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Research for the Lp theory of the solutions to nonlinear partial differential equations

非线性偏微分方程解的Lp理论研究

基本信息

批准号：
09640179
负责人：
OGAWA Takayoshi
金额：
$ 1.98万
依托单位：
KYUSHU UNIVERSITY (1998)Nagoya University (1997)
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 1998
项目状态：
已结题

项目摘要

1. Concerning a system of nonlinear dispersive equations arose from the water wave theory, T.Ogawa discovered a different kind of a smoothing effect mainly due to the special structures of nonlinear coupling and established the local Well-posedness of the solution in a weaker initial data.2. H.Kozono studied the uniqueness problem for the Leray -Hopff weak solution to the Navier-Stokes equation and showed the uniqueness holds for the critical case, C(O, T ; L^n), suppose that the solution satisfies the small gap condition.3. M.Kawashita considered unique existence of the strong solutions of the Cauchy prob- lems of the compressible Navier-Stokes equations. These equations are well known as explaining motions of fluid that density may change in time and space variables.4. K.Kato worked with Dr. P.Pipolo about the solitary wave solutions to general- ized Kadomtsev-Petviashvili equations (KP equations) and proved that solutions are real analytic. Also in a joint work with N.Hayashi and P.Naumkin he studies that there exist scattering states to small initial data for some nonlinear Schr_dinger equations and Hartree equations by using some class of Gevrey functions.5. S.Kawashima proved the existence and asymptotic stability of shock waves for the simplest model system of a radiating gas. Also, we showed the existence of global solutions to a class of hyperbolic-elliptic coupled systems and obtained the decay estimate of the solutions.6. Y.Kagei introduced a new approximation to the Oberbeck-Boussinesq equation and showed the existence and uniqueness of solution. Also the stability is discussed.

1. 对于由水波理论引起的非线性色散方程系统，T.Ogawa发现了一种不同的平滑效应，主要是由于非线性耦合的特殊结构，并在较弱的初始数据下建立了解的局部适定性。H.Kozono研究了Navier-Stokes方程的Leray -Hopff弱解的唯一性问题，并证明了当解满足小间隙条件时，对于临界情况C（O， T; L^n）的唯一性保持。考虑了可压缩Navier-Stokes方程柯西问题强解的唯一存在性。这些方程以解释密度随时间和空间变化而变化的流体运动而闻名。K.Kato与P.Pipolo博士共同研究了广义Kadomtsev-Petviashvili方程（KP方程）的孤波解，并证明了解是实解析解。在与N.Hayashi和P.Naumkin的联合工作中，他利用一类Gevrey函数研究了一些非线性Schr_dinger方程和Hartree方程的小初始数据存在散射态。S.Kawashima证明了辐射气体最简单模型系统激波的存在性和渐近稳定性。此外，我们还证明了一类双曲-椭圆耦合系统整体解的存在性，并得到了解的衰减估计。Y.Kagei引入了Oberbeck-Boussinesq方程的一种新的近似，并证明了解的存在唯一性。并对稳定性进行了讨论。