Asymptotic Analysis for Singularities of Solutions to Nonlinear Partial Differential Equations

非线性偏微分方程解奇异性的渐近分析

基本信息

批准号：
11440057
负责人：
OGAWA Takayoshi
金额：
$ 8.9万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440057/
关键词：
nonlinear dispersive equations analytic smoothing effect Navier-Stokes equations harmonic heat flow well posedness regularity criterion Besov spaces the critical Sobolev inequality Euler方程式半導体素子方程式実解析性渦度 KdV方程式正則性非圧縮性粘性流体

项目摘要

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

首席研究员T. Okawa与研究合作者之一K. Kato研究发现，在初始数据一定条件下，半线性色散方程的解具有很强的平滑效应，称为“解析平滑效应”该结果表明，从具有强单奇异性的初始数据（例如 Dirac delta 测度）开始，Korteweb-de Vries 方程的解将立即平滑到空间和时间变量上的真实解析。类似的效果可以是。显示为小川还与合作者H. Kozono 和Y. Taniuchi 证明了不可压缩Navier-Stokes 方程和Euler 方程的唯一性和正则性准则。谐波热流的解以 Besov 空间的形式给出。这些结果是通过改进 Sobolev 不等式的临界类型获得的。同时，利用 Lizorkin-Triebel 插值空间获得了涉及对数项的 Beale-Kato-Majda 型不等式的尖锐版本。对于半导体器件仿真中出现的方程，头部组织者小川和M. Kurokiba证明了该解在加权L-2空间中具有全局强解，并显示了一些守恒定律和正则性，此外，在特殊阈值条件下，该解具有一定的规律性。在有限时间内形成奇点。还表明，正解的阈值是尖锐的。联合研究员 S. Kawashima 研究了一般椭圆双曲系统（包括辐射气体方程）的解的渐近行为。渐近行为可以用系统的线性化部分来表征，并用通常的热核来表示。联合研究员 Y.Kagei 与联合研究员 T.Kobayashi 共同研究了他们研究了三维半空间中不可压缩纳维-斯托克斯解的渐近行为，研究了方程的恒定密度稳态的稳定性，并显示了 L- 意义上的扰动解的最佳可能衰减阶。 2.合作研究员K. Ito研究了中间表面扩散方程，发现当扩散系数很大时，解存在自相互作用。合作研究员N. Kita与T. Wada合作研究了当时间参数无穷大时非线性薛定谔方程解的渐近展开问题他们确定了当非线性具有长程相互作用的阈值指数时散射解的渐近轮廓的第二项。