Dissipative solutions for viscoelastic phase separation with degenerate mobility: a dynamic transport approach
Abstract
We study the convergence of minimising movement approximations to weak solutions of a fourth-order viscoelastic phase separation system with degenerate concave mobility. The underlying semi-implicit variational scheme combines a dynamic transport distance for the phase-field variable with a weighted L2-distance for a transformed stress variable. We establish subsequential convergence of the scheme and show that the limiting solutions satisfy the energy-dissipation inequality. To this end, we derive a coercivity estimate for the metric slope, which yields the required space-time regularity for the De Giorgi interpolant, and construct constitutive fluxes whose dissipation is controlled by the squared metric slope. These estimates allow us to pass to the limit in the discrete energy-dissipation inequality. The weak formulation of the evolution equation for the phase-field variable is derived using the flow interchange technique. Under additional assumptions on the data, we further show that the limiting solutions satisfy an entropy-dissipation inequality
Related
- A linear, second-order, unconditionally energy-stable scalar-auxiliary-variable scheme for the CHQ viscoelastic phase-separation model
- Nonlinear Stability of Linearly Expanding Goldreich-Weber Solutions for the Navier-Stokes-Poisson System with Degenerate Viscosity Under Radial Perturbations
- Solutions to fourth order degenerate parabolic equations obtained via weighted energy dissipation
- Weak Solutions for the Unregularised Hibler Momentum Equation with Degenerate Coefficients
- Global weak solutions to the Cahn-Hilliard equation with degenerate mobility and singular diffusion
BibTeX
@misc{gau2026dissipative,
title = {{Dissipative solutions for viscoelastic phase separation with degenerate mobility: a dynamic transport approach}},
author = {Moritz Immanuel Gau and Katharina Hopf and Matthias Liero},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11034},
eprint = {2610.11034},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11034},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dissipative solutions for viscoelastic phase separation with degenerate mobility: a dynamic transport approach
AU - Moritz Immanuel Gau
AU - Katharina Hopf
AU - Matthias Liero
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11034
UR - https://arxiv.org/abs/2610.11034
AB - We study the convergence of minimising movement approximations to weak solutions of a fourth-order viscoelastic phase separation system with degenerate concave mobility. The underlying semi-implicit variational scheme combines a dynamic transport distance for the phase-field variable with a weighted L^2-distance for a transformed stress variable. We establish subsequential convergence of the scheme and show that the limiting solutions satisfy the energy-dissipation inequality. To this end, we derive a coercivity estimate for the metric slope, which yields the required space-time regularity for the De Giorgi interpolant, and construct constitutive fluxes whose dissipation is controlled by the squared metric slope. These estimates allow us to pass to the limit in the discrete energy-dissipation inequality. The weak formulation of the evolution equation for the phase-field variable is derived using the flow interchange technique. Under additional assumptions on the data, we further show that the limiting solutions satisfy an entropy-dissipation inequality
ER -