三维Cahn-Hilliard方程的整体适定性

Global well-posedness of the 3D Cahn-Hilliard equation

主要研究三维Cahn-Hilliard方程的Cauchy问题。首先,利用傅里叶变换求出其相对应线性方程的形式解并证明形式解的光滑性;然后,构造压缩映射,应用Banach不动点定理证明其局部适定性;最后,通过连续性准则得到其在无任何小初值假设条件下的整体适定性。

In this paper,the Cauchy problem of the three-dimensional Cahn-Hilliard equations has been studied.Firstly,the formal solution of the corresponding linear system is obtained by the Fourier transform method,and the smoothness of the formal solution is proved.Then,the local well-posedness is proved by constructing the compression map and applying the Banach fixed point theorem.Finally,the global well-posedness is demonstrated via the continuation criterion without assumption of small initial data in Sobolev spaces.