Navier—Stokes方程稳定性研究（Ⅲ）

A Stability Study of Navier-Stokes Equations(Ⅲ)

本文给出Ｎａｖｉｅｒ—Ｓｔｏｋｅｓ方程某些初值问题存在Ｃ￣２解的必要条件，并给出其在｛ｔ＝０｝上的初值问题不适定的例证。Ｎａｖｉｅｒ—Ｓｔｏｋｅｓ方程的初值问题是研究这个方程的基础问题之一。国内外很多学者在这方面的研究曾取得了不同程度的结果。法国时Ｊ．Ｌｅｒａｙ教授就曾在某种意义下证明过Ｎａｖｉｅｒ—Ｓｔｏｋｅｓ方程某种初边值问题解的存在性￣［３］．本文根据Ｊ．Ｈａｄａｍａｒｄ的偏微分方程的基础理论￣［１］．给出某些关键问题的严格定义，叙述一个有关Ｎａｖｉｐｒ－Ｓｔｏｋｅｓ方程不稳定的基本定理。最后给出若干例证，其证明可参见［４］。

In this paper, the necessary conditions of the existence of C￣2 solutions in some initial problems of Navier- Stokes equatious are given,and examples of ins-tability of initial value (at t=0) problems are also given。The initial value problem of Navier-Stokes equation is one of the most fundamental problem for this equation and various authors studies this problem and contributed a nun1ber of resuIts.1.Leray,a FrenG,h professor,proved the existence of Navier一Stokes equa-tion ulder certain defind initial and boundary value conditions.In this paper,with certa in rigorously defined key concepts.based upon the basic theory of J.Hadamard partial deferential equations[1],gives a fundamental theory of insta-bility of Navier-Stokes eql1ations.Finally,many examples are given,proofs referring to reference[4].