Well-Posedness of Solutions for Sixth-Order Cahn-Hilliard Equation Arising in Oil-Water-Surfactant Mixtures

Abstract

In this paper, by using the $L_p$-$L_q$-estimates, regularization property of the linear part of $e^{-t\Delta^3}$ and successive approximations, we consider the existence and uniqueness of global mild solutions to the sixth-order Cahn-Hilliard equation arising in oil-water-surfactant mixtures in suitable spaces, namely $C^0([0,T];\dot{W}^{2,\frac{N(l-1)}2}(\Omega))$ when the norm $\|u_0\|_{\dot{W}^{2,\frac{N(l-1)}2}(\Omega)}$ is sufficiently small.