Relations between geometric invariant and singularity of solution in nonlinear evolution equations related to nonlinear waves

非线性波非线性演化方程中几何不变量与解奇异性的关系

• 批准号: 09640159
• 负责人: TSUTSUMI Yoshio
• 金额: $ 1.92万
• 依托单位: TOHOKU UNIVERSITY (1999)The University of Tokyo (1997-1998)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640159/
• 关键词: nonlinear wave equations; KdV equations; Cauchy problem; well-posedness; white noise; stochastic differential equations; KdV方程式; 確率微分方程式; ストリッカーツの評価式; 伝播速度の異なる波の相互作用; クライン-ゴルドン-ザハロフ方程式; 零条件; 非線形クライン-ゴルドン方程式; ヘルマンダー予想; 概大域的存在; 非線型シュレディンガー方程式

We had studied the following two subjects for the period of July, 1997-March, 2000.We first studied the well-posedness of the Cauchy problem for the system of nonlinear wave equations with different propagation speeds. One of the most important problems in the field of partial differential equations is to look for the largest possible function space in which the wave equations with quadratic nonlinearity is well-posed. This problem is closely related to the Lorentz invariant for the wave equation. When we consider the system of nonlinear wave equations with different propagation speeds, the discrepancy of propagation speeds breaks the Lorentz symmetry. We classified the quadratic nonlinear terms from a point of view of the time local well-posedness.Second, we studied the unique solvability of the Cauchy problem for the Korteweg-de Vries equation with stochastic forcing term. The stochastic forcing term is regarded as a nonsmooth perturbation from a mathematical point of view. Especially, in general, the inverse scattering method is inapplicable to the Korteweg-de Vries equation with forcing term. We investigated the time local existence of solution for a natural class of stochastic forces.

我们研究了1997年7月3月，2000年7月的两名主题。我们首先研究了凯奇问题的良好性，用于具有不同传播速度的非线性波方程系统。部分微分方程领域中最重要的问题之一是寻找最大的可能功能空间，在该空间中，具有二次非线性的波方程良好。这个问题与波动方程的洛伦兹不变密切相关。当我们考虑具有不同传播速度的非线性波方程系统时，传播速度的差异会破坏Lorentz的对称性。我们从局部稳定度的时间的角度对二次非线性项进行了分类。第二，我们研究了cauchy问题的独特解决性，用于使用随机强迫术语进行korteweg-de vries方程。从数学的角度来看，随机强迫项被认为是非平滑的扰动。尤其是，通常，反向散射方法不适合使用强迫项的Korteweg-de Vries方程。我们研究了自然类随机力的局部溶液的时间。

项目成果

A.de Bouard: "White noise driven Korteweg-de Vries equation"J. Funct. Anal.. 169. 532-558 (1999)

A.de Bouard：“白噪声驱动的 Korteweg-de Vries 方程”J。

T.Ozawa: "On the coupled system of nonlinear wave equations with different propagation speeds"Proceedings Series of Banach Center. (出版予定).

T. Ozawa：“不同传播速度的非线性波动方程的耦合系统”巴拿赫中心论文集系列（即将出版）。

T.Ozawa: "On the coupled system of nonlinear wave equations with different propagation speeds"Proceedings Series of Banach Center. 出版予定.

T. Ozawa：“不同传播速度的非线性波动方程的耦合系统”，巴拿赫中心论文集系列即将出版。

A.de bouard: "White noise driven Korteweq-de Vries equation"J. Funct. Anal.. 169. 532-558 (1999)

A.de Bouard：“白噪声驱动的 Korteweq-de Vries 方程”J。

K.Moriyama: "Almost global existence of solutions for the quadratic semilthnear Klein-Gordon equation in one space dimension" Funkcialaj Ekvacioj. 40. 313-333 (1997)

K.Moriyama：“一维二次半近 Klein-Gordon 方程的解几乎全局存在”Funkcialaj Ekvacioj。