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

首席研究员T. Ogawa与研究合作者之一K. Kato研究发现，在初始数据的一定条件下，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”。这一结果表明，从具有强单奇点的初始数据，如狄拉克δ测量，Korteweb-de Vries方程的解在空间和时间变量上都将立即平滑到真正的解析。对于非线性薛定谔方程和Benjamin-Ono方程的解也有类似的效果。Ogawa还与合作者H. Kozono和Y. Taniuchi一起证明了不可压缩Navier-Stokes方程和Euler方程的唯一性和正则性准则。此外，还给出了在Besov空间中谐波热流的解。这些结果是通过改进Sobolev不等式在Besov空间中的临界类型而得到的。同时，利用lizorkin - triiebel插值空间，得到了包含对数项的Beale-Kato-Majda型不等式的更尖锐的版本。对于半导体设计模拟中出现的方程，首席组织者Ogawa与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规律性。此外，在特殊的阈值条件下，解在有限时间内出现奇点。结果还表明，对于正解，阈值是尖锐的。共同研究者S. Kawashima研究了包含辐射气体方程的一般椭圆-双曲系统解的渐近行为。系统的渐近行为可以用系统的线性化部分来表征，用通常的热核来表示。共同研究员Y.Kagei与共同研究员T.Kobayashi研究了三维半空间中不可压缩Navier-Stokes解的渐近行为。他们研究了方程的等密度稳态的稳定性，并给出了在L-2意义上扰动解的最佳衰减顺序。共同研究员K. Ito对中间表面扩散方程进行了研究，表明当扩散系数趋于很大时，解具有自相互作用。N. Kita与T. Wada合作研究了非线性薛定谔方程在时间参数趋于无穷时解的渐近展开问题。当非线性具有长距离相互作用的阈值指数时，他们确定了散射解的渐近轮廓的第二项。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）。

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)

S.Kawashima and S.Nishibata: "A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics."Indiana Univ.Math.J. 49. (2000)

S.Kawashima 和 S.Nishibata：“辐射流体动力学中双曲椭圆耦合系统的奇异极限。”印第安纳大学数学杂志。