Global Well-Posedness of Solutions for the Sixth Order Convective Cahn-Hilliard Equation

DOI：10.3770/j.issn:2095-2651.2021.02.004

 作者 单位 赵晓朋 东北大学理学院, 辽宁 沈阳 110819 刘凤楠 大连理工大学数学科学学院, 辽宁 大连 116024 孟海潮 江南大学理学院, 江苏 无锡 214122

本文考虑了一个具小初值的六阶对流Cahn-Hilliard方程Cauchy问题光滑解的整体适定性问题. 我们首先构造了一类光滑局部解, 进一步假设初值的$L^1$模充分小且光滑非线性函数$f(u)$和$g(u)$在点$\bar{u}\in\mathbb{R}$处满足局部增长条件, 结合先验估计, 连续性准则, 我们将该局部光滑解延拓到$t>0$.

In this paper, we consider the global well-posedness of smooth solutions for the Cauchy problem of a sixth order convective Cahn-Hilliard equation with small initial data. We first construct a local smooth solution, then by combining some a priori estimates, continuity argument, the local smooth solution is extended step by step to all $t>0$ provided that the $L^1$ norm of initial data is suitably small and the smooth nonlinear functions $f(u)$ and $g(u)$ satisfy certain local growth conditions at some fixed point $\bar{u}\in\mathbb{R}$.