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与一项研究合作者K. Kato进行了研究，即半线性分散方程的解决方案在特定条件下在初始数据的一定条件下具有非常强的平滑效果，称为“分析平滑效应”。该结果表明，从具有强大的单个奇异性（例如Dirac Delta测量）的初始数据中，Korteweb-de Vries方程的解决方案立即将在时空和时段变量中平滑至真实分析。对于非线性Schroedinger方程和Benjamin-ono方程的溶液也可以显示类似的效果。此外，与合作者H. Kozono和Y. Taniuchi的解决方案，Ogawa表明，不可压缩的Navier-Stokes方程和Euler方程表明，唯一性和规律性标准。此外，还鉴于用BESOV空间提出了谐波热流的解决方案。这些结果是通过改善Sobolev Inqualit的关键类型来获得的……在BESOV空间中更多。同时，通过使用Lizorkin-Triebel插值空间获得了Beale-Kato-Majda型不平等，涉及对数期的不平等。此外，在特殊的阈值条件下，该溶液在有限的时间内发展出奇异性。还表明，对于积极的解决方案，阈值是尖锐的。研究人员S. kawashima研究了解决方案对通用椭圆形 - 热纤维系统的不对称行为，包括散热气的方程。不对称行为可以由系统的线性部分进行特征，并由通常的热核。Co-corearcher Y.Kagei与共同研究者T.Kobayashi进行了研究，介绍了解决方案对三个尺寸半空间中不可压缩的空语播种的不对称行为。他们研究了方程的恒定密度稳态稳定性的稳定性，并显示了l-2.co-co-coolearcher K. iTo研究的扰动解决方案的最佳衰减顺序，涉及中间表面扩散方程式，并表明该解决方案在扩散系数与较大的范围内相互作用时具有自我交互。在时间参数无穷大时，在非线性schroedinger方程的溶液上。当非线性具有远距离相互作用的阈值指数时，他们确定了散射溶液的不对称曲线的第二项。较少的

项目成果

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

K.Ito，Y.Kohsaka：“三角域中表面扩散的三相边界运动”数学进展。

K.Yoshitomi: "Eigenvalue problems on domains with cracks II"シンポジウム「スペクトル、散乱理論とその周辺」講究録,京都大学数理解析研究所. 1255. 76-97 (2001)

K. Yoshitomi：“裂纹域的特征值问题 II”研讨会“频谱、散射理论和相关主题”Kokyuroku，京都大学数学分析研究所 1255. 76-97 (2001)。

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)

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）。