Abstract:
We discuss biological aggregation in a heterogeneous environment and on noncovex, nonsmooth domains. We model the heterogeneous environment as a Riemannian manifold with boundary and develop gradient flow approach in the space of probability measures on the manifold endowed with Riemannian 2-Wasserstein metric. We use the gradient flow structure to show the well-posedness of a class of nonlocal interaction equations. We discuss how heterogeneity of the environment leads to new dynamical phenomena.
We also present a result generalizing the well-posedness of weak measure solutions to a class of nonlocal interaction equations on nonconvex and nonsmooth domains. We use particle approximations, solve the discrete ODE systems and pass to the continuum limit by stability property. The novelty here is that under mild regularity conditions on space(i.e. prox-regularity), we can show the well-posedness of the ODE systems and the stability properties with explicit dependence on the geometry of the space(prox-regular constant).
The talk is based on collaborations with J. A. Carrillo and D. Slepcev. |