Regularity via one vorticity component for the 3D axisymmetric MHD equations

In this paper, we investigate the regularity criteria of axisymmetric weak solutions to the three-dimensional (3D) incompressible magnetohydrodynamics (MHD) equations with nonzero swirl component. By making use of techniques of the Littlewood–Paley decomposition, we show that weak solutions to the 3D axisymmetric MHD equations become regular if the swirl component of vorticity satisfies that $w_{\theta }{e}_{\theta }\in L^1(0,T;{\dot{B}}_{\infty ,\infty }^0)$, which partially gives a positive answer to the marginal case for the regularity of MHD equations.     

Regularity criterion of axisymmetric weak solutions to the 3D Navier-Stokes equations

We consider the regularity of axisymmetric weak solutions to the Navier-Stokes equations in R³. Let u be an axisymmetric weak solution in R³×(0,T), w=curlu, and wθ be the azimuthal component of w in the cylindrical coordinates. Chae-Lee [D. Chae, J. Lee, On the regularity of axisymmetric solutions of the Navier-Stokes equations, Math. Z. 239 (2002) 645-671] proved the regularity of weak solutions under the condition wθ∈Lq(0,T;Lr), with , . We deal with the marginal case r=∞ which they excluded. It is proved that u becomes a regular solution if .     

Morrey-Campanato空间中三维 Navier—Stokes方程的正则性准则

利用能量不等式和一些临界空间中的不等式,在Morrey—Campanato空间获得了两个只涉及水平速度场的正则性准则,改进了一些已有的结果．     

Remark on the regularity for weak solutions to the magnetohydrodynamic equations

We study the regularity criteria for weak solutions to the incompressible magnetohydrodynamic (MHD) equations. Some regularity criteria, which are related only with u+B or u?B, are obtained for weak solutions to the MHD equations. Copyright © 2008 John Wiley & Sons, Ltd.     

Regularity criteria for the 3D MHD equations in term of velocity

In this paper, we consider three‐dimensional incompressible magnetohydrodynamics equations. By using interpolation inequalities in anisotropic Lebesgue space, we provide regularity criteria involving the velocity or alternatively involving the fractional derivative of velocity in one direction, which generalize some known results. Copyright © 2014 John Wiley & Sons, Ltd.     

Extension criterion on regularity for weak solutions to the 3D MHD equations

In this paper, we consider the regularity criteria for weak solutions to the 3D incompressible magnetohydrodynamic equations and prove some regularity criteria which are related only with u+B or u?B. This is an improvement of the result given by He and Wang (J. Differential Equations 2007; 238:1–17; Math. Meth. Appl. Sci. 2008; 31:1667–1684) and He and Xin (J. Differential Equations 2005; 213(2):235–254). Copyright © 2010 John Wiley & Sons, Ltd.     

On regularity criterion to the 3D axisymmetric incompressible MHD equations

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_re_r+u_θe_θ+u_ze_z and b = b_θe_θ, we prove that if |ru(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.     

Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations

In this paper, we study the local behavior of the solutions to the three-dimensional magnetohydrodynamic equations. we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions. It is shown that, in some neighborhood of (x₀,t₀), the gradients of the velocity field u and the magnetic field B are locally uniformly bounded in L^∞ norm as long as that either the scaled local L²-norm of the gradient or the scaled local total energy of the velocity field is small, and the scaled local total energy of the magnetic field is uniformly bounded. These estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory. As an immediately corollary we can derive an estimates of Hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid. Various partial regularity results are obtained as consequences of our blow-up estimates.     

Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary

We present some new regularity criteria for “suitable weak solutions” of the Navier-Stokes equations near the boundary in dimension three. We prove that suitable weak solutions are Hölder continuous up to the boundary provided that the scaled mixed norm with 3/p+2/q?2, 2<q?∞, (p,q)≠(3/2,∞) is small near the boundary. Our methods yield new results in the interior case as well. Partial regularity of weak solutions is also analyzed under some additional integral conditions.     

Remarks on the regularity criteria of weak solutions to the three-dimensional micropolar fluid equations

This paper is investigate the regularity criteria of weak solutions to the three-dimensional microp- olar fluid equations. Several sufficient conditions in terms of some partial derivatives of the velocity or the pressure are obtained.     

Regularity criterion of weak solutions to the 3D Navier-Stokes equations via two velocity components

This paper is concerned with the regularity criterion of Leray-Hopf weak solutions to the 3D Navier-Stokes equations with respect to Serrin type condition on two velocity filed components. It is shown that the weak solution u=(u₁,u₂,u₃) is regular on (0,T] if there exist two solution components, for example, u₂ and u₃, satisfying the condition

     

Blow‐up criteria of smooth solutions to the 3D Boussinesq equations

In this paper, we investigate three‐dimensional incompressible Boussinesq equations and establish some logarithmically improved blow‐up criteria of smooth solutions to the Cauchy problem for the incompressible Boussinesq equations. Copyright © 2011 John Wiley & Sons, Ltd.     

The regularity criterion of the weak solution to the 3D viscous Boussinesq equations in Besov spaces

In this paper, by using the Fourier localization technique and Bony's paraproduct decomposition, we give a regularity criterion of the weak solution to 3D viscous Boussinesq equations in Besov spaces. Copyright © 2010 John Wiley & Sons, Ltd.     

广义磁流体方程组轴对称解的正则准则

利用能量法研究广义磁流体方程组的轴对称弱解在三维空间中的正则性,得到了用方位角分量控制的正则准则,该准则表明方位角分量起主导作用.     

Regularity condition of solutions to the quasi-geostrophic equations in Besov spaces with negative indices

With a Hǒlder type inequality in Besov spaces, we show that every strong solution θ（t, x） on （0, T ） of the dissipative quasi-geostrophic equations can be continued beyond T provided that ⊥θ（t, x） ∈L 2γ/γ-2δ （（0, T ）; B^δ-γ/2 ∞∞（R^2）） for 0 〈 δ 〈 γ/2 .     

BMO and the regularity criterion for weak solutions to the magnetohydrodynamic equations

In this paper, we study the regularity criterion for weak solutions to the incompressible magnetohydrodynamic equations. We derive the regularity of weak solutions in the marginal class. Moreover, our result demonstrates that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solutions to the magnetohydrodynamic equations.