Studies on behaviors of solutions to hydrodynamical equations

(1)对于描述无界区域内不可压缩流体运动的Navier-Stokes方程的解，研究了定常解在无穷远处的衰减和非定常解在几种LP范数下的时间衰减，发现了方程的非线性对衰减的影响。(2)建立了Burgers-Helmholtz方程组古典解和Sobolev型整体解存在的充分条件，并得到了解的渐近性态.证明了行波的存在性和渐近稳定性。(3)建立了可压缩Euler-Helmholtz系统熵函数的数学定义。(4)本文证明了粘性不可压缩流体无限层定常流动在临界稳定性情形下的渐近稳定性。(5)对于Korteweg-de Vires方程的初值问题，发现了解析光滑效应. (6)引入了一类可以处理拟线性正对称双曲型方程组解的形式渐近展开式，并研究了它的基本性质。(7)提出了一种新的抽象Wiener空间的复化方法。建立了基于复化的变量复变方法，并将其应用于研究随机积分的渐近性态（稳定相法、鞍点法等）。

(1) As for solutions to Navier-Stokes equations, which describe motions of incompressible fluids in unbounded domains, decays at infinity of stationary solutions and time-decays of non-stationary measured in several LP-like norms were studied, and an influence of non- lineality of the equations on the decay was found out. (2) Sufficient conditions for classical and Sobolev type global solutions to Burgers-Helmholtz system were established, and asymptotic behavior of the solutions were obtained. The existence of traveling waves and theire asymptotic stability were seen. (3) A mathematical definition of entropy functions for compressible Euler-Helmholtz system was established. (4) The asymptotic stability of steady flows in infinite layers of viscous incompressible fluids in critical cases of stability has been verified. (5) For the initial value problems associated with Korteweg-de Vires equations, an analytic smoothing effect was found out. (6) A class where one can handle formal asymptotic expansions of solutions to quasilinear positive symmetric systems of hyperbolic equations was introduced, and its basic properties were studied. (7) A new complexification of an abstract Wiener space was proposed. A complex change of variable based on the complexification was established, and appled to study asymptotic behaviors of stochastic oscillatry integrals (methods of stationary phase, saddle point methods, and so on).