Leírás
This talk consists of two parts:
The first part deals with the geometry of minimizers for certain free energy functionals related to aggregation-diffusion equations.
As shown by Lim and McCann [ARMA, 2021], the minimizers of the interaction energy with certain mildly repulsive potentials are equi-distributed on $\Delta^n$, the vertices of the $n$-simplex. Employing the $\Gamma$-convergence, we show that the minimizers of the free energy functional are compactly supported and equi-distributed near $\Delta^n$ when the diffusion effect corresponds to the porous medium diffusion. This is a joint work with Tongseok Lim (Purdue).
The second part is devoted to the study of the asymptotic convergence of nonlinear PDEs described by gradient flows for nonconvex functionals. Our goal in this research is to establish the generalization of the Łojasiewicz-Simon theory to nonconvex Wasserstein gradient flows. This is a joint work in progress with Beomjun Choi and Seunghoon Jeong (POSTECH).