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

1.对于由水波理论产生的非线性色散方程组，小川发现了主要由于非线性耦合的特殊结构而产生的一种不同的光滑效应，并在较弱的初值条件下建立了解的局部适定性. H.Kozono研究了Navier-Stokes方程Leray-Hopff弱解的唯一性问题，证明了在临界情形C（O，T ; L^n）下，当解满足小间隙条件时，该解的唯一性成立. M.Kawashita考虑了可压缩Navier-Stokes方程Cauchy问题强解的唯一存在性。这些方程是众所周知的解释运动的流体密度可能会改变时间和空间变量. K.Kato和P.Pipolo博士一起研究了广义Kadomtsev-Petviashvili方程（KP方程）的孤立波解，并证明了解是真实的解析的。在与N.Hayashi和P.Naumkin的合作工作中，他利用Gevrey函数研究了某些非线性Schr_dinger方程和Hartree方程在小初值下的散射态. S.Kawashima证明了最简单的辐射气体模型系统激波的存在性和渐近稳定性。另外，我们还证明了一类双曲-椭圆耦合方程组整体解的存在性，并得到了解的衰减估计. Y.Kagei对Oberbeck-Boussinesq方程引入了一个新的近似，并证明了解的存在唯一性。并对稳定性进行了讨论。

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.