RESEARCH IN RESIDENCE — The Hele-Shaw flow with elasticity: Wasserstein flow and phase-field approximation

Our project focuses on understanding the interplay between elasticity and the classical Hele-Shaw model, with the long-term of simplifying the model and improving its suitability for numerical applications and algorithmic development. Our first aim is to show existence of solutions via the convergence of an implicit time-discretization using the Wasserstain distance. We expect the limit to satisfy the equation in a distributional sense as well as an optimal energy-dissipation relation. Similarly, we construct weak solutions to the Cahn-Hilliard equation with degenerate mobility and elastic misfits. Finally, we aim to prove that the sharp interface limit is a distributional solution to the elastically modified Hele–Shaw flow with optimal energy-dissipation rate.