# Research

## Supervision

Current:

**Alejandro Ramos Lora**. PhD student since 2019. Co-supervised with María José Cáceres.**Sebastian Throm**. Postdoctoral researcher funded by the German DFG. (2018-2020).

Past:

**Havva Yoldaş**. PhD student at Universidad de Granada / BCAM, finished in October 2019.**Daniel Balagué**. PhD student at Universitat Autònoma de Barcelona, finished in June 2013. Co-supervised with José A. Carrillo. The thesis is available at the Teseo online repository.

## Recent publications

Below you can find some recent papers. A full list can be found here.

### Preprints

**The scaling hypothesis for Smoluchowski’s coagulation equation with bounded perturbations of the constant kernel**. 2019.We consider Smoluchowski’s coagulation equation with a kernel of the form $K=2+\epsilon W$, where $W$ is a bounded kernel of homogeneity zero. For small $\epsilon$, we prove that solutions approach a universal, unique self-similar profile for large times, at almost the same speed as the constant kernel case (the speed is exponential when self-similar variables are considered). All the constants we use can be explicitly estimated. Our method is a constructive perturbation analysis of the equation, based on spectral results on the linearisation of the constant kernel case. To our knowledge, this is the first time the scaling hypothesis can be fully proved for a family of kernels which are not explicitly solvable.

### Some recent papers

**Hypocoercivity of linear kinetic equations via Harris’s Theorem**.*Accepted in Kinetic and Related Models*13(1), 2020.We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $(x,v)∈ \mathbb{T}^d \times \R^d$ or on the whole space $(x,v) ∈\R^d \times \R^d$ with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively $L^1$ or weighted $L^1$ norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris’s Theorem.

**Uniform moment propagation for the Becker-Döring equation**.*Proceedings of the Royal Society of Edinburgh, Section A: Mathematics*149(4):995–1015, 2019.We show uniform-in-time propagation of algebraic and stretched exponential moments for the Becker-Döring equations. Our proof is based upon a suitable use of the maximum principle together with known rates of convergence to equilibrium.

**Exponential equilibration of genetic circuits using entropy methods**.*Journal of Mathematical Biology*78(1-2):373–411, 2019.We analyse a continuum model for genetic circuits based on a partial integro-differential equation initially proposed in Friedman, Cai & Xie (2006) as an approximation of a chemical master equation. We use entropy methods to show exponentially fast convergence to equilibrium for this model with explicit bounds. The asymptotic equilibration for the multidimensional case of more than one gene is also obtained under suitable assumptions on the equilibrium stationary states. The asymptotic equilibration property for networks involving one and more than one gene is investigated via numerical simulations.

**Asymptotic behaviour of neuron population models structured by elapsed-time**.*Nonlinearity*32(2):464–495, 2019.We study two population models describing the dynamics of interacting neurons, initially proposed by Pakdaman, Perthame, and Salort (2010, 2014). In the first model, the structuring variable $s$ represents the time elapsed since its last discharge, while in the second one neurons exhibit a fatigue property and the structuring variable is a generic “state”. We prove existence of solutions and steady states in the space of finite, nonnegative measures. Furthermore, we show that solutions converge to the equilibrium exponentially in time in the case of weak nonlinearity (i.e., weak connectivity). The main innovation is the use of Doeblin’s theorem from probability in order to show the existence of a spectral gap property in the linear (no-connectivity) setting. Relaxation to the steady state for the nonlinear models is then proved by a constructive perturbation argument.

**Improved energy methods for nonlocal diffusion problems**.*Discrete and Continuous Dynamical Systems - A*38(3):1405–1425, 2018.We prove an energy inequality for nonlocal diffusion operators of the following type, and some of its generalisations: $Lu (x) := \int_{\mathbb{R}^N} K(x,y) (u(y) - u(x)) \, \mathrm{d}y$, where $L$ acts on a real function u defined on $\mathbb{R}^N$, and we assume that $K(x,y)$ is uniformly strictly positive in a neighbourhood of $x=y$. The inequality is a nonlocal analogue of the Nash inequality, and plays a similar role in the study of the asymptotic decay of solutions to the nonlocal diffusion equation $\partial_t u=Lu$ as the Nash inequality does for the heat equation. The inequality allows us to give a precise decay rate of the $L^p$ norms of $u$ and its derivatives. As compared to existing decay results in the literature, our proof is perhaps simpler and gives new results in some cases (particularly, and surprisingly, in dimensions $N=1,2$).

## Funding

Since 2015 I am co-director of a research project funded by the Spanish Ministerio de Economía y Competitividad (MTM2014-52056-P). Until 2015 I was the principal researcher of a Marie-Curie CIG project based at the University of Birmingham. Here is a short page with a summary of the aims and results of this project.

## Editorial work

I am currently an editor of Communications in Pure and Applied Analysis.