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In this short note, a logarithmically improved regularity criterion for the harmonic heat flow is established for arbitrary dimension. [ABSTRACT FROM AUTHOR]

In this paper, we establish a regularity criterion of solutions to the incompressible Navier-Stokes equations in three-dimensional infinite channel in terms of the derivatives of the vertical component of the velocity field. More precisely, we show that the strong solution exists on the time interval [0,T] provided that one of the following conditions holds: ∂u3∂xi∈Lq([0,T];Lp(Ω)), 1≤i≤3 where p>3, 1≤q<∞, 3p+2q≤12+32p for i=1,2, and p>2, 1≤q<∞, 3p+2q≤34+32p for i=3. [ABSTRACT FROM AUTHOR]

In this paper we prove the global-in-time existence of smooth solutions of the 2D generalized magnetohydrodynamic equations with zero viscosity, when the power of the Laplacian operator -Δ in the dissipation term of the induction equation is between 1 and 2. We also show a regularity criterion on the direction of the magnetic field b0:=b|b| when β=1. [ABSTRACT FROM AUTHOR]

In this paper, two regularity criteria to the two-dimensional incompressible generalized magnetohydrodynamics equations with zero magnetic diffusivity are given. It is proved that the solution remains smooth on [0,T] provided that the current j×b satisfies 0Tj(.,t) L 2λpλp-1 4αλp(2α-1) p + 2 dt<∞ with 1<α<2, 12≤λ≤1, 2≤p≤2λ, or the vorticity w×u satisfies0w(t,.)BMOdt<∞ with 1<α<2. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new regularity criterion for the 3D incompressible MHD equations involving partial components of the velocity gradient and magnetic fields. [ABSTRACT FROM AUTHOR]

This paper proves a new regularity criterion for the 3D generalized MHD system with fractional diffusion terms ( − Δ) u and ( − Δ) b with $0<\alpha <\frac 54\leqslant \beta $. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the incompressible Navier-Stokes equations in R³, and provide two sufficient conditions to ensure the smoothness of solutions. Both of them only involve two entries of the velocity Hessian tensor. [ABSTRACT FROM AUTHOR]

We establish some new regularity criteria for the three-dimensional incompressible Hall-magnetohydrodynamic equations. [ABSTRACT FROM AUTHOR]

Abstract: The aim of the paper is to establish regularity criteria for the weak solution of fluid passing through the porous media in . Our results show that if with , , then the weak solution is regular and unique; if with , , then the weak solution is regular and unique. [Copyright &y& Elsevier]

Abstract: This paper proves two regularity criteria for the density-dependent Hall-MHD system with positive initial density. We also prove a global nonexistence result for initial density with a high decrease at infinity. [Copyright &y& Elsevier]

In this paper, we consider sufficient conditions for the regularity of Leray-Hopf solutions of the 3D incompressible magnetohydrodynamic equations via two components of the velocity and magnetic fields in terms of BMO spaces. We prove that if [ABSTRACT FROM AUTHOR]

In this paper, we consider the three-dimensional generalized MHD equations, a system of equations resulting from replacing the Laplacian -δ in the usual MHD equations by a fractional Laplacian (δ)ɑ. We obtain a regularity criterion of the solution for the generalized MHD equations in terms of the summation of the velocity field u and the magnetic field b by means of the Littlewood-Paley theory and the Bony paradifferential calculus, which extends the previous result. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we consider sufficient conditions for the regularity of Leray Hoph Hopf solutions of the 3D incompressible magnetohydrodynamic equations via the velocity and magnetic fields in terms of spaces. We prove that if belongs to the space , then the solution is regular. This extends recent results contained in Gala (2011) [3]. [Copyright &y& Elsevier]

In this paper, we consider the three-dimensional Boussinesq equations, and obtain a logarithmically improved regularity criterion in terms of pressure. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies the three-dimensional density-dependent incompressible magnetohydrodynamic equations. First, a regularity criterion is proved which allows the initial density to contain vacuum. Then we establish another blow-up criterion in the Besov space when the positive initial density is bounded away from zero. Third, we prove a global nonexistence result for initial density with high decreasing at infinity. Fourth, we obtain a regularity criterion to the density-dependent incompressible magnetohydrodynamic equations in a bounded domain. Finally, we also give some remarks on the regularity criteria for the three-dimensional full compressible magnetohydrodynamic equations in a bounded domain and for the incompressible homogeneous magnetohydrodynamic equation in the whole space . [Copyright &y& Elsevier]

Several regularity criterions of Leray-Hopf weak solutions u to the 3D Navier-Stokes equations are obtained. The results show that a weak solution u becomes regular if the gradient of velocity component ∇hu (or ∇u3) satisfies the additional conditions in the class of Lq(0, T; Ḃp,rs (ℝ³)), where ∇h = (∂x1, ∂x2) is the horizontal gradient operator. Besides, we also consider the anisotropic regularity criterion for the weak solution of Navier-Stokes equations in ℝ³. Finally, we also get a further regularity criterion, when give the sufficient condition on ∂3u3. [ABSTRACT FROM AUTHOR]

In this paper, we study the regularity criterion of smooth solution to the Oldroyd model in ℝn (n = 2, 3). We obtain a Beale-Kato-Majda-type criterion in terms of vorticity in two and three space dimensions, namely, the solution (u(t,x), F(t,x)) does not develop singularity until t = T provided that ∇ x u ∈ L¹ (0, T; Ḃ∞,∞0 (ℝn)) in the case n = 2,3. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the incompressible Navier-Stokes equations in R, and provide a new regularity criterion involving only two entries of the Jacobian matrix of the velocity field. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we consider the regularity problem under the critical condition to the Boussinesq equations with zero heat conductivity. The Serrin type regularity criteria are established in terms of the critical Besov spaces. This improves a result established in a recent work by Geng and Fan (2012) [6]. [Copyright &y& Elsevier]

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]

In this paper, we study the regularity criterion of smooth solution to the Oldroyd model in ℝn(n = 2, 3). Firstly, we establish a regularity criterion in terms of the BMO norm of the gradient of columns of the deformation tensor in two space dimensions; secondly, we obtain a Beale-Kato-Majda-type criterion in terms of vorticity with the BMO norm in two and three space dimensions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the Cauchy problem for the 3D incompressible MHD equations. A regularity criterion by providing sufficient condition in terms of the pressure is established. More precisely, we show that if with and , then any weak solutions of the 3D MHD equations is regular on . [Copyright &y& Elsevier]

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 where denotes the critical Besov space, then the solution becomes a regular solution on . This result extends the recent regularity criterion obtained by Fan and Ozawa (2012) [11]. [Copyright &y& Elsevier]

In this paper, some sufficient conditions in terms of the magnetic field are established to guarantee global existence of solutions to a magnetohydrodynamic-omega model. [ABSTRACT FROM AUTHOR]

The paper studies regularity properties of set-valued mappings between metric spaces. In the context of metric regularity, nonlinear models correspond to nonlinear dependencies of estimates of error bounds in terms of residuals. Among the questions addressed in the paper are equivalence of the corresponding concepts of openness and 'pseudo-Hölder' behavior, general and local regularity criteria with special emphasis on 'regularity of order $$k$$', for local settings, and variational methods to extimate regularity moduli in case of length range spaces. The majority of the results presented in the paper are new. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the Cauchy problem for the 3D viscous MHD equations, and provide some regularity criteria involving only one directional derivative of the pressure, say . In particular, if then the solution remains smooth on . [Copyright &y& Elsevier]

We establish some regularity criteria for the incompressible Navier-Stokes equations in a bounded three-dimensional domain concerning the quotients of the pressure, the velocity field and the pressure gradient. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the incompressible Navier-Stokes equations in R, and provide two new regularity criteria involving only two entries of the Jacobian matrix of the velocity field. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we establish sufficient conditions for the regularity of solutions of Navier–Stokes equations based on one of the nine entries of the gradient tensor. We improve the recent results of C.S. Cao, E.S. Titi [C.S. Cao, E.S. Titi, Global regularity criterion for the 3D Navier–Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal. 202 (2011) 919–932] and Y. Zhou, M. Pokorný [Y. Zhou, M. Pokorný, On the regularity of the solutions of the Navier–Stokes equations via one velocity component, Nonlinearity 23 (2010) 1097–1107]. [Copyright &y& Elsevier]

In this paper, we consider the three-dimensional generalized Boussinesq equations, a system of equations resulting from replacing the Laplacian − Δ in the usual Boussinesq equations by a fractional Laplacian ( − Δ) α. We prove the local existence in time and obtain a regularity criterion of solution for the generalized Boussinesq equations by means of the Littlewood-Paley theory and Bony's paradifferential calculus. The results in this paper can be regarded as an extension to the Serrin-type criteria for Navier-Stokes equations and magnetohydrodynamics equations, respectively. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

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, whereis the multiplier space (a definition is given in the text) for 0 ≤ r ≤ 1, then the weak solution is actually regular. Since this spaceis 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]. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper we study the global regularity issue for the 3D incompressible MHD equations. It is proved that if the pressure satisfies , then the corresponding weak solution is regular on . [Copyright &y& Elsevier]

Abstract: This paper studies the Cauchy problem of the Euler–Landau–Lifshitz system arising in the nematic liquid crystal flows. We prove two logarithmically improved regularity criteria for local smooth solutions in the homogeneous Besov space. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate a hyperbolic–parabolic system arising from chemotaxis. A blow up criterion of local strong solutions to the model in is established based on the scaling invariance of the system. As a by-product, we prove a global-in-time well-posedness result when for large initial data. [Copyright &y& Elsevier]

In this paper, we consider the three-dimensional (3D) incompressible Boussinesq equations. We obtain some regularity criteria for the three-dimensional (3D) Boussinesq equations in the Morrey-Campanato space. [ABSTRACT FROM AUTHOR]

We consider the regularity criterion for the incompressible Navier-Stokes equations. We show that the weak solution is regular, provided for some T > 0, where Ẋr is the multiplier space. This extends a result of Kukavica and Ziane [14]. [ABSTRACT FROM AUTHOR]

Abstract: In the study of regularity criteria for the weak solutions of the 3D Navier–Stokes equations, an improved regularity criterion is obtained. More precisely, it is proved that if the pressure satisfies the critical growth condition then the weak solution is regular on . The finding is mainly based on the innovative function decomposition methods together with Besov space techniques. [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]

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]. [ABSTRACT FROM AUTHOR]

Abstract: For the incompressible Navier–Stokes equations in , a regularity criterion for weak solutions is proved under the assumption that the pressure belongs to the scaling invariant Lorentz space with small norm, while corresponding results for the velocity field were proved by Sohr. The main theorem continues and extends a previous result given by the author. [Copyright &y& Elsevier]

Abstract: An improved regularity criterion for a weak solution of three-dimensional magnetohydrodynamic equations is obtained. Employing the Fourier localization technique, it is proved that weak solutions become regular on if the summation of velocity field and magnetic field belong to the largest critical spaces: . This obviously extends the previous results. [Copyright &y& Elsevier]

Abstract: We prove that a smooth solution of the 3D Boussinesq system with zero viscosity in a bounded domain breaks down, if a certain norm of vorticity blows up at the same time. Here this norm is weaker than the bmo-norm. [Copyright &y& Elsevier]

In this paper, we establish some improved regularity conditions for the 3D incompressible magnetohydrodynamic equations via only two components of the velocity and magnetic fields. This is an improvement of the result given by Ji and Lee [8]. [ABSTRACT FROM AUTHOR]