The Comprehensive Study of Differential Equations

本研究计划拟涵盖常微分方程与偏微分方程两个领域。实际活动是在1989年12月20日至22日在早稻田大学举行的大会以及为该项目组织或与之有关的许多其他专题讨论会和研讨会上进行的。由于这些活动，在微分方程的各个分支中获得了许多有意义和重要的结果，如下所列：(1)实域内常微分方程和泛函微分方程的稳定性、振荡等定性理论；(2)复域常微分方程和偏微分方程的解析理论；(3)超几何函数等特殊函数的理论；(4)哈密顿系统理论；控制理论；(5)双曲型、抛物型、椭圆型线性偏微分方程（或方程组）与线性薛定谔方程解的存在唯一性和定性行为问题；(6)亚椭圆性；伪微分算子理论；波动方程的散射理论；演化方程理论；(7)双曲型、抛物型、椭圆型非线性偏微分方程和非线性薛定谔方程的正全局解、（非）有界解、爆破解、粘滞解等的存在唯一性和渐近性问题；(8)流体力学和其他应用科学中出现的非线性偏微分方程的基本问题，如Navier-Stokes方程、Boltzmann方程等。

This research project is, as proposed, aimed to cover the fields of both ordinary differential equations and partial differential equaitons. The actual activities have been done at the general meeting on 20-22/12/1989 at Waseda University and at many other symposia and seminars organized for or related to this project. As a consequence of these activities a great number of significant and important results have been obtained in various branches of differential equations as listed below :(1) qualitative theory, such as the stability, oscillation, etc., of ordinary and functional differential equations in the real domain ;(2) analytic theory of ordinary and partial differential equations in the complex domain ;(3) theory for the special functions such as hypergeometric functions ;(4) theory of Hamiltonian systems ; and the control theory ;(5) the problem of existence, uniqueness, and the qualitative behaviors of solutions for linear partial differential equations (or systems) of hyperbolic, parabolic and elliptic types and linear Schrodinger equations ;(6) hypoellipticity ; theory of pseudo-differential operator ; scattering theory for wave equations ; theory of evolution equation ;(7) the problem of existence, uniqueness and asymptotic behaviors of positive global solutions, (non) bounded solution, blow-up solutions, viscosity solutions, etc., of nonlinear partial differential equations of hyperbolic, parabolic and elliptic types and of nonlinear Schrodinger equations ;(8) fundamental problems for nonlinear partial differential equations arising in fluid dynamics and other applied sciences, such as Navier-Stokes equations, Boltzmann equations, etc.