Energy-variational solutions for geodynamical two-phase flows -- From logarithmic to double-obstacle potentials by variational convergence

TL;DR

This paper introduces energy-variational solutions for geodynamical two-phase flows, compares them with dissipative solutions, and explores the convergence from logarithmic to double-obstacle potentials, enabling better handling of pure phases.

Contribution

It develops and analyzes energy-variational solutions for two-phase flow models, demonstrating their advantages for variational convergence over dissipative solutions.

Findings

Energy-variational solutions include an energy variable that bounds the system energy.

Comparison shows energy-variational solutions are more suitable for variational convergence.

The study establishes the limit from logarithmic to double-obstacle potentials, allowing pure phases.

Abstract

In [Cheng, Lasarzik, Thomas 2025 ARXIV-Preprint 2509.25508], we studied a Cahn--Hilliard two-phase model describing the flow of two viscoelastoplastic fluids in the framework of dissipative solutions using a logarithmic potential for the phase-field variable. This choice of potential has the effect that the fluid mixture cannot fully separate into two pure phases. The notion of dissipative solutions is based on a relative energy-dissipation inequality featuring a suitable regularity weight. In this way, this is a very weak solution concept. In the present work, we study the well-posedness of the geodynamical two-phase flow in the notion of energy-variational solutions. They feature an additional scalar energy variable that majorizes the system energy along solutions and they are further characterized by a variational inequality that combines an energy-dissipation estimate with the weak formulation of the system adding an error term that accounts for the mismatch between the energy variable and the system energy multiplied by a suitable regularity weight. We give a comparison of these two concepts. We further study different phase-field potentials for the geodynamical two-phase flow model. In particular, we address the variational limit from a potential with a logarithmic contribution to a double-obstacle potential, then also allowing for the emergence of pure phases. This study underlines that, thanks to its structure, the energy-variational solution is better suited for variational convergence methods than the dissipative solution.