Solutions to fourth order degenerate parabolic equations obtained via weighted energy dissipation
Abstract
We apply the variational method of Weighted Energy Dissipation (WED) to obtain a global-in-time approximation of solutions to fourth order degenerate parabolic equations of Cahn-Hilliard type. Differently from the standard approach to the existence theory, WED induces an elliptic regularization in time, not in space. The confinement on the phase field is guaranteed already for the approximation, a priori estimates follow without modification of Lyapunov functionals, and the approximations are weakly differentiable in time --- even twice in the case of linear mobility. While WED has been widely used for the construction of gradient flows in Hilbert spaces, this appears to be the first application of WED to a metric gradient flow beyond second order PDEs of Wasserstein type.
Related
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BibTeX
@misc{matthes2026solutions,
title = {{Solutions to fourth order degenerate parabolic equations obtained via weighted energy dissipation}},
author = {Daniel Matthes and Vera Pazukhina},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05293},
eprint = {2609.05293},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05293},
}
RIS
TY - UNPB
TI - Solutions to fourth order degenerate parabolic equations obtained via weighted energy dissipation
AU - Daniel Matthes
AU - Vera Pazukhina
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05293
UR - https://arxiv.org/abs/2609.05293
AB - We apply the variational method of Weighted Energy Dissipation (WED) to obtain a global-in-time approximation of solutions to fourth order degenerate parabolic equations of Cahn-Hilliard type. Differently from the standard approach to the existence theory, WED induces an elliptic regularization in time, not in space. The confinement on the phase field is guaranteed already for the approximation, a priori estimates follow without modification of Lyapunov functionals, and the approximations are weakly differentiable in time --- even twice in the case of linear mobility. While WED has been widely used for the construction of gradient flows in Hilbert spaces, this appears to be the first application of WED to a metric gradient flow beyond second order PDEs of Wasserstein type.
ER -