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.     

Some regularity criteria for the 3D incompressible magnetohydrodynamics

In this paper, we consider regularity criterion for the three-dimensional incompressible magnetohydrodynamic equations. We present some sufficient integrability conditions on some components of the velocity and magnetic fields for the regularity of the weak solutions.     

Remarks on the regularity criteria for generalized MHD equations

We study the Cauchy problem for the generalized MHD equations, and prove some regularity criteria involving the integrability of ∇u in the Morrey, multiplier spaces.     

A new scaling invariant regularity criterion for the 3D MHD equations in terms of horizontal gradient of horizontal components

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.     

A note on the regularity criteria for the Navier-Stokes equations

We consider the regularity problem for 3D Navier-Stokes equations in a bounded domain with smooth boundary. A new sufficient condition which guarantees the regularity of weak solutions on the quotient ∇p/(1+|u|^δ₁+|∇u|^δ₂) for the Navier-Stokes equations is established.     

Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces

The aim of this paper is to establish some logarithmically improved regularity criteria in term of the multiplier spaces to the Navier-Stokes equations.     

三维Navier-Stokes方程组关于速度梯度两个分量的正则性准则的一个改进

该文考虑三维Navier-Stokes方程组的Cauchy问题,得到了一个改进的、各向异性的、关于速度梯度的两个分量的正则性准则.此准则改进了文献[24]中的结果.     

三维磁微极流体方程弱解的正则性

本文给出了磁微极流体方程弱解的一个新的正则性准则:如果u满足uz ∈Lq(0,T;Lp(R3)),其中p≥3且满足3/p+2/q≤1,那么弱解(u,ω,b)在(0,T)是光滑解.     

Geometric depletion of vortex stretch in 3D viscous incompressible flow

New geometric constraints on vorticity are obtained which suppress possible development of finite-time singularity from the nonlinear vortex stretching mechanism. We find a new condition on the smoothness of the direction of vorticity in the vortical region which yields regularity. We also detect a regularity condition of isotropy type on vorticity in the intensive vorticity region via a new cancellation principle. This is in contrast with the one of isotropy type on the curl of vorticity obtained recently by A. Ruzmaikina and Z. Gruji? [A. Ruzmaikina, Z. Gruji?, On depletion of the vortex-stretching term in the 3D Navier-Stokes equations, Comm. Math. Phys. 247 (2004) 601-611]. We improve as well all of their results by eliminating their assumption that the initial vorticity ω₀ is required to be in L¹.     

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．     

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

     

Geometric constrains for global regularity of 2D quasi-geostrophic flows

We study the two-dimensional quasi-geostrophic equations (2D QG) in Sobolev spaces. We first prove a new analytic condition for global regularity which is both sufficient and necessary. We then prove several new results on the geometric constraints on the 2D QG active scalar which suppress the development of singularity from the nonlinear stretching mechanism. We focus mainly on the case with critical dissipation. Our results are also relevant to the inviscid case.     

A regularity criterion for the solutions of 3D Navier-Stokes equations

In terms of two partial derivatives of any two components of velocity fields, we give a new criterion for the regularity of solutions of the Navier-Stokes equation in R³. More precisely, let u=(u¹,u²,u³) be a weak solution in (0,T)×R³. Then u becomes a classical solution if any two functions of _∂1u¹, _∂2u² and _∂3u³ belong to Lθ(0,T;Lr(R³)) provided with , .     

A regularity criterion for 3D micropolar fluid flows

In this work, we prove a regularity criterion for micropolar fluid flows in terms of the pressure in Besov space.     

Regularity criteria of axisymmetric weak solutions to the 3D magnetohydrodynamic equations

In this paper, we study the regularity criteria for axisymmetric weak solutions to the MHD equations in ?³. Let ω _θ , J _θ and u _θ be the azimuthal component of ω, J and u in the cylindrical coordinates, respectively. Then the axisymmetric weak solution (u, b) is regular on (0, T) if (ω _θ , J _θ ) ∈ L ^q (0, T; L ^p ) or (ω _θ , ▽(u _θ e _θ )) ∈ L ^q (0, T; L ^p ) with $\tfrac{3} {p} + \tfrac{2} {q} \leqslant 2$ , $\tfrac{3} {2} < p < \infty$ . In the endpoint case, one needs conditions $\left( {\omega _\theta ,J_\theta } \right) \in L^1 \left( {0,T;\dot B_{\infty ,\infty }^0 } \right)$ or $\left( {\omega _\theta ,\nabla \left( {u_\theta e_\theta } \right)} \right) \in L^1 \left( {0,T;\dot B_{\infty ,\infty }^0 } \right)$ .