| Project/Area Number |
11440057
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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 | KYUSHU UNIVERSITY |
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Principal Investigator |
OGAWA Takayoshi Faculty of Math. Kyushu University, Associate Prof., 大学院・数理学研究院, 助教授 (20224107)
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| Co-Investigator(Kenkyū-buntansha) |
KOZONO Hideo Inst. Of Math. Tohoku University, Prof., 大学院・理学研究科, 教授 (00195728)
KAGEI Yoshiyuki Faculty of Math. Kyushu University, Associate Prof., 大学院・数理学研究院, 助教授 (80243913)
KAWASHIMA Shuichi Faculty of Math. Kyushu University, Prof., 大学院・数理学研究院, 教授 (70144631)
KOBAYASHI Takayuki Dept. of Eng. Kyushu Inst. Tech., Associate Prof., 工学部, 助教授 (50272133)
KATO Keiichi Dept. of Math. Science University Tokyo, Associate Prof., 理学部一部, 助教授 (50224499)
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| Project Period (FY) |
1999 – 2002
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| Keywords | nonlinear dispersive equations / analytic smoothing effect / Navier-Stokes equations / harmonic heat flow / well posedness / regularity criterion / Besov spaces / the critical Sobolev inequality |
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| Research Abstract |
The head investigator, T. Ogawa researched with one of the research collaborator K. Kato that the solution of the semi-linear dispersive equation has a very strong type of the smoothing effect called "analytic smoothing effect" under a certain condition for the initial data. This result says that from an initial data having a strong single singularity such as the Dirac delta measure, the solution for the Korteweb-de Vries equation is immediately going to smooth up to real analytic in both space and time variable. Similar effect can be shown for the solutions of the nonlinear Schroedinger equations and Benjamin-Ono equations.
Also with collaborators H. Kozono and Y. Taniuchi, Ogawa showed that the uniqueness and regularity criterion to the incompressible Navier-Stokes equations and Euler equations. Besides, it is also given that the solution to the harmonic heat flow is presented in terms of the Besov space. Those result is obtained by improving the critical type of the Sobolev inequalit
… More
ies in the Besov space. On the same time, the sharper version of the Beale-Kato-Majda type inequality involving the logarithmic term was obtained by using the Lizorkin-Triebel interpolation spaces.
For the equation appeared in the semiconductor devise simulation, the head organizer Ogawa showed with M. Kurokiba that the solution has a global strong solution in a weighted L-2 space and showed some conservation laws as well as the regularity. Besides, under a special threshold condition, the solution develops a singularity within a finite time.
It is also shown that the threshold is sharp for a positive solutions.
Co-researcher S. Kawashima investigated the asymptotic behavior of the solutions to a general elliptic-hyperbolic system including the equation for the radiation gas. The asymptotic behavior can be characterized by the linearized part of the system and it is presented by the usual heat kernel.
Co-researcher Y.Kagei researched with co-researcher T.Kobayashi about the asymptotic behavior of the solutions to the incompressible Navier-Stokes in the three dimensional half space. They studied on the stability of the constant density steady state for the equation and the showed the best possible decay order of the perturbed solution in the sense of L-2.
Co-researcher K. Ito studied about the intermediate surface diffusion equation and showed that the solution has the self interaction when the diffusion coefficients are going to very large.
Co-researcher N. Kita with T. Wada collaborates on the problem of the asymptotic expansion on the solution of the nonlinear Schroedinger equation when the time parameter goes infinity. They identified the second term of the asymptotic profile of the scattering solution when the nonlinearity has the threshold exponent of the long range interaction. Less
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