Nonlinear aggregation-diffusion equations: (reverse) HLS inequalities and equilibration.
speaker: José A. Carrillo De La Plata (Mathematical Institute, University of Oxford)abstract: We analyse under which conditions equilibration between two competing effects, repulsion modelled by nonlinear diffusion and attraction modelled by nonlocal interaction, occurs. I will discuss the regimes that appear in aggregation diffusion problems with homogeneous kernels. We will discuss the main qualitative properties in terms of stationary states and minimizers of the free energies. In particular, all the porous medium cases are critical while the fast diffusion are not in this regime. In the second part, I will discuss the diffusion dominated case for porous medium cases in which this balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrisation techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the fast diffusion regime we are able to find conditions for the existence of global minimizers in a different range by means of reversed HLS inequalities. Concentration at the origin in part of this range is not ruled out. This talk is based on works in collaboration with S. Hittmeir, B. Volzone, Y. Yao, V. Calvez, F. Hoffmann, E. Mainini, J. Dolbeault, M. Delgadino, and R. Frank.
timetable:
Tue 15 Jan, 11:00 - 12:00, Aula Magna Bruno Pontecorvo
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