三维薄区域上MHD方程的渐进分析

在强解全局存在的基础上, 得到了三维薄区域上MHD方程的解(u,h)对任意时间t≥ 0的渐进分析. 当区域厚度ε小时, MHD方程的强解(u,h)可形式展开为u=ū(t)+up+U, h=h(t)+hp+H, t ≥0或u=ū(t)+us+U＊,h=h(t)+hs+H＊,t ≥0,其中(u,h) 是2D-3C MHD 方程的解, (u_p,h_p) 是P-S MHD 方程的解, u,h 分别是两个Stokes方程的解, (U,H),(U＊,H＊)是仅依赖于初始数据的两个函数对. (U,H)和(U＊,H＊)关于区域厚度\varepsilon是小的, (u_p,h_p)和u,h更小;证明了上述形式展开的收敛性.

Asymptotic Analysis for MHD Equations on Thin Domains

Based on the global existence of strong solution of MHD equations on three dimensional thin domain, asymptotic expansion for the strong solution (u,h) ofMHD equations is obtained, and this expansion holds uniformly forall the time t≥0. When ε, the thickness of thedomain, is small, the strong solution (u,h) of MHD equations canbe formally expressed asu=\bar{u}(t)+u_p+U,\quad h=\bar{h}(t)+h_p+H,\quad \forall t≥0,oru=\bar{u}(t)+u_s+U^{\star},\quad h=\bar{h}(t)+h_s+H^{\star},\quad \forall t≥0,where (\bar{u},\bar{h}) is a solution of 2D-3C MHD equations, (u_{p},h_p) is a solution of P-S MHD equations, u_s,h_s are respectively solutions of two Stokes equations, (U,H),(U^\star,H^\star)are two function pairs depending only on the initial data. (U,H) and (U^\star,H^\star)are small with respect to\varepsilon. (u_{p},h_p) and u_{s},h_s are smaller. And the convergence of the expansion is proved.