José A. CañizoResearch · Publications · Teaching · Other

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.

My office is number 12, second floor, Escuela Técnica Superior de Ingeniería Informática y Telecomunicación. The mailing address is:

Avenida de Fuentenueva S/N
Spain

Email: | Phone: (+34) 958 24 08 27

Preprints & recent publications

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

Preprints

1. José A. Cañizo and Alexis Molino. 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 $N=1,2$).

2. José A. Cañizo and Francesco Patacchini. 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.

3. María J. Cáceres and José A. Cañizo. 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)

1. José A. Cañizo and Bertrand Lods. 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.

2. Alethea B. T. Barbaro, José A. Cañizo, José A. Carrillo and Pierre Degond. 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.

3. Marzia Bisi, José A. Cañizo and Bertrand Lods. 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.