| Project/Area Number |
09640177
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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 |
解析学
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| Research Institution | Science University of Tokyo |
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Principal Investigator |
TACHIKAWA Atsushi Science University of Tokyo(S.U.T.), Faculty of Science and Technology, Associate Professor, 理工学部, 助教授 (50188257)
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| Co-Investigator(Kenkyū-buntansha) |
TAMIYA Takanori Science University of Tokyo(S.U.T.), Fac.of Sci.and Tech.nology, Lecturer, 理工学部, 講師 (60183472)
TANAKA Ryuichi Science University of Tokyo(S.U.T.), Fac.of Sci.and Tech.nology, Lecturer, 理工学部, 講師 (10112898)
KOBAYASHI takao Science University of Tokyo(S.U.T.), Fac.of Sci.and Tech.nology, Associate Professor, 理工学部, 助教授 (90178319)
NAGASAWA Takeyuki Tohoku Univ., Mathematical Institute, Associate Professor, 理学研究科, 助教授 (70202223)
KOTANI Kouichi Science University of Tokyo(S.U.T.), Fac.of Sci.and Tech.nology, Assistant, 理工学部, 助手 (80183341)
細尾 敏男 東京理科大学, 理工学部, 助手 (30130339)
浜畑 芳紀 東京理科大学, 理工学部, 講師 (90260645)
吾郷 孝視 東京理科大学, 理工学部, 教授 (60112893)
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| Project Period (FY) |
1997 – 1999
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| Project Status |
Completed (Fiscal Year 1999)
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| Budget Amount *help |
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 1999: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1998: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1997: ¥1,100,000 (Direct Cost: ¥1,100,000)
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| Keywords | Variational Method / Discretization / Harmonic Maps / 変分法 / 差分法 / 調和写像 |
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| Research Abstract |
The aim of this research is to study the solutions of the nonlinear partial differential equations which are closely related to variational problems by using calculus of variations and time-discretization schemes. More precisely, for some variational functional F(u) we treat the partial differential equation ィイD7∂u(/)∂tィエD7-(the Euler-Language equation of F) = 0. To construct weak solutions for the above equation we proceed as follows. We consider the functional GィイD2nィエD2(u) = ∫ィイD7|u-uィイD2nィエD2-1|(/)2hィエD7dx+F(u), and define uィイD2nィエD2 as a minimizer of GィイD2nィエD2(u) successively. Combining {uィイD2nィエD2} by line segments we construct a approximate solution uィイD2hィエD2(x, t). Finally, under some conditions, we prove that uィイD2hィエD2(x, t) converge to a weak solution. On the other hand, id the limit of the sequence of maps {uィイD2hィエD2}converges to some map u(x, t), u(x, t) will be closely related to minimizing movement which is a new notion introduced by E.De Giorgi.
Using the above method
… More
, Tachikawa constructed weak solutions of the heat-type equations for harmonic maps (Eells-Sampson equation) from noncompact Riemannian manifolds into the n-dimensional spheres (ィイD7∂u(/)∂tィエD7-Δu-u|Du|ィイD12ィエD1 = 0). Moreover, he proved that the weak solutions are minimizing movements of the energy functionals.
Related to the above problem, Nagasawa and Tachikawa studied harmonic maps between noncompact complete Riemannian manifolds. Especially, they considered harmonic maps with a certain non-degeneracy condition and get the following nonexistence result. "Let N be a Handamard manifold whose sectional curvatures at a point p do not decay faster than distィイD1-2ィエD1"(p, pィイD20ィエD2) for some fixed point pィイD20ィエD2. Then there is no entire harmonic maps from RィイD1mィエD1 into N which satisfies a certain non-degeneracy condition."
Nagasawa constructed a weak solution of the Navier-Strokes equation on a Riemannian manifold using the above method. Moreover, he sharpened the energy estimates on the weak solutions constructed as above and got a new partial regularity estimates. He constructed weak solutions of the hyperbolic Ginzburg Landau equations too and studied them numerically. Less
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