2006 Fiscal Year Final Research Report Summary
Reseach for the singularities and regularity of solutions to crtical nonlinear partial differential equations
| Project/Area Number |
15340056
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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 | Tohoku University (2004-2006)
Kyushu University (2003) |
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Principal Investigator |
OGAWA Takayoshi Tohoku University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (20224107)
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| Co-Investigator(Kenkyū-buntansha) |
ISHIGE Kazuhiro Tohoku University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (90272020)
NAKAMURA Makoto Tohoku University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (70312634)
KAWASHIMA Shuichi Kyushu University, Graduate School of Mathematics, Professor, 大学院数理学研究院, 教授 (70144631)
KOBAYASHI Takayuki Saga University, Department of Mathematics, Professor, 理工学部, 教授 (50272133)
KAGEI Yoshiyuki Kyushu University, Graduate School of Mathematics, Professor, 大学院数理学研究院, 教授 (80243913)
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| Project Period (FY) |
2003 – 2006
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| Keywords | nonl-local parabolic equations / drift-diffusion equations / semiconductor simulation / Chemotaxis / nonlinear damped wave equations / Naveir-Stokes equations / Critical Sobolev inequality / mean curvature flow |
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| Research Abstract |
The main researcher, Prof.Ogawa obtained the following results. He researched for the Sobolev type inequality of the critical type, especially for the real interpolation spaces such as Besov and Triebel-Lirzorkin spaces and generalized it for the abstract Besov and Lorentz space. Those inqualities involving the logarithmic interpolation order can be applied for the regularity and uniqueness criterion of the seimilinear partial differential equation. In a series of collaboration with the research colabolators, he shows that the reguarlity and uniquness criterion for the weak solution of the 3 dimensional Navier-Stokes equations and break down condition for the Euer equation. In a similar method, he also showed the regularity criterion for the smooth solution of the 2 dimensional harmonic heat flow into a sphere. In particular, for the weak solution of the harmonic heat flow, the similar regularity criterion is also holds. The result is obtained by establishing the "monotonicity formula"
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for the mean oscillation of the energy density of the solutions.
He also consider the asymptotic behavior of the solution for the semi-lineear parabolic equation of the non-local type. Those system appeared in a various Physical scaling such as semi-conductor simulation model, Chemotaxis model and the birth of star in Astronomy. The system is involving Poisson equation as the field generated by the dencity of the charge or mucous ameba and the non-local effect is essential for the analysis of the solution. He particulariy investigated to the critical situation, 2-dimensional case, and showed that there exists a time local solution in the critical Hardy space, time global solution upto the threshold initial density and finite time blow-up for the system of forcusing drift-diffusion case. Besides, the asymootitic behavior of the solution for small data is characterized by the heat kernel. Moreover if the field equation is purterbed in a certain nonlinear way, then there exist two solutions for the same initial data in a radially symmetric case.
He also studied for the asymptotic behavior of the solution for the semi-linear damped wave equation in whole and half spaces and exterior domains and show the small solution is going to be decomposed into the solutions of the linear heat equation, some combination of linear wave equation with nonlinear effect. This was shown for 1 and 3 dimensional cases before, however the mothod there could not be applicable for the 2dimensional case. Less
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Research Products
(115 results)
