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DEPELOPMENTS IN OPERATOR THEORY TOWARDS EVOLUTION EQUATIONS

演化方程算子理论的发展

基本信息

批准号：
14540187
负责人：
OKAZAWA Noboru
金额：
$ 0.96万
依托单位：
TOKYO UNIVERSITY OF SCIENCE
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540187/
关键词：
HYPERGEOMETRIC FUNCTIONS OF OPERATORS FRACTIONAL POWERS OF NONNEGATIVE OPERATORS COMPLEX GINZBURG-LANDAU EQUATION INITIAL-BOUNDARY VALUE PROBLEMS LOCAL LIPSCHITZ CONTINUITY OF NONLINEAR TERM COMPLEX COEFFICIENTS AND CONITINUITY OF SOLUTION OPERA

项目摘要

We have considered(1)hypergeometric functions of non-negative operators, and(2)the weliposedness of initial-boundary value problems for the complex Ginzburg-Landau equation.(1) The Gauss hypergeometric function F(α,β,γ ; -z) is first defined by the power series in the unit disk of the complex plane. If 0 < Re α < Re γ, then an analytic continuation of F outside the unit disk is given by the integral representation which makes sense on C＼ (-∞, 1].Replacing the complex variable with a class of closed linear operators, we obtain the corresponding formula for the operator-valued functions. Noting that the power and logarithmic functions ~log z are written down in terms of F(α,β,γ ; -z)and F(α',β',γ' ; -z^<-1>) on C＼(-∞, 0], we can define in a unified way the fractional powers and logarithm of a non-negative operator (with inverse) in terms of operator-valued hypergeometric functions.(2)The existence and uniqueness of global strong solutions to the initial-boundary value problem for the complex Ginzburg-Landau equation with L^2-initial data(smoothing effect on the initial data)is established under a condition on the power of the nonlinear term without any restriction on the complex coefficients.Moreover, we have shown that the solution operator forms a nonlinear semigroup of locally Lipschitz continuous operators on L^2.This improves and extends partially the previous result which asserts that the solution operator forms a nonlinear semigroup of quasi-contractions on L^2 under strict restriction on the complex coefficient of the nonlinear term.

本文研究了（1）非负算子的超几何函数，（2）复Ginzburg-Landau方程初边值问题的适定性。(1)首先在复平面的单位圆上用幂级数定义高斯超几何函数F（α，β，γ ; -z）.若0 < Re α < Re γ，则F在单位圆盘外的解析延拓由在C-∞，1]上有意义的积分表示给出，用一类闭线性算子代替复变量，得到了相应的算子值函数公式.注意到幂函数和对数函数~log z分别用C\（-∞，0]上的F（α，β，γ ; -z）和F（α '，β'，γ' ; -z^<-1>）表示，我们可以用算子值超几何函数统一定义非负算子（带逆算子）的分数次幂和对数.（2）具有L^2-初值的复Ginzburg-Landau方程初边值问题整体强解的存在唯一性（对初始数据的平滑效果）是在对复系数没有任何限制的非线性项的幂的条件下建立的。我们证明了解算子在L^上形成局部Lipschitz连续算子的非线性半群2.这改进和部分推广了以前的结果，即在严格限制非线性项的复系数的条件下，解算子在L^2上形成一个拟压缩的非线性半群.