A new blowup criterion for strong solutions to the three‐dimensional compressible magnetohydrodynamic equations with vacuum in a bounded domain

In this paper, we establish a new blowup criterions for the strong solution to the Dirichlet problem of the three‐dimensional compressible MHD system with vacuum. Specifically, we obtain the blowup criterion in terms of the concentration of density in BMO norm or the concentration of the integrability of the magnetic field at the first singular time. The BMO‐type estimate for the Lam system 2.6 and a variant of the Brezis‐Waigner's inequality 2.3 play a critical role in the proof. Copyright © 2017 John Wiley & Sons, Ltd.     

On the suitable weak solutions to the Boussinesq equations in a bounded domain

In this paper we construct the suitable weak solution for the initial-boundary value problem of the Boussinesq equations and obtain some properties for these solutions. Also in the case of a two dimensional space the uniqueness of weak solution is proved.     

A note on global existence of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force

In this note, for the case of , we prove the existence of global-in-time finite energy weak solution of the equations of a two-dimensional magnetohydrodynamics with Coulomb force, where γ denotes the adiabatic exponent. The value is the optimal lower bound of γ to establish global-in-time finite energy weak solution under current frame.     

Energy dissipation for weak solutions of incompressible MHD equations

In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D(u, B) that stems from an eventual lack of smoothness in the solution, and then obtain a local equation of energy for weak solutions of 3D MHD equations. Finally, we consider the 2D case at the end of this article.     

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.     

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.     

Blow-up of smooth solutions to the compressible magnetohydrodynamic flows

In this paper, we prove the blow-up phenomena of smooth solutions to the Cauchy problem for the full compressible magnetohydrodynamic equations and isentropic compressible magnetohydrodynamic equations with constant and degenerate viscosities under some restrictions on the initial data. In particular, our results do not require that the initial data have compact support or contain vacuum in any finite region.     

Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations

In this paper we derive some new equations and we call them MHD-Leray-alpha equations which are similar to the MHD equations. We put forward the concept of weak and strong solutions for the new equations. Whether the 3-dimensional MHD equations have a unique weak solution is unknown, however, there is a unique weak solution for the 3-dimensional MHD-Leray-alpha equations. The global existence of strong solution and the Gevrey class regularity for the new equations are also obtained. Furthermore, we prove that the solutions of the MHD-Leray-alpha equations converge to the solution of the MHD equations in the weak sense as the parameter ε in the new equations converges to zero.     

Remarks on Leray's self‐similar solutions to the 3D magnetohydrodynamic equations

In this note, we examine small self‐similar solutions of the 3D incompressible magnetohydrodynamic equations in homogenous function spaces and show the uniqueness of self‐similar solutions when the velocity field is small in some homogeneous function spaces. This extend the recent results given by C. He and Z. Xin. “On the self‐similar solutions of the magneto‐hydro‐dynamic equations. Acta Mathematica Scientia, série B English Edition 2009; 29 (3): 583–598”. It is a natural way to extend the space widely and improve the previous results. Copyright © 2013 John Wiley & Sons, Ltd.     

Global well‐posedness of classical solutions to 3D compressible magnetohydrodynamic equations with large potential force

We consider the Cauchy problem of isentropic compressible magnetohydrodynamic equations with large potential force in . When the initial data (ρ₀,u₀,H₀) is of small energy, we investigate the global well‐posedness of classical solutions where the flow density is allowed to contain vacuum states.     

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.     

Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force

In this paper, we consider the equations of Magnetohydrodynamics with Coulomb force which is of hyperbolic–parabolic–elliptic mixed type. By constructing the approximate solutions to the modified system with an artificial pressure term added, global existence of finite energy weak solutions is established via the weak convergence method. More careful argument has been paid to overcome the new difficulty arising from the Poisson term of Coulomb force in two dimensions when the adiabatic exponent is close to one. We also investigate the large-time behavior of such weak solutions after discussing the regularity and uniqueness of solutions to the stationary problem.     

Existence of partially regular weak solutions to Landau-Lifshitz-Maxwell equations

In this paper, the authors establish the existence of partially regular weak solutions to the Landau-Lifshitz equations coupling with static Maxwell systems in 3 dimensions by Ginzburg-Landau approximation. It is proved that the Hausdorff measure of the singular set is locally finite. This extends the similar results of Ding and Guo [S. Ding, B. Guo, Hausdorff measure of the singular set of Landau-Lifshitz equations with a nonlocal term, Comm. Math. Phys. 250 (1) (2004) 95-117] from the stationary solutions to weak solutions and the results of Wang [C. Wang, On Landau-Lifshitz equations in dimensions at most four, Indiana Univ. Math. J. 55 (5) (2006) 1615-1644] from Landau-Lifshitz equations to Landau-Lifshitz-Maxwell equations.     

On the regularity criteria for the 3D magnetohydrodynamic equations via two components in terms of BMO space

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 belongs to the space , then the solution (u,b) is regular. This extends recent results contained by Gala, Ji E and Lee J. Copyright © 2013 John Wiley & Sons, Ltd.     

Weak and smooth solutions to incompressible Navier-Stokes-Landau-Lifshitz-Maxwell equations

Considering the Navier-Stokes-Landau-Lifshitz-Maxwell equations, in dimensions two and three, we use Galerkin method to prove the existence of weak solution. Then combine the a priori estimates and induction technique, we obtain the existence of smooth solution.     

Strong solutions to 3D compressible magnetohydrodynamic equations with Navier‐slip condition

We consider the short time strong solutions to the compressible magnetohydrodynamic equations with initial vacuum, in which the velocity field satisfies the Navier‐slip condition. The Navier‐slip condition differs in many aspects from no‐slip conditions, and it has attracted considerable attention in nanoscale and microscale flows research. Inspired by Kato and Lax's idea, we use the Lax–Milgram theorem and contraction mapping argument to prove local existence. Moreover, under the Navier‐slip condition, we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of L^∞ norm of the deformation tensor D(u). Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of weak solutions to the three-dimensional steady compressible magnetohydrodynamic equations

The purpose of this paper is to prove the existence of a spatially periodic weak solution to the steady compressible isentropic MHD equations in R3 for any specific heat ratio γ 1.The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density,and the method of weak convergence.According to the author's knowledge,it is the first result that treats in three dimensions the existence of weak solutions to the steady compressible MHD equations with γ 1.