2007 Fiscal Year Final Research Report Summary
Mathematical Study of the Nonlinear Partial Differential Equations Arising in the Statistical Mechanics
| Project/Area Number |
16340047
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Osaka University |
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Principal Investigator |
SUZUKI Takashi Osaka University, Graduate School of Engineering Science, Full Professor (40114516)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUMURA Akitaka Osaka University, Graduate School of Information Science, Full Professor (60115938)
ODANAKA Shinji Osaka University, Cyber Media Center, Full Professor (20324858)
NAWA Hayato Osaka University, Graduate School of Science, Full Professor (90218066)
WADA Takeshi Kumamoto University, Faculty of Fngineering, Associate Professor (70294139)
NOBE Atsushi Osaka University, Graduate School of Engineeriig Science, Assistant Professor (80397728)
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| Project Period (FY) |
2004 – 2007
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| Keywords | nonlinear problems / mean field hierarchy / blowup of the solution / variational structure / chemotaxis / self-interacting fluid / self-dual eauee theory / stationary turbulence |
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| Research Abstract |
In this research project, we provided a unified analysis to the critical phenomena, arising in the solution to the partial differential equation provided with the nonlinearity due to the self-interaction and the non-equilibrium. We formulate the mathematical principle across the hierarchy to control several phenomena common to many nonlinear problems. These problems are the mean field of stationary turbulence in high energy and of gauge field, Ricci flow, nonlinear parabolic equations, self-interacting fluids, material-energy transport, tumor growth, and nonlinear thermodynamics. Among them, we formulated the system of chemotaxis, derived in the context of th the formation of spores of the cellular slime molds, as the fundamental equation of the material transport subject to the mass conservation and the decrease of the free energy in the thermodynamics called Smoluchowski-Poisson equation and clarified the quantized blowup mechanism by developing various new methods of analysis. Then, we obtained the notions of the blowup envelope, formulation of the stationary and non-stationary states by the dual variation, hierarchical control of the stationary states upon the non-stationary states, which motivates the study on the structure and the stability of the set of stationary solutions of phenomenological equations concerning the critical phenomena arising in the non-equilibrium thermodynamics, formation of sub-collapses and the collision of collapses in the mean field equation arising in the gauge theory and turbulent theory and the semilinear parabolic equation with the critical Sobolev exponent, deformed quantization in the nonlinear parabolic equation with non-local term and the normalized Ricci flow, and mass quantization in higher dimensions.
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Research Products
(134 results)
