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

• 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

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

首席调查员T。小川与研究合作者之一K。Kato指出，在一定的初值条件下，半线性色散方程的解具有一种很强的光滑效应，称为“解析光滑效应”。这个结果表明，从一个具有强单奇点的初始数据，如狄拉克δ测度，柯尔特韦布-德弗里斯方程的解立即平滑到空间和时间变量的真实的解析。对于非线性Schroedinger方程和Benjamin-Ono方程的解也可以得到类似的结果。Kozono和Y. Taniuchi，Ogawa等人证明了不可压Navier-Stokes方程和Euler方程的唯一性和正则性准则。此外，还给出了在Besov空间中求解调和热流的方法。这些结果是通过改进Sobolev不等式的临界型得到的 ...更多信息 在Besov空间。同时，利用Lizorkin-Triebel插值空间得到了包含对数项的Beale-Kato-Majda型不等式的更尖锐形式. Kurokiba等人证明了该解在加权L-2空间中存在整体强解，并证明了一些守恒律和正则性。另外，在一个特殊的阈值条件下，解在有限时间内会出现奇异性，并且对于正解，阈值是尖锐的。Kawashima研究了一般椭圆-双曲方程组解的渐近行为，其中包括辐射气体方程。渐近行为可以用系统的线性化部分来表征，它由通常的热核来表示。共同研究员Y.Kagei与共同研究员T.小林研究了三维半空间中不可压缩Navier-Stokes方程解的渐近行为。他们研究了该方程的常密度定态的稳定性，并证明了扰动解在L-2意义下的最佳可能衰减阶。Ito研究了中间表面扩散方程，证明了当扩散系数很大时，解具有自相互作用。Kita与T。和田合作的问题的渐近展开的解决方案的非线性薛定谔方程时，时间参数趋于无穷大。当非线性具有长程相互作用的阈值指数时，他们确定了散射解的渐近分布的第二项。少

项目成果

H.Kozono, T.Ogawa, Y.Taniuchi: "The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations."Math.Z.. 242. 251-278 (2002)

H.Kozono、T.Okawa、Y.Taniuchi：“Besov 空间中的临界 Sobolev 不等式和一些半线性演化方程的正则性准则。”Math.Z.. 242. 251-278 (2002)

K.Kato、P.N. Pipolo: "Analyticity of solitary wave solutions to generalized Kadomtesv-Pteviashvili equations,"Proc. Roy. Soc. Edinburg. (to appear). (2000)

K. Kato，P. N. Pipolo：“广义 Kadomtesv-Pteviashvili 方程的分析性”，Proc. Edinburg（2000 年）。

T.Wada: "A remark on long range scattering for the Hartree type equation,"Kyushi J. Math.. 54(in press). (1999)

T.Wada：“关于 Hartree 型方程的长程散射的评论”，Kyushi J. Math.. 54（印刷中）。

K.Ito, Y.Kohsaka: "Three phase boundary motion by surface diffusion in triangular domain"Advances in Math.Sci.Appl.. 11. 753-779 (2001)

K.Ito, Y.Kohsaka：“三角域中表面扩散的三相边界运动”Math.Sci.Appl. 进展. 11. 753-779 (2001)

Takayoshi Ogawa,Yasushi Taninchi: "Remarks on uniqueness and blow-up criterion to the Euler equations in the generalized Besov spaces."J.Korean Math.Soc.. 37. 1021-1029 (2000)

Takayoshi Okawa、Yasushi Taninchi：“关于广义 Besov 空间中欧拉方程的唯一性和爆炸准则的评论。”J.Korean Math.Soc.. 37. 1021-1029 (2000)