Global Well-Posedness of Solutions for the Sixth Order Convective Cahn-Hilliard Equation
Received:March 06, 2020  Revised:September 07, 2020
Key Words: Global smooth solution   sixth order convective Cahn-Hilliard equation   Cauchy problem   local existence  
Fund Project:Supported by the Fundamental Research Funds for the Central Universities (Grant No.N2005031).
Author NameAffiliation
Xiaopeng ZHAO College of Sciences, Northeastern University, Liaoning 110819, P. R. China 
Fengnan LIU School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China 
Haichao MENG School of Science, Jiangnan University, Jiangsu 214122, P. R. China 
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      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 $\bar{u}\in\mathbb{R}$.
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