Comprehensive study of differential equations

1.研究了薛定谔方程，得到了(1)散射波动算子的L p-有界性；(2)基本解的光滑有界性和次二次扰动下的稳定性；(3)阐明了时间周期系统Floquet算子的局部衰减性与谱之间的关系；(4)建立了相空间隧道效应的一般理论及其应用；(5)定义了随机算子的积分态密度，证明了Wegner估计和Lifschitz奇性。Kiyoomi Kataoka研究了线性偏微分方程组的一般理论，(1)给出了ε、Rx-模的全纯解复的初等定义，(2)将Fuchsian椭圆边值问题的奇性分支简化为ODE.3的继续分支。Yoshikazu Giga研究了非线性偏微分方程组，(1)给出了一阶非线性偏微分方程粘性解的数值计算方法，(2)定义了适当的粘性解并证明了解的存在唯一性，(3)证明了具有非滞后初始条件的Navier-Stokcs方程解的存在性。Yoshio TsutSumi证明了(1)具有不同速度的非线性耦合波动方程的适定性，(2)非线性质量Klein-Gordon方程常解的稳定性。Hideyuki Majima从超渐近分析的角度给出了关于渐近展开的新的基本定理。Ikawa研究了波动方程在多个凸体上的散射理论，得到了散射矩阵极点的精确渐近公式。

We carried out an comprehensive study on linear and nonlinear partial and ordinary differential equations and obtained among others the following results:1. Kenji Yajima and Shu Nakamura studied Schrodinger equations and obtained (1) the L^p-boundedness of wave operators of scattering and (2) the stability under subquadratic perturbations of the smooth and boundedness of fundamental solution; (3) clarified the relation between the local decay and the spectrum of Floquet operator for time periodic system; (4) constructed general theory of tunnenling in phase space and gave its applications; (5) defined the integrated density of states for random operators and proved Wegner estimates and the Lifschitz singularities.2. Kiyoomi Kataoka studied the general theory of linear PDE and (1) gave an elementary definition of the holomorphic solutions complex of ε^R_x-modules and (2) reduced the branching of the singularities for Fuchsian elliptic boundary problems to that of the continuation of ODE.3. Yoshikazu Giga studied nonlinear PDE and (1) gave a way to numerically compute the viscous solution of first order nonlinear PDE, (2) defined the proper visocous solutions and proved the existence and the uniquess of solutions and (3) proved the existence of solution of Navier-Stokcs equation with non-deaying initial conditions.4. Yoshio Tsutsumi proved (1) the well-posedness of nonlinearly couples wave equations with different speeds, (2) the stability of constant solutions of nonlinear massive Klein-Gordon equation.5. Hideyuki Majima produced new fundamental theorem on the asymptotic expansions from the point of view the super-asymptotic analysis.6. Mitsuru Ikawa studied the scattering theory of wave equation by several convex bodies and obtained a precise asymptotic, formula for the poles of scattering matrix.