Seminar series
Date
Thu, 27 Feb 2025
12:00
Location
C6
Speaker
Alejandro Fernández-Jiménez
Organisation
University of Oxford

On this talk we will focus on the family of aggregation-diffusion equations

 

ρt=div(m(ρ)(U(ρ)+V)).

 

Here, m(s) represents a continuous and compactly supported nonlinear mobility (saturation) not necessarily concave. U corresponds to the diffusive potential and includes all the porous medium cases, i.e. U(s)=1m1sm for m>0 or U(s)=slog(s) if m=1. V corresponds to the attractive potential and it is such that V0, VW2,.

 

Taking advantage of a family of approximating problems, we show the existence of C0-semigroups of L1 contractions. We study the ω-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. Furthermore, since this problem has a formal gradient-flow structure, we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the L-constrained gradient flow of probability densities. Finally, we explore the properties of a corresponding implicit finite volume scheme introduced by Bailo, Carrillo and Hu.

 

The talk presents joint work with Prof. J.A. Carrillo and Prof. D.  Gómez-Castro.

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