Energy-variational solutions for geodynamical two-phase flows -- From logarithmic to double-obstacle potentials by variational convergence
Fan Cheng, Robert Lasarzik, Marita Thomas
TL;DR
The paper develops an energy-variational solution (EVS) framework for a geodynamical two-phase flow described by a coupled Cahn–Hilliard–type phase-field model and Maxwell-type stress evolution, incorporating a non-constant mobility. It introduces an auxiliary energy variable $E$ and a regularity weight $\mathcal{K}$ to obtain a relative energy-dissipation inequality, and proves existence of EVS for both regularized ($\gamma>0$) and non-regularized ($\gamma=0$) settings with a logarithmic phase-field potential. A core contribution is the variational limit from a logarithmic potential to a double-obstacle potential via Mosco convergence, enabling the phase-field to reach pure phases while remaining well-posed under variational convergence methods. The EVS framework is shown to be compatible with, and indeed finer than, dissipative solutions, preserving a semi-flow property and enabling robust limit passages; this yields a consistent pathway to double-obstacle models and supports variational convergence analyses in geophysical multiphase flows.
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.