广义磁流体力学方程组的部分正则性

Partial Regularity of Generalized MHD Equations

研究带分数次扩散项(-Δ)a和(-Δ)β的广义磁流体力学方程组(GMHD)的正则性。这一方程包含了Navier-Stokes方程与通常的磁流体力学方程组(MHD)。本文采用能量积分方法,研究GMHD方程的解用速度向量的分量来判定正则性,并且结果并不依赖于磁场函数。本文主要讨论α=β的情形。设u=(u1,u2,u3),=(u1,u2,0),0<α=β<2/3初始速度与初始磁场满足u0,b0∈H1(R3)。在上述条件下,本文指出,如果∈LP(0,T;Lq(R3))且(2α/p+3/q≤2a,或者∈L2((2α)/(2α-r))(0,T;(R3)),0≤r≤α,那么方程的解在[0,T]上依然是光滑的。

In this paper, the regularity of 3D generalized magneto-hydrodynamics （GMHD） equations with fractional dissipative terms （ -△）αand （-△）β is studied. The equations contain the well-known Navier-Stokes equations and magneto-hydrodynamics equations. The regularity problem of Navier-Stokes equation and generalized type, such as the magneto-hydrodynamics equation and the generalized Navier-Stokes equations, were studied extensively. But the problem is still unsettled now. Some researchers turned to study the regularities criterion of Naveir-Stokes equations in terms of the components of velocity, and got some of useful results. Since MHD and GMHD are far more difficult than Navier-Stokes equations, there are few similar results of these two equations. Using energy integral method, the regularity of GMHD equations in terms of two components of velocity is studied, and the results does not depend on the magnetic field. case α =β is considered in the paper. Let u = （u1, u2, u3 ） ,u = （u1, u2 ,0） ,0 〈 α =β 〈 3/2, the initial velocity and magnetic field satisfied uo ,bo ∈ H^1（R^3）. Under these conditions, it is proved that if △↓∈LP（0,T;Lq（R3）） with 2α/p＋3/q≤2α on ~0, T],or if △↓∈L2（（2α）/（2α-r））（0,T;（R3））with0≤r≤α the solution remains smooth on[0, T].