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.     

On the regularity criteria for weak solutions to the magnetohydrodynamic equations

We study the regularity criteria for weak solutions to the incompressible magnetohydrodynamic equations. Some regularity criteria are obtained for weak solutions to the magnetohydrodynamic equations, which generalize the results in [C. He, Z. Xin, On the regularity of solutions to the magneto-hydrodynamic equations, J. Differential Equations 213 (2) (2005) 235-254]. Our results reveal 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.     

Remarks on partial regularity for suitable weak solutions of the incompressible magnetohydrodynamic equations

In this paper, we are concerned with the partial regularity for suitable weak solutions of the tri-dimensional magnetohydrodynamic equations. With the help of the De Giorgi iteration method, we obtain the results proved by He and Xin (C. He, Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal. 227 (2005) 113–152), namely, the one dimensional parabolic Hausdorff measure of the possible singular points of the velocity field and the magnetic field is zero.     

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.     

The combined inviscid and non-resistive limit for the nonhomogeneous incompressible magnetohydrodynamic equations with Navier boundary conditions

In this paper, we establish the existence of the global weak solutions for the nonhomogeneous incompressible magnetohydrodynamic equations with Navier boundary conditions for the velocity field and the magnetic field in a bounded domain ? ? R3. Furthermore,we prove that as the viscosity and resistivity coefficients go to zero simultaneously, these weak solutions converge to the strong one of the ideal nonhomogeneous incompressible magnetohydrodynamic equations in energy space.     

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.     

Limiting case for the regularity criterion of the Navier-Stokes equations and the magnetohydrodynamic equations

Any weak solution u to the Navier-Stokes equations is showed to be regular under the assumption that ||u|| L 2w (0,T ;L ∞ ( R 3 )) is sufficiently small, which is a limiting case of the regularity criteria derived by Kim and Kozono. Our result gives a positive answer to the question proposed by Kim and Kozono. For the incompressible magnetohydrodynamic equations, we also show the regularity of weak solution only under the assumption that ||u|| L 2w (0,T ;L ∞ ( R 3 )) is sufficiently small.     

Global Regularity for the 2D Magneto-Micropolar Equations with Partial Dissipation

This paper studies the global existence and regularity of classical solutions to the 2D incompressible magneto-micropolar equations with partial dissipation. The magneto-micropolar equations model the motion of electrically conducting micropolar fluids in the presence of a magnetic field. When there is only partial dissipation, the global regularity problem can be quite difficult. We are able to single out three special partial dissipation cases and establish the global regularity for each case. As special consequences, the 2D Navier-Stokes equations, the 2D magnetohydrodynamic equations, and the 2D micropolar equations with several types of partial dissipation always possess global classical solutions. The proofs of our main results rely on anisotropic Sobolev type inequalities and suitable combination and cancellation of terms.     

The Zero Mach Number Limit of the Three-Dimensional Compressible Viscous Magnetohydrodynamic Equations

This paper is concerned with the zero Mach number limit of the three-dimension- al compressible viscous magnetohydrodynamic equations. More precisely, based on the local existence of the three-dimensional compressible viscous magnetohydrodynamic equations, first the convergence-stability principle is established. Then it is shown that, when the Mach number is sufficiently small, the periodic initial value problems of the equations have a unique smooth solution in the time interval, where the incompressible viscous magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number goes to zero. Moreover, the authors prove the convergence of smooth solutions of the equations towards those of the incompressible viscous magnetohydrodynamic equations with a sharp convergence rate.     

Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations

In this paper, we study the incompressible limit of the three-dimensional compressible magnetohydrodynamic equations, which models the dynamics of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. Based on the convergence-stability principle, we show that, when the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient are sufficiently small, the initial-value problem of the model has a unique smooth solution in the time interval where the ideal incompressible magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient go to zero. Moreover, we obtain the convergence of smooth solutions for the model forwards those for the ideal incompressible magnetohydrodynamic equations with a sharp convergence rate.     

A New Regularity Criterion for the Navier-Stokes Equations in Terms of the Direction of Vorticity

In this paper, we prove a new regularity criterion in terms of the direction of vorticity for the weak solution to 3-D incompressible Navier-Stokes equations.     

A New Regularity Criterion for the Navier-Stokes Equations in Terms of the Direction of Vorticity

In this paper, we prove a new regularity criterion in terms of the direction of vorticity for the weak solution to 3-D incompressible Navier-Stokes equations.     

A regularity criterion for the density‐dependent magnetohydrodynamic equations

In this paper, regularity criterion for the 3‐D density‐dependent magnetohydrodynamic equation is considered. It is proved that the solution keeps smoothness only under an integrable condition on the velocity field in multiplier spaces. Hence, it turns out that the velocity field plays a dominant role in the regularity criteria of the weak solutions to this nonlinear coupling problem even with density. Copyright © 2009 John Wiley & Sons, Ltd.     

On the incompressible limits for the full magnetohydrodynamics flows

In this article the incompressible limits of weak solutions to the governing equations for magnetohydrodynamics flows on both bounded and unbounded domains are established. The governing equations for magnetohydrodynamic flows are expressed by the full Navier-Stokes system for compressible fluids enhanced by forces due to the presence of the magnetic field as well as the gravity and with an additional equation which describes the evolution of the magnetic field. The scaled analogues of the governing equations for magnetohydrodynamic flows involve the Mach number, Froude number and Alfven number. In the case of bounded domains the establishment of the singular limit relies on a detail analysis of the eigenvalues of the acoustic operator, whereas the case of unbounded domains is being treated by their suitable approximation by a family of bounded domains and the derivation of uniform bounds.     

Partial regularity of suitable weak solutions to the four‐dimensional incompressible magneto‐hydrodynamic equations

In this paper, we study the partial regularity of suitable weak solutions to the incompressible magneto‐hydrodynamic equations in dimension four by borrowing and improving the arguments given by Caffarelli, Kohn, and Nirenberg for incompressible Navier–Stokes equations. The so‐called ε‐regularity criteria are established for suitable weak solutions. As an application, an estimate on Hausdorff dimension of the possible singular points set for a suitable weak solution is given. Finally, we present further information on distribution of the possible singular points if the given initial data decay sufficiently rapidly or are not too singular at the origin, in some sense. Copyright © 2012 John Wiley & Sons, Ltd.     

Boussinesq方程解的部分正则性

本文考虑Boussinesq方程一类合适弱解的部分正则性.我们先运用广义能量不等式和奇异积分理论得到一些无维量的估计;再通过合适弱解满足的等式,运用迭代技巧,推导出温度场的小性估计;最后由尺度分析(scaling arguments)得到了一类合适弱解的部分正则性.     

A Beale–Kato–Majda criterion for the 3D viscous magnetohydrodynamic equations

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.     

Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion

Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.     

Control of magnetohydrodynamic flow in the formation of a magnetic field with a prescribed configuration

Control problems for the viscous incompressible magnetohydrodynamic equations are considered. The goal is to produce a magnetic field with a prescribed structure at a given time by applying electromotive forces so as to minimize Joule heating or the work done on the conduction currents. Based on estimates obtained for the solution of a subdifferential Cauchy problem for the Navier-Stokes-type equations, the possibility of producing the required magnetic field is proved and the solvability conditions for the control problems are found.     

Mixed boundary value problems for steady-state magnetohydrodynamic equations of viscous incompressible fluid

The inhomogeneous boundary value problem for the steady-state magnetohydrodynamic equations of viscous incompressible fluid under the Dirichlet conditions for the velocity and mixed boundary conditions for the electromagnetic field is considered. Sufficient conditions for the data that ensure the global solvability of this problem and the local uniqueness of its solution are found.