Aggregation-diffusion equations with saturation
On this talk we will focus on the family of aggregation-diffusion equations
$$
\frac{\partial \rho}{\partial t} = \mathrm{div}\left(\mathrm{m}(\rho) \nabla (U'(\rho) + V) \right).
$$
Here, $\mathrm{m}(s)$ 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. $U(s) = \frac{1}{m-1} s^m$ for $m > 0$ or $U(s) = s \log (s)$ if $m = 1$. $V$ corresponds to the attractive potential and it is such that $V \geq 0$, $V \in W^{2, \infty}$.
For this problem, we discuss: Existence using a suitable regularised approximation of the problem, we prove that the problem admits an $L^1$-contractive $C_0$-semigroup; $L^1$-local minimisers of the associated free-energy functional in the corresponding class of measures; and the long-time behaviour of the constructed solutions in view of its gradient flow structure. Furthermore, we observe saturation effects leading to "freezing" behavior, i.e. free boundaries at the saturation level. 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.