# 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
18015 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

**The Fokker-Planck equation for bosons in 2D: well-posedness and asymptotic behaviour**. 2015.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**. 2015.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.

**Trend to Equilibrium for the Becker-Döring Equations: An Analogue of Cercignani’s Conjecture**. 2015.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 $t$.

**Phase transitions in a kinetic flocking model of Cucker-Smale type**. 2015.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.

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

**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.

**Existence of Compactly Supported Global Minimisers for the Interaction Energy**.*Archive for Rational Mechanics and Analysis*217(3):1197–1217, 2014.The existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses. The main assumption on the potential is that it is catastrophic or not H-stable, which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics. The proof is based on a uniform control on the local mass around each point of the support of a global minimiser, together with an estimate on the size of the ’holes’ that a minimiser may have. The class of potentials for which we prove existence of minimisers includes power-law potentials and, for some range of parameters, Morse potentials, widely used in applications. Finally, using Euler-Lagrange conditions on local minimisers we give a link to classical obstacle problems in the calculus of variations.