Weak solutions for the functionalized Cahn–Hilliard equation with degenerate mobility

Abstract

The functionalized Cahn–Hilliard free energy describes phase separation in mixtures of amphiphilic molecules in solvent. Applications to highly amphiphilic molecules such as lipids requires degenerate diffusion that eliminates bulk diffusion, resulting in surface driven diffusion. We study the existence of weak solutions of a gradient flow of the functionalized Cahn–Hilliard equation that incorporates degenerate mobility, capturing solutions as limits of the weak solution for equations with non-degenerate mobilities.

COMMUNICATED BY:

• M. Mei

Keywords:

• Weak solutions
• the functionalized Cahn–Hilliard equation
• degenerate mobility

2010 Mathematics Subject Classifications:

• 35G20
• 35G31

Acknowledgments

KP recognizes support from the NSF DMS under award 1813203.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the US National Science Foundation (NSF) through grants DMS-1802863 and DMS-1815746 (Shibin Dai), DMS-1813203 (Keith Promislow), and by NSF of Guangdong Province, China through grant 2016A030313048 (Qiang Liu).