homogeneous Besov spaces. Moreover, we apply this method to a model arising from electro-hydrodynamics, which is a strongly coupled system of the Navier–Stokes equations and the Poisson–Nernst–Planck equations through charge transport and external forcing terms. We show that some weighted negative Besov norms of solutions are preserved along time evolution, and obtain the optimal time decay rates of the higher-order spatial derivatives of solutions by the Fourier splitting approach and the interpolation techniques. [ABSTRACT FROM AUTHOR]

In recent paper [7] , Y. Du and K. Wang (2013) proved that the global-in-time Koch–Tataru type solution ( u , d ) to the n -dimensional incompressible nematic liquid crystal flow with small initial data ( u 0 , d 0 ) in BMO − 1 × BMO has arbitrary space–time derivative estimates in the so-called Koch–Tataru space norms. The purpose of this paper is to show that the Koch–Tataru type solution satisfies the decay estimates for any space–time derivative involving some borderline Besov space norms. More precisely, for the global-in-time Koch–Tataru type solution ( u , d ) to the nematic liquid crystal flow with initial data ( u 0 , d 0 ) ∈ BMO − 1 × BMO and ‖ u 0 ‖ BMO − 1 + [ d 0 ] BMO ≤ ε for some small enough ε > 0 , and for any positive integers k and m , one has ‖ t k + m 2 ( ∂ t k ∇ m u , ∂ t k ∇ m ∇ d ) ‖ L ˜ ∞ ( R + , B ˙ ∞ , ∞ − 1 ) ∩ L ˜ 1 ( R + ; B ˙ ∞ , ∞ 1 ) ≤ ε . Furthermore, we shall give that the solution admits a unique trajectory which is Hölder continuous with respect to space variables. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the regularity criterion for the solution of nematic liquid crystal flows in three dimensions ( ) and in two dimensions ( ). We prove that the solution is smooth up to time provided that there exists a positive constant such that (i) for , and (ii) for , [Copyright &y& Elsevier]

Abstract: In this paper, we study a hydrodynamical system modeling the deformation of the vesicle membrane under external incompressible viscous flow fields. The system is in the Eulerian formulation and is governed by the coupling of the incompressible Navier–Stokes equations with a phase field equation. In the three dimensional case, we establish two logarithmically improved blow-up criteria for local smooth solutions of this system in terms of the vorticity field only in the homogeneous Besov spaces. [Copyright &y& Elsevier]

Abstract: This paper proves a logarithmically improved Beale–Kato–Majda’s criterion for the 3D nematic liquid crystal flows, which says that if then the solution is smooth up to the time . [Copyright &y& Elsevier]