Dissipative solutions for fibred Wasserstein gradient flows of Cahn-Hilliard-type energies

Abstract

We study fourth-order evolution equations with a fibred Wasserstein gradient-flow structure, describing spatial transport of finitely many or a continuum of types. The driving Cahn-Hilliard-like energy combines a diagonal gradient contribution with nonlinear interactions between types. We prove subsequential convergence of the fibred Jordan-Kinderlehrer-Otto scheme to solutions satisfying an energy-dissipation inequality. For general type distributions, the absence of compactness in the type variable is addressed through defect-based solution concepts. On $\mathbb R^d, d\le 3$, we construct dissipative solutions satisfying an additional entropy-dissipation inequality. On smooth bounded domains in any space dimension, a coercivity estimate for the metric slope allows us to construct energy-dissipating solutions without relying on flow interchange, thereby covering non-convex domains endowed with the intrinsic length distance. In both cases, a variational comparison of the piecewise constant and De Giorgi interpolants yields a common limiting energy and strong convergence of the difference to zero without requiring strong compactness. For finitely many types, the defects vanish and the construction yields weak solutions. We further establish weak-strong uniqueness of dissipative solutions in every space dimension and of energy-dissipating solutions in dimensions $d\le5$. Combined with the respective existence results, this yields strong convergence of the variational approximations whenever a sufficiently regular, strictly positive solution exists.