Diffuse interface model for two-phase flows on evolving surfaces with different densities: global well-posedness Diffuse interface model for two-phase flows on evolving surfaces

• Published in: Calculus of variations and partial differential equations Vol. 64; no. 5
• Main Authors: Abels, Helmut, Garcke, Harald, Poiatti, Andrea
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2025
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical; 35D35; 35Q35; 76D05; 35Q30; 76T06
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-025-03001-w

We show global in time existence and uniqueness on any finite time interval of strong solutions to a Navier–Stokes/Cahn-Hilliard type system on a given two-dimensional evolving surface in the case of different densities and a singular (logarithmic) potential. The system describes a diffuse interface model for a two-phase flow of viscous incompressible fluids on an evolving surface. We also establish the validity of the instantaneous strict separation property from the pure phases. To show these results we use our previous achievements on local well-posedness together with suitable novel regularity results for the convective Cahn-Hilliard equation. The latter allows to obtain higher-order energy estimates to extend the local solution globally in time. To this aim the time evolution of energy type quantities has to be calculated and estimated carefully.