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

Cheng Fan, Robert Lasarzik+1 more|arXiv (Cornell University)|Jan 13, 2026

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TL;DR

The paper introduces energy-variational solutions for geodynamical two-phase flows and analyzes the variational limit from a logarithmic to a double-obstacle potential, highlighting advantages over dissipative solutions.

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.

Motivation & Objective

• Motivate and model two-phase geodynamical flows of incompressible viscoelastoplastic fluids with diffuse interfaces.
• Establish the well-posedness framework via energy-variational solutions that include an auxiliary energy and a regularity-weighted error term.
• Study variational convergence from logarithmic to double-obstacle phase-field potentials and its impact on sharp-interface limits.
• Compare energy-variational solutions with dissipative solutions and elucidate advantages for variational convergence methods.

Proposed method

• Define energy-variational solutions with an auxiliary energy variable E(t) majorizing the system energy.
• Introduce a regularity weight K that yields a lower semicontinuous defect term in the energy balance.
• Use Gamma-convergence and graph-convergence to pass to the limit in weakly convergent approximating sequences.
• Apply a time-discrete scheme to the geodynamical two-phase system and obtain existence with stress diffusion (gamma>0).
• Pass to the limit gamma→0 to obtain energy-variational solutions for the non-regularized system.
• Perform the variational limit alpha→0 from a logarithmic to a double-obstacle potential in the Cahn–Hilliard component and analyze the resulting energy-variational solution.

Experimental results

Research questions

• RQ1 How can energy-variational solutions be formulated for geodynamical two-phase flows with non-smooth dissipation?
• RQ2 Does the energy-variational framework provide a robust convergence method under weak limits, compared to dissipative solutions?
• RQ3 What is the effect of passing from a logarithmic phase-field potential to a double-obstacle potential on solution properties and convergence?
• RQ4 Can one recover pure phases or allow phase boundaries to reach pure states within a variational limit?
• RQ5 How does the variational limit interact with non-constant mobility and stress diffusion in the model?

Key findings

• Energy-variational solutions are shown to be always dissipative and to satisfy a semi-flow property.
• Existence results are established for the system with a logarithmic phase-field potential.
• A limit gamma→0 is performed to obtain energy-variational solutions without stress diffusion.
• A variational limit alpha→0 from logarithmic to double-obstacle potential is analyzed within the energy-variational framework.
• The approach highlights that energy-variational solutions are well-suited for variational convergence methods, unlike dissipative solutions.