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

TL;DR

This paper investigates the pressure jump and stationary solutions of the degenerate Cahn-Hilliard equation, providing insights into interface behavior and limits in incompressible flow models.

Contribution

It introduces a method to compute pressure jumps in stationary radial solutions and characterizes incompressible stationary states, including the incompressible limit and convergence results.

Findings

Computed pressure jump in small dispersion regime.

Characterized compactly supported stationary solutions.

Proved convergence of parabolic problems to stationary states.

Abstract

The Cahn-Hilliard equation with degenerate mobility is used in several areas including the modeling of living tissues. We are interested in quantifying the pressure jump at the interface in the case of incompressible flows. To do so, we include an external force and consider stationary radial solutions. This allows us to compute the pressure jump in the small dispersion regime. We also characterize compactly supported stationary solutions in the incompressible case, prove the incompressible limit and prove convergence of the parabolic problems to stationary states.