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A one-component regularity criterion for the non-resistive axially symmetric Hall-MHD system is given in this paper. More precisely, we show that strong solutions to a family of axially symmetric initial data, whose velocity and current density have trivial swirl components, could be smoothly extended beyond a possible blow-up time T ∗ if and only if the swirl component of the magnetic field h θ satisfies a Beale-Kato-Majda-type criterion. See (1.5) below. This criterion is not trivial even if the velocity field vanishes since the quantity h θ r satisfies a partial Burgers' equation in this situation. [ABSTRACT FROM AUTHOR]

In this paper, we consider Cauchy problem of the axially symmetric Magnetohydrodynamic (MHD) system. By using energy method, we establish some regularity criteria of the solutions for the axisymmetric solutions of the three dimensional incompressible MHD system. [ABSTRACT FROM AUTHOR]

In this paper we establish a regularity criterion for the 2D full compressible MHD equations with zero heat conductivity and initial vacuum in a bounded domain. [ABSTRACT FROM AUTHOR]

Abstract In this paper we consider the 3D compressible MHD system in a bounded domain. We first prove a regularity criterion and then use it and the bootstrap argument to show the uniform-in- η global small solutions. Here η is the resistivity coefficient. [ABSTRACT FROM AUTHOR]

This paper establishes a regularity criterion for the strong solution to the Leray‐α‐MHD in Campanato‐Morrey space. We show that if the magnetic field b satisfies the following condition: ∫0T‖∇b(τ,·)‖M˙qp2p2p−3+‖D2b(τ,·)‖M˙qp2p2p−3dτ<∞, the strong solution (v,b) to the 3‐D Leray‐α MHD equation can be extended beyond T > 0. We also prove that if the magnetic field b satisfies ∫0T‖∇b(τ,·)‖M˙qppp−1dτ<∞, the strong solution (v,b) to the 2‐D Leray‐α MHD equation can be extended beyond T > 0, which extends the results in previous studies. [ABSTRACT FROM AUTHOR]

Abstract This work establishes a new regularity criterion for the 3D incompressible magneto-hydrodynamical (MHD) equations in terms of one directional derivative of the velocity (i.e., ∂ 3 u) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that if ∂ 3 u satisfies ∫ 0 T ‖ ‖ ∂ 3 u (τ) ‖ L x 3 α ‖ L x 1 x 2 β q d τ < + ∞ , where 2 q + 1 α + 2 β = k ∈ [ 1 , 3 2 ] and 3 k < α ≤ β ≤ 1 k − 1 , for some T > 0 , then the corresponding solution (u , b) to the 3D MHD equations is regular on (0 , T) × R 3. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we establish a new regularity criterion for the two-dimensional incompressible generalized magnetohydrodynamics equations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the two-dimensional incompressible generalized magnetohydrodynamics equations. We obtain some regularity criteria to the corresponding equations in terms of the current density j or the vorticity ω. Our approach is based on a key observation on the nice structure of the incompressible magnetohydrodynamics equations. [ABSTRACT FROM AUTHOR]

In this paper, we prove a logarithmic improvement of regularity criterion only in terms of the pressure in Morrey–Campanato spaces for the Cauchy problem to the incompressible MHD equations. [ABSTRACT FROM AUTHOR]

We establish a regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity and vacuum in a bounded domain. [ABSTRACT FROM AUTHOR]

The paper is dedicated to study of the regularity criterion for weak solutions to the 3D incompressible MHD equations. Employing the Littlewood–Paley decomposition, we show that if ∇ ˜ u ˜ = ( ∂ 1 u ˜ , ∂ 2 u ˜ ) ∈ L s 1 ( [ 0 , T ) ; B ˙ r 1 , 2 r 1 3 0 ( R 3 ) ) , 2 s 1 + 3 r 1 = 2 , 3 2 < r 1 ≤ ∞ and ∇ ˜ b ˜ = ( ∂ 1 b ˜ , ∂ 2 b ˜ ) ∈ L s 2 ( [ 0 , T ) ; B ˙ r 2 , 2 r 2 3 0 ( R 3 ) ) , 2 s 2 + 3 r 2 = 2 , 3 2 < r 2 ≤ ∞ , then the solutions to the MHD actually is smooth on ( 0 , T ) . [ABSTRACT FROM AUTHOR]

In this paper, we consider the three-dimensional incompressible Hall–magnetohydrodynamic equations, and establish an improved regularity criterion of local in time classical solutions involving only the vorticity field. Consequently, this improves the previous result. [ABSTRACT FROM AUTHOR]

This paper proves a regularity criterion for 3D density-dependent incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient. In particular, we build a blowup criterion just in terms of the gradient of density and velocity, and the initial density need not be strictly positive. [ABSTRACT FROM AUTHOR]

In this paper we consider the three dimensional (3D) incompressible Hall-magnetohydrodynamic equations (Hall-MHD). It is proved that the local smooth solution can be continued beyond timeTprovided that the velocityuand the magnetic fieldBsatisfy for. As a consequence, this result extend the previous result. [ABSTRACT FROM PUBLISHER]

In this paper, we are concerned with the regularity criterion for weak solutions to the 3D incompressible MHD equations. We show that if any two groups functions of $$(\partial _{1}u_{1}, \partial _{1}b_{1})$$ , $$(\partial _{2}u_{2}, \partial _{2}b_{2})$$ and $$(\partial _{3}u_{3}, \partial _{3}b_{3})$$ belong to the space $$L^{\theta }([0,T);L^{r}(\mathbb {R}^3)), \frac{2}{\theta }+\frac{3}{r}=2, \frac{3}{2}

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 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)]. [ABSTRACT FROM AUTHOR]

This paper proves a regularity criterion for a 3D generalized Hall-MHD system in terms of velocity gradient in negative Besov spaces. [ABSTRACT FROM AUTHOR]

The aim of the paper is to establish regularity criteria for the weak solution of fluid passing through the porous media in R^{3}. We show that if (grad_{h}u,\partial_{3}b_{3}) belongs to L^{2alpha,2gamma} with 2/alpha+3/gamma<=3, 1<=gamma<=infinity, then the weak solution is regular and unique; if (grad_{h}u, grad_{h}b) belongs to L^{2alpha,2gamma} with 2/alpha+3/gamma}<=3, 1<=gamma<=infinity, then the weak solution is regular and unique; if (partial_{3}u, grad u_{3}) belongs to L^{2alpha,2gamma} and (u_{3}, b, \partial_{3} b, grad b_{3}) belongsL^{4alpha,4gamma} with 2/alpha+3/gamma}<=3, 1<=gamma<=infinity, then the weak solution is regular and unique. Here we use the notation grad_{h}=(partial_{1}, partial_{2}). [ABSTRACT FROM AUTHOR]

In this paper, we prove that suitable weak solution ( u, b) of the 3-D MHD equations can be extended beyond T if u ∈ L (0, T; L (ℝ)) and the horizontal components b of the magnetic field satisfies the well-known Ladyzhenskaya-Prodi-Serrin condition, which improves the corresponding regularity criterion by Mahalov-Nicolaenko-Shilkin. [ABSTRACT FROM AUTHOR]

This paper concerns with the regularity criteria for the 3 D axisymmetric MHD system. It is proved that the control of swirl component of vorticity can ensure the smoothness of the solution. [ABSTRACT FROM AUTHOR]

This paper proves a regularity criterion ∇ u , ∇ b ∈ L ∞ ( 0 , T ; L ∞ ) for 3D ideal density-dependent MHD system. [ABSTRACT FROM AUTHOR]

In this paper we investigate the regularity criterion for the local-in-time classical solution to the three-dimensional (3D) incompressible Hall-magnetohydrodynamic equations (Hall-MHD). It is proved that the control of the vorticity alone can ensure the smoothness of the solution. [ABSTRACT FROM AUTHOR]

We study the global regularity of classical solution to two-and-half-dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion. We prove that any possible finite time blow-up can be controlled by the L ∞-norm of the vertical components. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper studies the global regularity of classical solutions to 2D magneto-micropolar fluid equations with only micro-rotational velocity dissipation and magnetic diffusion. Here the micro-rotational velocity dissipation and magnetic diffusion are given by − Δ Ω and ( − Δ ) β b . Making use of several combined quantities, maximal regularity of heat operator and Littlewood–Paley decomposition theory, we establish a regularity criterion in terms of magnetic field for the case β = 1 and the global regularity for β > 1 . The regularity criterion given here is also new even for the 2D magnetohydrodynamic equations. In addition, to prove these two main results, as preparation we establish a new global a priori estimate for magnetic field, namely Δ b ∈ L ∞ ( 0 , T ; L p ( R 2 ) ) with p ≥ 2 which also holds for the 2D magnetohydrodynamic equations as a particular case. [ABSTRACT FROM AUTHOR]

Our main object is to establish a regularity criterion [ABSTRACT FROM AUTHOR]

In this paper, we consider the generalized Hall-MHD system in dimension three. A new regularity criterion is established for β ∈ ( 1 2 , 1 ] . [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 proves a regularity criterion for the 3D generalized Hall-MHD system. [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 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]

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 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 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]

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]

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]

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]

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 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 concerns the conditional global regularity of incompressible MHD equations with horizontal dissipation and horizontal magnetic diffusion in two dimension. When only horizontal dissipation and horizontal magnetic diffusion are present, there is no control on the vertical derivatives of velocity field and magnetic field, which is the main difficulty to establish the global regularity. In this paper, we establish a global regularity criterion in terms of one entry of the velocity gradient tensor or one entry of the magnetic field gradient tensor, which extends the recent work (Fan and Ozawa, 2014). [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]

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 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]

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]