Study on the global behavior of solutions for the fluid equation

我们主要开展了以下两个方面的研究工作：(1)分析了Kuramoto-Sivashinsky(KS)方程和/或Blasius方程的稳态解的结构。这两个方程都与流体力学有关，并且具有相似的三阶微分算子。利用单调性，将三阶方程化为二阶方程。在此基础上，证明了KS方程定态解的存在性。到目前为止，这种解决方案还没有在文献中被提及。此外，还给出了Blasius方程爆破解的初等存在性证明，这可能有助于阐明Blasius方程本身对于Prandtl边界层理论的有效性。(2)自由边界问题广泛存在于各种非线性科学中，包括单相流体流动问题。这里我们关注的是美式看跌期权的定价。我们给出了解的精确积分公式。

We have undertaken our research projects mainly on the following two subjects.(1) Results are obtained for the analysis on the structure of solutions to the steady state of the Kuramoto-Sivashinsky (KS) equation and/or to the Blasius equation. Both equations are related to the fluid dynamics and have the similar third-order differential operator. By use of the monotonicity, the reduction of the third-order equation into the second-order one is performed. In view of this reduction, the existence of blowing-up solutions for the steady state of the KS equation is proved. These kind of solutions have not been mentioned in the literature so far. Moreover, an elementary existence proof of blowing-up solutions for the Blasius equation is also given, which may shed light on the validity of the Blasius equation itself with regard to the Prandtl boundary layer theory.(2) Free boundary problems arise in a wide variety of nonlinear sciences, including one-phase fluid flow problem. Here we are concerned with the pricing of American put option. We present an exact integral formula for the solution.