# José A. Cañizo - Homepage

# José A. Cañizo

I am a researcher at the Department of Applied Mathematics of the University of Granada. I work mainly on existence theory and asymptotic behaviour of kinetic equations and related models, including coagulation and fragmentation processes and nonlocal partial differential equations.

## Address

My office is number 14, third floor, Escuela Técnica Superior de Ingeniería Informática y Telecomunicación. The mailing address is:

*Departamento de Matemática Aplicada
Facultad de Ciencias
Avenida de Fuentenueva S/N
18071 Granada
Spain*

Email: <canizo@ugr.es> | Phone: (+34) 958 24 08 27

## Upcoming activities

### Seminario de Ecuaciones Diferenciales

Próxima charla:

**Resumen.**Caused by the development of renewable energies in recent years, power networks experience a considerable change from few major generators to smart grids of small producers. A fundamental question for such systems is whether they achieve synchronisation to a common frequency on maybe even very large network structures. In this talk, we study this problem for a second-order Kuramoto-type rotator model. More precisely, we will present a theory of continuum limit for this model, both for deterministic and random networks. Based on these limits, the stability properties of synchronised states and their dependence on the topological properties of the underlying network are examined.

### CIMPA 2019

CIMPA Summer School to take place in La Habana, Cuba, in 2019.

## Editorial work

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

## Preprints & 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.

**Asymptotic behaviour of neuron population models structured by elapsed-time**. 2018.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.

### Some recent publications (see full list here)

**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*462(1):801–839, 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 $t$.

**Close-to-equilibrium behaviour of quadratic reaction-diffusion systems with detailed balance**.*Nonlinear Analysis*159:62–84, 2017.We study general quadratic reaction-diffusion systems with detailed balance, in space dimension $d \leq 4$. We show that close-to-equilibrium solutions (in an $L^2$ sense) are regular for all times, and that they relax to equilibrium exponentially in a strong sense. That is: all detailed balance equilibria are exponentially asymptotically stable in all $L^p$ norms, at least in dimension $d \leq 4$. The results are given in detail for the four-species reaction-diffusion system, where the involved constants can be estimated explicitly. The main novelty is the regularity result and exponential relaxation in Lp norms for p > 1, which up to our knowledge is new in dimensions 3 and 4.

**The Fokker-Planck equation for bosons in 2D: well-posedness and asymptotic behaviour**.*Nonlinear Analysis*137:291–305, 2016.We show that solutions of the 2D Fokker-Planck equation for bosons are defined globally in time and converge to equilibrium, and this convergence is shown to be exponential for radially symmetric solutions. The main observation is that a variant of the Hopf-Cole transformation relates the 2D equation in radial coordinates to the usual linear Fokker-Planck equation. Hence, radially symmetric solutions can be computed analytically, and our results for general (non radially symmetric) solutions follow from comparison and entropy arguments. In order to show convergence to equilibrium we also prove a version of the Csiszár-Kullback inequality for the Bose-Einstein-Fokker-Planck entropy functional.

**Exponential trend to equilibrium for the inelastic Boltzmann equation driven by a particle bath**.*Nonlinearity*5(29):1687–1715, 2016.We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres (with constant restitution coefficient $\alpha \in (0,1)$) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove that the solution to the associated initial-value problem converges exponentially fast towards the unique equilibrium solution. The proof combines a careful spectral analysis of the linearised semigroup as well as entropy estimates. The trend towards equilibrium holds in the weakly inelastic regime in which $\alpha$ is close to $1$, and the rate of convergence is explicit and depends solely on the spectral gap of the elastic linearised collision operator.