由Hrmander向量场构成的抛物方程的W_*^(1,p)正则性

W_*^(1,p)Regularity for Parabolic Equations Structured on Hrmander's Vector Fields

设X_1,…,X_q(q<n)是有界区域ΩR^n上的一组光滑Hrmander向量场.考虑下面的散度型抛物方程:u_t+X_i~*(a_(ij)(x,t)X_ju)=X_i~*f_i,其中X_i~*是X_i的形式伴随,系数a_(ij)(x,t)(i,j=1,2,…,q)是定义在Ω_T=Ω×(0,T)∈R^(n+1)上满足一致椭圆性条件的有界可测函数.在系数属于VMO_(loc)∩L~∞函数空间的情况下,得到了抛物方程弱解的内W_*^(1,p)正则性.

Let W1,p＊ be a family of real smooth vector fields satisfying HSrmander＇s condition in a bounded domain Ω Rn. We are concerned with the following divergence parabolic equation： ut＋x＊i（aij（x,）Xju）=X＊i fi where X＊i is the formal adjoint of Xi, the coefficients aij（x,t） （i,j = 1,2,… ,q）valued bounded measurable functions defined in ΩT=Ω×（0,T）∈Rn＋1 , satisfying the uniform ellipticity condition. Under the assumption that the coefficients aij（x, t） belong to the space VMOloc∩L∞, the interior WI＇p regularity for weak solutions of the equation above is established.