Pressure jump and radial stationary solutions of the degenerate Cahn–Hilliard equation

Abstract

The Cahn–Hilliard equation with degenerate mobility is used in several areas including the modeling of living tissues, following the theory of mixtures. We are interested in quantifying the pressure jump at the interface between phases in the case of incompressible flows. To do so, we depart from the spherically symmetric dynamical compressible model and include an external force. We prove existence of stationary states as limits of the parabolic problems. Then we prove the incompressible limit and characterize compactly supported stationary solutions. This allows us to compute the pressure jump in the small dispersion regime and in particular the force dependent curvature effect.

Keywords

• Degenerate Cahn–Hilliard equation
• Asymptotic Analysis
• Incompressible limit
• Hele–Shaw equations
• Surface tension
• Pressure jump