# José A. Cañizo - Homepage

# José A. Cañizo

I am associate professor at the Department of Applied Mathematics of the University of Granada. I work on mathematical models in biology and physics, mainly based on partial differential equations. This includes kinetic equations, coagulation and fragmentation models, and nonlocal PDE in several contexts. I am interested in analytic properties of these models, their asymptotic behaviour, and related mathematical techniques.

I am subdirector of the Institute of Mathematics of the University of Granada (IMAG).

Here’s a summary of things on this site:

**Research.**Some updates on my research, a full list of preprints and publications, some informal notes on various topics, and other things!**Teaching / docencia.**List of courses I have taught and some publicly accessible lecture notes and material (most of them in Spanish).**Photo collection.**Some photos from conferences and places.**Other things.**Some LaTeX templates for papers and posters, technical details on LaTeX things, and anything else which doesn’t fit elsewhere.**Contact information.**My email is canizo@ugr.es and my phone (+34) 958 24 08 27. See here for mailing address and other details.

## Upcoming activities

### CIMPA School on Partial Differential Equations and Calculus of Variations

To take place from August 29 to September 9, 2022 at Dangbo, Benin. More information at the CIMPA website or at the school’s website.

### El Enigma 2022

The amazing yearly Enigmatic competition taking place in November at Facultad de Ciencias, Universidad de Granada. Not yet announced, but upcoming! You can also see the last edition here, and here the one from 2019. The 2021 edition was cancelled due to Enigma being busy with some dangerous experiments with human embryos. But this year she told us she is planning something.

## Preprints & 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 publications (see full list here)

**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.

**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 $N=1,2$).

**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.