José A. CañizoResearch · Publications · Teaching · Other

# Research

## Recent publications

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

### Preprints

1. José A. Cañizo, Amit Einav and Bertrand Lods. On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials. 2017.

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.

2. José A. Cañizo, Bertrand Lods and Amit Einav. Uniform moment propagation for the Becker-Döring equation. 2017.

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.

### Some recent papers

1. José A. Cañizo, Amit Einav and Bertrand Lods. Trend to Equilibrium for the Becker-Döring Equations: An Analogue of Cercignani’s Conjecture. Analysis & PDE, 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 $t$.

2. María J. Cáceres and José A. Cañizo. 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.

3. José A. Cañizo, José A. Carrillo, Philippe Laurençot and Jesús Rosado. 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.

4. 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.

## Funding

Since 2015 I am co-director of a research project funded by the Spanish Ministerio de Economía y Competitividad (MTM2014-52056-P). Until 2015 I was the principal researcher of a Marie-Curie CIG project based at the University of Birmingham. Here is a short page with a summary of the aims and results of this project.

## Editorial work

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