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.     

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.     

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.     

Interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations

We present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three: a suitable weak solution is regular near an interior point z if the scaled -norm of the velocity with 1?3/p+2/q?2, 1?q?∞ is sufficiently small near z and if the scaled -norm of the magnetic field with 1?3/l+2/m?2, 1?m?∞ is bounded near z. Similar results are also obtained for the vorticity and for the gradient of the vorticity. Furthermore, with the aid of the regularity criteria, we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations.     

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.     

Partial regularity of solutions to the magnetohydrodynamic equations

We study the local smoothness of solutions to the magnetohydrodynamic equations

Local partial regularity theorems for suitable weak solutions of a class of degenerate systems

A partial regularity theorem is established for a particular class of weak solutions to the systemu/t– div(K(u)u)=(u)¦¦², div((u))=0 on a bounded domain inR ^N . Under our assumptions, (u) may exhibit exponential decay, and thus the system may be degenerate. Our proof is based upon a blow-up argument.This work was supported in part by NSF Grant DMS9424448.     

On the regularity of weak solutions to the magnetohydrodynamic equations

In this paper, we study the regularity of weak solution to the incompressible magnetohydrodynamic equations. We obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations, which is similar to that of incompressible Navier-Stokes equations. Moreover, our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto-hydrodynamic equations.     

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.     

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 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.     

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.     

Partial regularity of weak solutions to a PDE system with cubic nonlinearity

In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points.     

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 $\left( {\omega _\theta ,\nabla \left( {u_\theta e_\theta } \right)} \right) \in L^1 \left( {0,T;\dot B_{\infty ,\infty }^0 } \right)$ .     

Partial regularity of finite Morse index solutions to the Lane-Emden equation

We prove regularity and partial regularity results for finite Morse index solutions u∈H¹(Ω)∩Lp(Ω) to the Lane-Emden equation −Δu=|u|^p−1u in Ω.     

Partial regularity and singular set of solutions to second order parabolic systems

In this paper we deal with the study of regularity properties of weak solutions to nonlinear, second-order parabolic systems of the type

     

Existence and uniqueness of steady solutions to the magnetohydrodynamic equations of compressible flows

We establish the existence and uniqueness of a strong solution to the steady magnetohydrodynamic equations for the compressible barotropic fluids in a bounded smooth domain with a perfectly conducting boundary, under the assumption that the external force field is small.