LOCAL REGULARITY CRITERIA OF A SUITABLE WEAK SOLUTION TO MHD EQUATIONS

We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are HSlder continuous near boundary provided that the scaled mixed Lx,t^p,q-norm of the velocity vector field with 3/p ＋ 2/q 〈 2, 2 〈 q 〈 ∞ is sufficiently small near the boundary. Also, we will investigate that for this 3 2〈3 solution U ∈ Lx,t^p,q with 1 〈 3＋p ＋2/q＋≤3/2, 3 〈 p 〈 ∞, the Hausdorff dimension of its singular set is no greater than max{p, q}（3/q＋2/q- 1）.

PARTIAL REGULARITY FOR THE NAVIER-STOKES-FOURIER SYSTEM

This paper addresses a nonstationary flow of heat-conductive incompressible Newtonian fluid with temperature-dependent viscosity coupled with linear heat transfer with advection and a viscous heat source term, under Navier/Dirichlet boundary conditions. The partial regularity for the velocity of the fluid is proved for each proper weak solution, that is, for such weak solutions which satisfy some local energy estimates in a similar way to the suitable weak solutions of the Navier-Stokes system. Finally, we study the nature of the set of points in space and time upon which proper weak solutions could be singular.

On Partial Regularity of Suitable Weak Solutions to the Stationary Fractional Navier-Stokes Equations in Dimension Four and Five

In this paper, we investigate the partial regularity of suitable weak solutions to the multi- dimensional stationary Navier-Stokes equations with fractional power of the Laplacian （-△）^α （n/6 ≤α〈1 and a ≠ 1/2）. It is shown that the n ＋ 2 - 6α （3 ≤ n ≤5） dimensional Hausdorff measure of the set of the possible singular points of suitable weak solutions to the system is zero, which extends a recent result of Tang and Yu [19] to four and five dimension. Moreover, the pressure in ε-regularity criteria is an improvement of corresponding results in [1, 13, 18, 20].

Partial Regularity of Suitable Weak Solutions to the System of the Incompressible Shear-thinning Flow

This paper is devoted to the partial regularity of suitable weak solutions to the system of the incompressible shear-thinning flow in a bounded domainΩ■R^(n),n≥2.It is proved that there exists a suitable weak solution of the shear-thinning fluid in the n-D smooth bounded domain(for n≥2).For 3 D model,it is proved that the singular points are concentrated on a closed set whose 1 dimensional Hausdorff measure is zero.

On the Motion of Shear-Thinning Heat-Conducting Incompressible Fluid-Rigid System

The full Navier-Stokes-Fourier system with mixed boundary condition that describes the motion of shear-thinning and incompressible viscous fluid in a rotating multi-screw extruder is investigated. The viscosity is assumed to depend on the shear rate and the temperature. The global existence of suitable weak solutions is established. The fictitious domain method which consists in filling the moving rigid screws with the surrounding fluid and taking into account the boundary conditions on these bodies by introducing a well-chosen distribution of boundary forces is used.

Non Blow-up Criterion for the 3-D Magneto-hydrodynamics Equations in the Limiting Case

In this paper, we prove that suitable weak solution （u,b） of tne 3-D MHD equations can be extended beyond T if u E L∞（0,T; La（R3）） and the horizontal components bh of the magnetic field satisfies the well-known Ladyzhenskaya-Prodi-Serrin condition, which improves the corresponding regularity criterion by Mahalov-Nicolaenko-Shilkin.