2000 Fiscal Year Final Research Report Summary
Applications of Besov spaces and Sobolev spaces to non-linear problems in mathematical physics
| Project/Area Number |
11640184
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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 | CHUO UNIVERSITY |
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Principal Investigator |
MURAMATU Tosinobu Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (60027365)
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| Co-Investigator(Kenkyū-buntansha) |
YAMAMOTO Makoto Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (10158305)
MATSUYAMA Yoshio Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (70112753)
IWANO Masahiro Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (70087013)
YOSHINO Masafumi Chuo University, Faculty of Economics, Professor, 経済学部, 教授 (00145658)
MITSUMATU Yoshihiko Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (70190725)
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| Project Period (FY) |
1999 – 2000
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| Keywords | non-linear problem / Besov spaces / parabolic evolution equation / E E of hyperbolic type / operational calculus / non-linear Schrodinger equation |
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| Research Abstract |
We investigated first precise linear theories which are bases of study on non-linear problems, then applied them to non-linear problems.
1. Study on parabolic evolution equations in Banach spaces
We have constructed a theory which extensively improved H.Tanabe's clsssical one. We replaced Holder continuity with modulus of continuity. We also proved solvability of the initial value problem with initial data in Besov spaces of order 0. This is the best assumption on initial data, which enable us to get better results on non-linear problems.
2. Study on evolution equations of hyperbolic type in Banach spaces.
We have reformed Tosio Kato's theory. By our result we can directly apply the abstract theory to regularly hyperbolic equations, which is useful in study on non-linear problems.
3. Operational calculus of non-negative operators
This is a generalization on fractional powers of non-netative operators, which are very useful tools in study of Navier-Stokes equations.
4. Applicatons of Besov type norms to non-linear partial differential eqations
We studied non-linear Schrodinger equation of one space dimention with quadratic non-linearities. We proved that the initial value problems with initial values in Besov spaces of order-3/4, exponent 2 and subexponent 1 has time-locally solvable. We also found that our method is applicable to KdV equation.
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