Existence results shown for a viscoelastoplastic two-phase flow model

What the study found: The authors established existence results for a Cahn–Hilliard model of two viscoelastoplastic fluids, including a regularized version with stress-diffusion and a non-regularized version using dissipative solutions.
Why the authors say this matters: The study suggests this is relevant for a two-phase flow model arising in geodynamics, where the authors conclude that dissipative solutions help extend existence results beyond the regularized setting.
What the researchers tested: The researchers studied a Cahn–Hilliard two-phase model for incompressible flow, using a phase-field variable to describe the mixture and a volume-averaged velocity formulation. The momentum balance included a Stokes-like viscous term and an internal stress tensor with a nonlinear Zaremba-Jaumann time derivative and a subdifferential of a non-smooth plastic potential.
What worked and what didn't: With stress-diffusion regularization, they obtained existence of Leray-Hopf-type weak solutions. They also introduced dissipative solutions based on a relative energy estimate, discussed their general properties, proved their existence in the stress-diffusion setting, and then used a limit passage in the relative energy inequality to obtain an existence result for the non-regularized model.
What to keep in mind: The abstract does not describe experimental data, numerical tests, or applications beyond the stated geodynamics setting. The summary is limited to well-posedness and existence results reported in the abstract.

Key points

• The paper studies a Cahn–Hilliard two-phase model for two viscoelastoplastic fluids.
• The model includes a phase-field variable, volume-averaged velocity, and an internal stress tensor for elastoplastic behavior.
• Existence of Leray-Hopf-type weak solutions was obtained with stress-diffusion regularization.
• The authors introduced dissipative solutions based on a relative energy estimate.
• An existence result for the non-regularized model was concluded by passing to the limit as stress-diffusion vanishes.