# 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 12, second 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

## Preprints & recent publications

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

### Preprints

**Improved energy methods for nonlocal diffusion problems**. 2016.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**. 2016.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.

**Close-to-equilibrium behaviour of quadratic reaction-diffusion systems with detailed balance**. 2016.We study the four-species reaction-diffusion system on a bounded domain, in space dimension $d ≤ 4$. We show that close-to-equilibrium solutions (in an $L^2$ sense) are regular, and that they relax to equilibrium exponentially in a strong sense. The results can be extended to general quadratic reaction-diffusion systems with detailed balance. The main novelty is the regularity result and exponential relaxation in $L^p$ norms for large $p$, which up to our knowledge is new in dimensions 3 and 4.

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

**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.

**Phase transitions in a kinetic flocking model of Cucker-Smale type**.*Multiscale Modelling and Simulation*14(3):1063–1088, 2016.We consider a collective behavior model in which individuals try to imitate each others’ velocity and have a preferred speed. We show that a phase change phenomenon takes place as diffusion decreases, bringing the system from a “disordered” to an “ordered” state. This effect is related to recently noticed phenomena for the diffusive Vicsek model. We also carry out numerical simulations of the system and give further details on the phase transition.

**Entropy dissipation estimates for the linear Boltzmann operator**.*Journal of Functional Analysis*269(4):1028–1069, 2015.We prove a linear inequality between the entropy and entropy dissipation functionals for the linear Boltzmann operator (with a Maxwellian equilibrium background). This provides a positive answer to the analogue of Cercignani’s conjecture for this linear collision operator. Our result covers the physically relevant case of hard-spheres interactions as well as Maxwellian kernels, and we always work with a cut-off assumption. For Maxwellian kernels, the proof of the inequality is surprisingly simple and relies on a general estimate of the entropy of the gain part operator due to Villani (1998) and Matthes and Toscani (2012). For more general kernels, the proof relies on a comparison principle. Finally, we also show that in the grazing collision limit our results allow to recover known logarithmic Sobolev inequalities.