# Research

## Recent publications

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

### Preprints

**Exponential equilibration of genetic circuits using entropy methods**. 2018.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.

### Some recent papers

**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 $N=1,2$).

**Discrete minimisers are close to continuum minimisers for the interaction energy**.*Calculus of Variations & PDE*57(24), 2018.Under suitable technical conditions we show that minimisers of the discrete interaction energy for attractive-repulsive potentials converge to minimisers of the corresponding continuum energy as the number of particles goes to infinity. We prove that the discrete interaction energy $\Gamma$-converges in the narrow topology to the continuum interaction energy. As an important part of the proof we study support and regularity properties of discrete minimisers: we show that continuum minimisers belong to suitable Morrey spaces and we introduce the set of empirical Morrey measures as a natural discrete analogue containing all the discrete minimisers.

**On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials**.*Journal of Mathematical Analysis and Applications*, 2018.In this work we present several quantitative results of convergence to equilibrium for the linear Boltzmann operator with soft potentials under Grad’s angular cutoff assumption. This is done by an adaptation of the famous entropy method and its variants, resulting in explicit algebraic, or even stretched exponential, rates of convergence to equilibrium under appropriate assumptions. The novelty in our approach is that it involves functional inequalities relating the entropy to its production rate, which have independent applications to equations with mixed linear and non-linear terms. We also briefly discuss some properties of the equation in the non-cutoff case and conjecture what we believe to be the right rate of convergence in that case.

**Uniform moment propagation for the Becker-Döring equation**.*Proceedings of the Royal Society of Edinburgh, Section A: Mathematics*, 2018.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.

**Trend to Equilibrium for the Becker-Döring Equations: An Analogue of Cercignani’s Conjecture**.*Analysis & PDE*10(7):1663–1708, 2017.In this work we investigate the rate of convergence to equilibrium for subcritical solutions to the Becker-Döring equations with physically relevant coagulation and fragmentation coefficients and mild assumptions on the given initial data. Using a discrete version of the log-Sobolev inequality with weights we show that in the case where the coagulation coefficient grows linearly and the detailed balance coefficients are of typical form, one can obtain a linear functional inequality for the dissipation of the relative free energy. This results in showing Cercignani’s conjecture for the Becker-Döring equations and consequently in an exponential rate of convergence to equilibrium. We also show that for all other typical cases one can obtain an ’almost’ Cercignani’s conjecture that results in an algebraic rate of convergence to equilibrium. Additionally, we show that if one assumes an exponential moment condition one can recover Jabin and Niethammer’s rate of decay to equilibrium, i.e. an exponential to some fractional power of $t$.

## 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.