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This paper proves some regularity criteria for some simplified non-isothermal models for nematic liquid crystals. [ABSTRACT FROM AUTHOR]

In this note, we prove a logarithmic regularity criterion for the 2D MHD equations without magnetic diffusion in terms of the magnetic field in homogeneous Besov space B . ∞ , ∞ 0 . [ABSTRACT FROM AUTHOR]

This paper establishes a new regularity criterion for the Navier-Stokes equations in terms of two velocity components. We show that if the two velocity components ũ = (u1, u2, 0) satisfy ∫T0 ∥ũ(s)∥2B0∞,∞ds < ∞, then the solution can be smoothly extended after t = T. This gives an answer to an open problem in [B. ǫ. Dong, Z. Zhang, Nonlinear Anal. Real World Appl. 11(2010), 2415-2421]. [ABSTRACT FROM AUTHOR]

In this article, we show an Osgood type regularity criterion for the three-dimensional Newton-Boussinesq equations, which improves the recent results in [4].

In this article, we consider the regularity for weak solutions to the Navier-Stokes equations in $mathbb{R}^3$. It is proved that if the horizontal derivatives of the two velocity components $$ abla _hwidetilde{u}in L^{2/(2-r)}(0,T;dot{mathcal{M}}_{2,3/r} (mathbb{R}^3)),quad hbox{for }0

In this note, we study the regularity of Leray-Hopf weak solutions to the Navier-Stokes equation, with the condition $$ abla (u_{1},u_{2},0) in L^{frac{2}{1-r}}(0,T; dot{mathcal{M}}_{2,3/r} (mathbb{R}^3) , $$ where $dot{mathcal{M}}_{2,3/r}(mathbb{R}^3)$ is the Morrey-Campanato space for $0

We consider the regularity of weak solutions to the Navier-Stokes equations in $mathbb{R}^3$. Let $u:=(u_1,u_2,u_3)$ be a weak solution and $widetilde{u}:=(u_1,u_2,0)$. We prove that $u$ is strong solution if $ ablawidetilde{u}$ satisfy Serrin's type criterion.

We give a further investigation on the regularity criteria for the 3D nematic liquid crystal flows. It is show that the obtained anisotropic integrability regularity criterion to the 3D nematic liquid crystal flows involving ∂ i u j , i , j = 1 , 2 , 3 is an improvement of Qian (2016). Besides, we also prove the solution is regular if some components of the velocity field and the vectorial field modeling the orientation of the crystal molecules satisfy the Prodi–Serrin type conditions. [ABSTRACT FROM AUTHOR]

This paper proves a regularity criterion for the 3D generalized Hall-MHD system. [ABSTRACT FROM AUTHOR]

This paper is concerned with the regularity criterion for a class of axisymmetric solutions to 3D incompressible magnetohydrodynamic equations. More precisely, for the solutions that have the form of u = u r e r+ u θ e θ+ u z e z and b = b θ e θ, we prove that if | r u( x, t)|≤ C holds for −1≤ t < 0, then ( u, b) is regular at time zero. This result can be thought as a generalization of recent results in for the 3D incompressible Navier-Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we consider the two-dimensional (2D) Boussinesq equations with partial dissipation. The issue of whether the 2D Boussinesq equations always possess global (in time) classical solutions can be difficult when there is only partial dissipation. Therefore, the goal of this paper is to establish some regularity criteria of the corresponding systems. [ABSTRACT FROM AUTHOR]

In this paper we study the three-dimensional (3D) incompressible Hall-magnetohydrodynamic (Hall-MHD) equations and obtain its regularity criterion only in terms of the vorticity field. As a consequence, it improves the known results. [ABSTRACT FROM AUTHOR]

This paper establishes a regularity criterion of strong solutions to the two-dimensional compressible magnetohydrodynamic (MHD) flows. The criterion depends on the density, but is independent of the velocity and the magnetic field. More precisely, once the strong solutions blow up, the L ∞ -norm for the density tends to infinity. In particular, the vacuum in the solutions is allowed. [ABSTRACT FROM AUTHOR]

This paper proves a new regularity criterion for a 3D density-dependent incompressible liquid crystals model with positive initial density. [ABSTRACT FROM AUTHOR]

This paper studies the regularity problem for the 3D incompressible resistive viscous Hall-magneto-hydrodynamic (Hall-MHD) system. The Kolmogorov 41 phenomenological theory of turbulence [14] predicts that there exists a critical wavenumber above which the high frequency part is dominated by the dissipation term in the fluid equation. Inspired by this idea, we apply an approach of splitting the wavenumber combined with an estimate of the energy flux to obtain a new regularity criterion. The regularity condition presented here is weaker than conditions in the existing criteria (Prodi–Serrin type criteria) for the 3D Hall-MHD system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the case of Prodi–Serrin type regularity criterion involving u 3 and ∂ 3 u h . More precisely, it is shown that Leray’s weak solutions of the three-dimensional Navier–Stokes equations become regular if the third component of velocity (or the gradient of the velocity field) satisfies the additional end-point Prodi–Serrin type condition. [ABSTRACT FROM AUTHOR]

In this paper we consider an initial value problem of the 2D MHD equations with velocity dissipation and without magnetic diffusion. We establish a new magnetic regularity criterion in terms of the magnetic field. In contrast to the magnetic regularity criterion $\nabla b\in L^{1}(0,T; BMO(\mathbb {R}^{2}))$, our regularity criterion $\int_{0}^{T} (\Vert b\otimes b(s)\Vert _{B_{\infty,1}^{0}(\mathbb {R}^{2})} +\Vert b\otimes b(s)\Vert _{L^{2}(\mathbb {R}^{2})} )\,ds<\infty$ is different; for example, our simplified regularity criterion $\int_{0}^{T}\Vert b(s)\Vert ^{2}_{B_{\infty,1}^{\varepsilon}(\mathbb {R}^{2})}\,ds<\infty$ requires higher time integrability and lower regularity of space. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish a logarithmically improved the regularity criterion in terms of the homogeneous Besov spaceto the Boussinesq equations. We prove the solutionis smooth up to timeprovided that for someand. [ABSTRACT FROM AUTHOR]

In this paper, we prove the following regularity criterion ω : = curl u ∈ L 1 ( 1 , T ; B M O ) for the 3D Boussinesq system with zero heat conductivity in a bounded domain. Here ω is the vorticity of the fluid velocity field u and BMO denotes the function space of bounded mean oscillation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the density-dependent Boussinesq equations, and provide a regularity criterion involving the velocity field in BMO space. [ABSTRACT FROM AUTHOR]

In this paper, we consider the classical three-dimensional (3D) incompressible Navier–Stokes equations. By making use of a new anisotropic Sobolev inequality, we improve a regularity criterion for the weak solution to the 3D Navier–Stokes equations involving the gradient of one velocity component. This result improves the previous work of Zhou and Pokorný (2010). [ABSTRACT FROM AUTHOR]

We prove some regularity criteria for the incompressible magnetohydrodynamic equations with partial viscosity. We study three cases: incompressible magnetohydrodynamic equations with zero viscosity in a bounded domain, incompressible magnetohydrodynamic equations with zero resistivity in a bounded domain, and the density-dependent magnetohydrodynamic equations with zero heat conductivity and zero resistivity in the whole space . Our results extend and improve some known results. [ABSTRACT FROM AUTHOR]

Smooth solutions to the axially symmetric Navier–Stokes equations obey the following maximum principle: ‖ r u θ ( r , z , t ) ‖ L ∞ ≤ ‖ r u θ ( r , z , 0 ) ‖ L ∞ . We first prove the global regularity of solutions if ‖ r u θ ( r , z , 0 ) ‖ L ∞ or ‖ r u θ ( r , z , t ) ‖ L ∞ ( r ≤ r 0 ) is small compared with certain dimensionless quantity of the initial data. This result improves the one in Zhen Lei and Qi S. Zhang [10] . As a corollary, we also prove the global regularity under the assumption that | r u θ ( r , z , t ) | ≤ | ln ⁡ r | − 3 / 2 , ∀ 0 < r ≤ δ 0 ∈ ( 0 , 1 / 2 ) . [ABSTRACT FROM AUTHOR]

In this paper, we establish a regularity criterion in terms of the temperature for the 2D fractional Boussinesq equations with zero heat conductivity. [ABSTRACT FROM AUTHOR]

In this article, we study the global regularity of the 3D Navier–Stokes equations in terms of one velocity component. In particular, we establish a new version of regularity criterion in the framework of the anisotropic Lebesgue space. It is a generalization of the notable work of C. Cao and E.S. Titi (2011) [3] . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the regularity criteria for the 3D nematic liquid crystal flows in terms of the components of the velocity u or the gradient of the velocity field ∇ u . More precisely, the anisotropic integrability regularity criterion to the 3D nematic liquid crystal flows involving ∂ i u , i = 1 , 2 , 3 is obtained firstly, and then it is proved that the solution is regular if u 3 , ω 3 or ∂ 3 u 3 , ω 3 satisfies the Prodi–Serrin type conditions. [ABSTRACT FROM AUTHOR]

We will prove some regularity criteria for harmonic heat flow, biharmonic heat flow and a surface growth model. [ABSTRACT FROM AUTHOR]

We prove a logarithmic regularity criterion for the 3D generalized magnetohydrodynamics (MHD) system with diffusion terms -?u and (-?)ßb, with 5/2 ≤ ∞. [ABSTRACT FROM AUTHOR]

We prove a new scaling invariant regularity criterion for the 3D MHD equations via horizontal gradient of horizontal components of weak solutions. This result improves a recent work by Ni et al. (2012), in the sense that the assumption on the horizontal gradient of the vertical components is removed. As a byproduct, a scaling invariant regularity criterion involving vertical components of vorticity and current density is also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new regularity criterion for the 3D Navier–Stokes equations via partial components of the fractional derivative. The main result shows that regularity criterion established on the fractional derivatives of u can be almost replaced by several of them. [ABSTRACT FROM AUTHOR]

This paper concerns the 3D Navier-Stokes equations and prove an almost Serrin-type regularity criterion in terms of one directional derivative of the pressure. [ABSTRACT FROM AUTHOR]

This paper proves some regularity criteria for the Hall-MHD system in terms of the pressure and the magnetic field. [ABSTRACT FROM AUTHOR]

In this paper we establish the regularity criteria of solutions to the three space dimensions generalized viscous Hall-magnetohydrodynamics. Moreover, we also prove that there exist classical solutions globally in time to the generalized viscous Hall-MHD if the initial data is small enough or the fractional Laplacian dissipation powers α and β are suitably large. These results are some improvements of very recent work (Chae and Lee, 2014) established by Chae and Lee. [ABSTRACT FROM AUTHOR]

In this paper, we establish the global weak solutions for two-dimensional Boussinesq equations with mixed partial dissipation and thermal diffusivity. Moreover, we give global regularity criterion for weak solutions to the two-dimensional Boussinesq equations with mixed partial dissipation and thermal diffusivity. [ABSTRACT FROM AUTHOR]

We establish a new regularity criterion for the 2D full compressible magnetohydrodynamic system in a bounded domain. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we provide a regularity criterion for the Navier–Stokes equations involving the diagonal entry of the velocity gradient, which says that if ∂ 3 u 3 ∈ L ∞ ( 0 , T ; L 2 ( R 3 ) ) , then the weak solution is smooth on ( 0 , T ) . This is an important case in the sense that the strong solution is in this class. Moreover, we verify the limiting case of one regularity criterion in Cao and Titi (2011). [ABSTRACT FROM AUTHOR]

In this paper, we establish some regularity criteria for both the simplified and the general Ericksen–Leslie system in R 3 . [ABSTRACT FROM AUTHOR]

In this paper we investigate the regularity criterion for the local in time classical solution to the 3D incompressible magnetohydrodynamic equations. We prove that if ∇ × u belongs to L 2 ( 0 , T ; B ̇ ∞ , ∞ − 1 ) , then the local solution ( u , B ) can be extended beyond time T . As a consequence, this result extends several previous works. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for the 3D viscous incompressible magnetohydrodynamic equations and establish a Beale-Kato-Majda regularity criterion of smooth solutions in terms of the velocity vector in the homogeneous bounded mean oscillations space. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper proves the global existence of strong solutions of the 2D liquid crystal model when ν1 = k2 = 0, ν2 = k1 = 1 or ν1 = k2 = 1, ν2 = k1 = 0. We also prove some regularity criteria when ν1 = k1 = 1, ν2 = k2 = 0 or ν1 = k1 = 0, ν2 = k2 = 1. [ABSTRACT FROM AUTHOR]