不可压MHD系统的零耗散极限中的边界层

Boundary layer problems in the vanishing viscosity-diffusion limits for the incompressible MHD system

本文研究通常的Dirichlet物理边界条件下带有小而变化的黏性和磁扩散系数的不可压磁流体(MHD)方程组的初边值的极限问题;发现了一类非平凡的初值,对于这类初值能建立其Prandtl型边界层的一致稳定性,并且严格证明了理想的MHD方程组的解和Pandtl型边界层矫正子的叠加是黏性扩散不可压MHD方程的解的一致逼近.这里的主要困难是处理和控制由耗散的MHD系统和理想MHD系统边界条件差异产生的Prandtl型的奇异边界层.关键的观察是对于本文研究的初值,其解的速度场和磁场的边界层的主要奇异项存在有抵消现象.这使得我们能基于精细的能量方法来使用这个特殊结构带来的好处,从而克服在研究这类问题中通常不能解决的困难.此外,在黏性系数为固定的正常数情形,对于一般初值,也能建立磁场的扩散边界层的稳定性以及零磁扩散极限中解的一致收敛性.

In this paper,we study boundary layer problem for the incompressible magneto-hydrodynamical（MHD） system in the presence of physical boundaries with the standard Dirichlet boundary conditions with small generic viscosity and diffusion coefficients.We identify a non-trivial class of initial data for which we can establish the uniform stability of the Prandtl＇s type boundary layers and prove rigorously that the solutions to the viscous and diffusive incompressible MHD system converges strongly to the superposition of the solution to the ideal MHD system with a Prandtl＇s type boundary layer corrector.One of the main difficulties is to deal with the effect of the difference between viscosity and diffusion coefficients and to control the singular boundary layers resulting from the Dirichlet boundary conditions for both the viscosity and the magnetic fields.One key derivation here is that for the class of initial data we identify here,there exist cancelations between the boundary layers of the velocity field and that of the magnetic fields so that one can use an elaborate energy method to take advantage of this special structure.In addition,in the case of fixed positive viscosity,we also establish the stability of diffusive boundary layer for the magnetic field and convergence of solutions in the limit of zero magnetic diffusion for general initial data.