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

Abstract:

      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 $\bar{u}\in\mathbb{R}$.