One of my lines of research is the study of the so-called Aggregation-Diffusion Equations

$\tag{ADE} \label{eq:ADE} \frac{\partial \rho}{\partial t} = \mathrm{div} \Big ( \rho \nabla ( U'(\rho) + V + {W * \rho } ) \Big)$

Typically, we are interested in the Porous-Medium / Fast-Diffusion range $$U(\rho) = \tfrac{1}{m-1} \rho^m$$, and radially increasing $$V$$ and $$W$$.

For a discussion on the modelling for this problem follow:
Modelling of Aggregation-Diffusion Equations

It is easy to prove formally that if $$W(x) = W(-x)$$, then \eqref{eq:ADE} is the gradient-flow in Wasserstein space of following free-energy functional

$\mathcal F [\rho] = \int_{\mathbb R^d} U(\rho) + \int_{\mathbb R^d} V \rho + \frac 1 2 \int_{\mathbb R^d} \int_{\mathbb R^d} \rho(x) W(x-y) \rho(y) \, dx dy$

When $$\mathcal F$$ is “suitably convex” and bounded below, then the gradient flow is well-defined and $$\rho(t)$$ converges to the unique minimiser of $$\mathcal F$$ as $$t \to \infty$$. A broad range of applicability of this theory is given in [1].

However, it well known that the heat equation ($$U(\rho) = \rho \log \rho, V = W = 0$$) diffuses like the heat kernel

$K(t,x) = \frac{1}{(4\pi t)^{\frac d 2}} \exp\left({- \frac{|x|^2}{4t}}\right) .$

Similar behaviour happens for the Porous Medium Equation (see [2]).

This two possible opossing behaviours give rise to a reach variety of results which apply different techniques. I have focused in the following two open problems.

## Concentration phenomena

It was recently proven that minimisers of these types of functionals may contain Dirac deltas. In that case, the usual notion of solution of a diffusion equation does not hold. In this direction we point to [3]. However, it is not clear whether this are actually the stationary states of the evolution problem.

Going back to the theory of Viscosity Solutions, in [4] (jointly with J.A. Carrillo and J.L. Vázquez) we were able to by elaborate compactness arguments that, when $$W =0$$ and $$V$$ is suitable, the expected explicit minimisers are actually achieved. The bulk of our analysis is performed in bounded domains.

The study of the case $$W \ne 0$$ is work in progress with a co-supervised Ph.D. student: Alejandro Fernández-Jiménez.

## Asymptotic simplification

At the other end of the spectrum, many authors were interested in studying under which hypothesis does it happen for $m = 1$ that $$\rho(t,x) \sim K(t,x)$$ as $$t \to \infty$$. Partial results were available for a decade (see [5]).

In [6] (joint work with J.A. Carrillo, Y. Yao, and C. Zheng) we provide almost-sharp hypothesis so that this is true, and provide rates of converge

$\| \rho(t,x) - K(t,x) \|_{L^1(\mathbb R^d)} \le C t^{-\alpha}.$

Our analysis is based on relative entropy arguments, and a priori estimates in Sobolev and Hölder spaces.

## References

1. Ambrosio, Luigi and Gigli, Nicola and Savare, Giuseppe. (2005) Gradient Flows. Birkhäuser-Verlag. Link
2. Vázquez, Juan Luis. (2006) The Porous Medium Equation. Oxford University Press. Link
3. Carrillo, José A. and Delgadino, Matías G. and Dolbeault, Jean and Frank, Rupert L. and Hoffmann, Franca. (2019) Reverse Hardy–Littlewood–Sobolev inequalities. J. des Math. Pures Appl.. Link
4. Carrillo, J.A. and Gómez-Castro, D. and Vázquez, J.L.. (2022) Infinite-time concentration in aggregation–diffusion equations with a given potential. Journal de Mathématiques Pures et Appliquées. Link
5. Cañizo, José A. and Carrillo, José A. and Schonbek, Maria E.. (2012) Decay rates for a class of diffusive-dominated interaction equations. J. Math. Anal. Appl.. Link
6. Carrillo, José A. and Gómez-Castro, D. and Yao, Yao and Zeng, Chongchun. (2021) Link