一类两相流体扩散界面模型的适定性研究

Navier-Stokes-Cahn-Hilliard(NS-CH) system is a diffuse interface model for two-phase flows of viscous, incompressible fluids, in this item, we will investigate global well-posedness of this system in two different free energy case. Mathematically, this sysetem is a class of coupled equations in form which have the second order parabolic equations and the fourth order parabolic equation. As is well-known, it is invalid to apply comparison principle for the forth order parabolic equation, which is our main difficulty in this item. As for a classe of NS-CH system with a singular (logarithmic tye) free enengy, since there is a singular property in system, it is difficlut to get higher regular estimates; we need to consider an approximation equation for the convective Cahn-Hilliard equation , by separating the solutions from the singular valuse of the potential, and then we will disscusses the global well-posedness in two-dimensions. As for a class of NS-CH system with a smooth (polynomial type) free energy, this thesis discusses axisymmetric solutions in three-dimensional total space, by using coordinate transformation to get a equivalence problem, and then we will consider the global well-posedness by energy method.

Navier-Stokes-Cahn-Hilliard(NS-CH)系统是一类粘性不可压缩两相流体扩散界面的数学模型，本项目考虑在两种不同的自由能量情形下该系统的整体适定性问题。数学上来讲，这是一类形式上的二阶抛物方程组耦合四阶抛物方程的数学模型。众所周知，经典的比较原理对四阶抛物型方程失效，这给我们的研究带来了一定的困难。对于一类具有奇异（对数型）自由能量的NS-CH系统，因为系统具有奇异性，所以要得到其更高的正则估计是比较困难的；我们需要先考虑对流Cahn-Hilliard方程的一个二阶逼近方程，得到势函数远离奇异值点的结果，从而再考虑二维情形下整个系统的整体适定性。对于一类具有光滑（多项式型）自由能量的NS-CH系统，我们考虑在三维全空间中其轴对称解的整体适定性，通过轴坐标变换得到等价问题，从而通过能量方法去考虑该系统的整体适定性。

自上个世纪80年代以来，越来越多的数学工作者开始关注耦合流体方程的研究。.Navier-Stokes-Cahn-Hilliard(NS-CH)系统是一类粘性不可压缩两相流体扩散界面的.数学模型，本项目考虑在两种不同的自由能量情形下该系统的整体适定性问题。数学上来.讲，这是一类形式上的二阶抛物方程组耦合四阶抛物方程的数学模型。众所周知，经典的.比较原理对四阶抛物型方程失效，这给我们的研究带来了一定的困难。对于一类具有奇异.（对数型）自由能量的NS-CH系统，因为系统具有奇异性，所以要得到其更高的正则估计.是比较困难的；我们需要先考虑对流Cahn-Hilliard方程的一个二阶逼近方程，得到势函.数远离奇异值点的结果，从而再考虑二维情形下整个系统的整体适定性。对于一类具有光.滑（多项式型）自由能量的扩散界面模型，我们考虑在三维全空间中解的整体适定.性，首先通过逼近方法得到局部适定性，进而通过能量方法去考虑该系统的整体适定性。