2001 Fiscal Year Final Research Report Summary
Comprehensive study of differential equations
| Project/Area Number |
11304006
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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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 | The University of Tokyo |
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Principal Investigator |
YAJIMA Kenji The University of Tokyo , Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (80011758)
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| Co-Investigator(Kenkyū-buntansha) |
MAJIMA Hideyuki Ochanomizu University, Faculty of Sciences, Professor, 理学部, 教授 (50111456)
IKAWA Mitsuru Kyoto University, Graduate School of Sciences, Professor, 大学院・数理科学研究科, 教授 (80028191)
NAKAMURA Shu The University of Tokyo , Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (50183520)
TSUTSUMI Yoshio Tohoku University, Graduate School of Sciences, Professor, 大学院・理学研究科, 教授 (10180027)
GIGA Yoshikazu Hokkaido University, Graduate School of Sciences, Professor, 大学院・理学研究科, 教授 (70144110)
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| Project Period (FY) |
1999 – 2001
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| Keywords | Shrodinger equation / nonlinear PDE / linear PDE / nonlinear evolution equation / scattering theory / viscous solution / Navier-Stokes equation / Klein-Gordon equation |
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| Research Abstract |
We carried out an comprehensive study on linear and nonlinear partial and ordinary differential equations and obtained among others the following results:
1. Kenji Yajima and Shu Nakamura studied Schrodinger equations and obtained (1) the L^p-boundedness of wave operators of scattering and (2) the stability under subquadratic perturbations of the smooth and boundedness of fundamental solution; (3) clarified the relation between the local decay and the spectrum of Floquet operator for time periodic system; (4) constructed general theory of tunnenling in phase space and gave its applications; (5) defined the integrated density of states for random operators and proved Wegner estimates and the Lifschitz singularities.
2. Kiyoomi Kataoka studied the general theory of linear PDE and (1) gave an elementary definition of the holomorphic solutions complex of ε^R_x-modules and (2) reduced the branching of the singularities for Fuchsian elliptic boundary problems to that of the continuation of ODE.
3. Yoshikazu Giga studied nonlinear PDE and (1) gave a way to numerically compute the viscous solution of first order nonlinear PDE, (2) defined the proper visocous solutions and proved the existence and the uniquess of solutions and (3) proved the existence of solution of Navier-Stokcs equation with non-deaying initial conditions.
4. Yoshio Tsutsumi proved (1) the well-posedness of nonlinearly couples wave equations with different speeds, (2) the stability of constant solutions of nonlinear massive Klein-Gordon equation.
5. Hideyuki Majima produced new fundamental theorem on the asymptotic expansions from the point of view the super-asymptotic analysis.
6. Mitsuru Ikawa studied the scattering theory of wave equation by several convex bodies and obtained a precise asymptotic, formula for the poles of scattering matrix.
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