2003 Fiscal Year Final Research Report Summary
DEPELOPMENTS IN OPERATOR THEORY TOWARDS EVOLUTION EQUATIONS
| Project/Area Number |
14540187
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | TOKYO UNIVERSITY OF SCIENCE |
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Principal Investigator |
OKAZAWA Noboru OKAZAWA,Noboru, 理学部第1部, 教授 (80120179)
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| Co-Investigator(Kenkyū-buntansha) |
YOKOTA Tomomi YOKOTA,Tomomi, 理学部第1部, 助手 (60349826)
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| Project Period (FY) |
2002 – 2003
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| Keywords | 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 OPERATORS / SEMIGROUPS OF CONTRACTIONS AND LIPSCHITZ OPERATORS / YOSIDA APPROXIMATIONS OF SUBDIFFERENTIAL OPERATORS |
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| Research Abstract |
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.
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