# Research

## Supervision

Current:

**Alejandro Gárriz Molina**. Postdoctoral researcher (March-July 2021).**Alejandro Ramos Lora**. PhD student since 2019. Co-supervised with María José Cáceres.**Antonio Camunga Tchikoko**. PhD student since 2020. Co-supervised with Jesús Montejo.

Past:

**Sebastian Throm**. Postdoctoral researcher funded by the German DFG. (2018-2020).**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

**Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups**. 2021.We provide simple and constructive proofs of Harris-type theorems on the existence and uniqueness of an equilibrium and the speed of equilibration of discrete-time and continuous-time stochastic semigroups. Our results apply both to cases where the relaxation speed is exponential (also called geometric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive estimates in the subgeometric case and discrete-time statements which seem both to be new. The method of proof also differs from previous works, based on semigroup and interpolation arguments, valid for both geometric and subgeometric cases with essentially the same ideas. In particular, we present very simple new proofs of the geometric case.

### Some recent papers

**The scaling hypothesis for Smoluchowski’s coagulation equation with bounded perturbations of the constant kernel**.*Journal of Differential Equations*270:285–342, 2021.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.

**Contractivity for Smoluchowski’s coagulation equation with solvable kernels**.*Bulletin of the London Mathematical Society*53(1):248–258, 2021.We show that the Smoluchowski coagulation equation with the solvable kernels $K(x,y)$ equal to $2$, $x+y$ or $xy$ is contractive in suitable Laplace norms. In particular, this proves exponential convergence to a self-similar profile in these norms. These results are parallel to similar properties of Maxwell models for Boltzmann-type equations, and extend already existing results on exponential convergence to self-similarity for Smoluchowski’s coagulation equation.

**Spectral gap for the growth-fragmentation equation via Harris’s Theorem**.*SIAM Journal on Mathematical Analysis*53(5):5185–5214, 2021.We study the long-time behaviour of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show the existence of a spectral gap under conditions that generalise those in the literature by using a method based on Harris’s theorem, a result coming from the study of equilibration of Markov processes. The difficulty posed by the non-conservativeness of the equation is overcome by performing an $h$-transform, after solving the dual Perron eigenvalue problem. The existence of the direct Perron eigenvector is then a consequence of our methods, which prove exponential contraction of the evolution equation. Moreover the rate of convergence is explicitly quantifiable in terms of the dual eigenfunction and the coefficients of the equation.

**Hypocoercivity of linear kinetic equations via Harris’s Theorem**.*Kinetic and Related Models*13(1):97–128, 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.

## Some informal mathematical notes

## 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.