带粗糙初始值向列型液晶流的适定性

Well-Posedness for the Nematic Liquid Crystal Flow with Rough Initial Data

考虑了R^n上n(n≥2)维向列型液晶流(u,d)当初值属于Q_α^(-1)(R^n,R^n)×Q_α(R^n,S^2)(其中α∈(0,1))时Cauchy问题的适定性,这里的Q_α(R^n)最早由Essen,Janson,Peng和Xiao(见[Essen M,Janson S,Peng L,Xiao J.Q space of several real variables,Indiana Univ Math J,2000,49:575-615])引入,是指由R^n中满足的所有可测函数f全体所组成的空间.上式左端在取遍Rn中所有以l(I)为边长且边平行于坐标轴的立方体I的全体中取上确界,而Q_α^(-1)(R^n):=▽·Q_α(R^n).最后证明了解(u,d)在类C([0,T);Q_(α,T)^(-1)(R^n,R^n))∩L_(loc)~∞((0,T);L~∞(R^n,R^n))×C([0,T);Q_α,T(R^n,S^2))∩L_(loc)~∞((0,T);W^(1,∞)(R^n,S^2))(其中0<T≤∞)中是唯一的.

The author investigates the well-posedness of the Cauchy problem of the ndimensional （n ≥ 2） hydrodynamic flow （u, d） of nematic liquid crystal materials on R^n with the initial data in Qα^-1（R^n, R^n） × Qα（R^n, S^2） with α ∈ （0, 1）. Here, Qα（R^n）, introduced by Essen, Janson, Peng and Xiao （see [Essen M, Janson S, Peng L, Xiao J. Q space of several real variables, Indiana Univ Math J, 2000, 49：575-615]）, is the space of all measurable functions f on R^n, satisfying supI（（l（I））^2α-n∫I∫I｜f（x）-f（y）｜^2/｜x-y｜^n＋2αdxdy）^1/2〈∞where the supremum is taken over all cubes I with the edge length t（I） and tne edges parallel to the coordinate axes in R^n, and Qα^-1（R^n） ：= △↓· Qα（R^n）. Moreover, for the nematic liquid crystal flow （u,d）, it is shown that the solution is unique in the class C（[0, T）; C（[0,T）;Qa^-1,T（R^n,R^n））∩loc^∞（（0,T）;L^∞（R^n,R^n））×C（[0,T）;Qα,T（R^n,S^2））∩Lloc^∞（（0,T）;W^1,∞（R^n,S^2））for 0〈T≤∞