# José Alfredo Cañizo

I am currently a Research Fellow at the Department of Pure Mathematics and Mathematical Statistics of the University of Cambridge. I work mainly on existence theory and asymptotic behavior of kinetic equations, especially for coagulation and fragmentation models.

My office is E.207 at the Center for Mathematical Sciences.

DPMMS, Centre for Mathematical Sciences

Tel. +44 1223 3 37982
Email: j.a.canizo@dpmms.cam.ac.uk

## Publications

The following are research papers. I also have some notes on several math-related topics.

### Preprints

• With Daniel Balagué and Pierre Gabriel, Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates, Preprint, (2012)

We are concerned with the long-time behavior of the growth-fragmentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first-order behavior close to 0 and $+\infty$. Using these estimates we prove a spectral gap result by following the technique in [Cáceres, Cañizo, Mischler 2011, JMPA], which implies that solutions decay to the equilibrium exponentially fast. The growth and fragmentation coefficients we consider are quite general, essentially only assumed to behave asymptotically like power laws.

• With Ricardo Alonso, Irene Gamba and Clément Mouhot, A new approach to the creation and propagation of exponential moments in the Boltzmann equation, Preprint, (2012)

We study the creation and propagation of exponential moments of solutions to the spatially homogeneous $d$-dimensional Boltzmann equation. In particular, when the collision kernel is of the form $|v-v_*|^\beta b(\cos(\theta))$ for $\beta \in (0,2]$ with $\cos(\theta)= |v-v_*|^{-1}(v-v_*)\cdot \sigma$ and $\sigma \in \mathbb{S}^{d-1}$, and assuming the classical cut-off condition $b(\cos(\theta))$ integrable in $\mathbb{S}^{d-1}$, we prove that there exists $a > 0$ such that moments with weight $\exp(a \min{t,1} |v|^\beta)$ are finite for $t>0$, where $a$ only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.

### Published or accepted papers

• With José A. Carrillo and Sílvia Cuadrado, Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, (2012)

We give a direct proof of well-posedness of solutions to general selection-mutation and structured population models with measures as initial data. This is motivated by the fact that some stationary states of these models are measures and not L1 functions, so the measures are a more natural space to study their dynamics. Our techniques are based on distances between measures appearing in optimal transport and common arguments involving Picard iterations. These tools provide a simplification of previous approaches and are applicable or adaptable to a wide variety of models in population dynamics.

• With J. A. Carrillo and M. Schonbek, Decay rates for a class of diffusive-dominated interaction equations, Journal of Mathematical Analysis and Applications, Vol. 389, No. 1, pp. 541-557 (May 2012)

We analyse qualitative properties of the solutions to a mean-field equation for particles interacting through a pairwise potential while diffusing by Brownian motion. Interaction and diffusion compete with each other depending on the character of the potential. We provide sufficient conditions on the relation between the interaction potential and the initial data for diffusion to be the dominant term. We give decay rates of Sobolev norms showing that asymptotically for large times the behavior is then given by the heat equation. Moreover, we show an optimal rate of convergence in the L1 norm towards the fundamental solution of the heat equation.

• With M. Bisi and B. Lods, Uniqueness in the weakly inelastic regime of the equilibrium state of the inelastic Boltzmann equation driven by a particle bath, SIAM Journal on Mathematical Analysis, Vol. 43, No. 6, pp. 2640-2674 (2011)

We consider the spatially homogeneous Boltzmann equation for inelastic hard-spheres (with constant restitution coefficient α ∈ (0,1)) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove uniqueness of the stationary solution (with given mass) in the weakly inelastic regime; i.e., for any inelasticity parameter α ∈ (α0,1), with some constructive α0 ∈ [0, 1). Our analysis is based on a perturbative argument which uses the knowledge of the stationary solution in the elastic limit and quantitative estimates of the convergence of stationary solutions as the inelasticity parameter goes to 1. In order to achieve this we give an accurate spectral analysis of the associated linearized collision operator in the elastic limit. Several qualitative properties of this unique steady state Fα are also derived; in particular, we prove that Fα is bounded from above and from below by two explicit universal (i.e. independent of α) Maxwellian distributions.

• With F. Bolley and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Applied Mathematics Letters, Vol. 25, No. 3, pp. 339-343 (2011)

We consider the continuous version of the Vicsek model with noise, proposed as a model for collective behavior of individuals with a fixed speed. We rigorously derive the kinetic mean-field partial differential equation satisfied when the number N of particles tends to infinity, quantifying the convergence of the law of one particle to the solution of the PDE. For this we adapt a classical coupling argument to the present case in which both the particle system and the PDE are defined on a surface rather than on the whole space. As part of the study we give existence and uniqueness results for both the particle system and the PDE.

• With M. J. Cáceres and S. Mischler, Rate of convergence to self-similarity for the fragmentation equation in L1 spaces, Communications in Applied and Industrial Mathematics, Vol. 1, No. 2, pp. 299-308 (2011)

In a recent result by the authors (ref. [1]) it was proved that solutions of the self-similar fragmentation equation converge to equilibrium exponentially fast. This was done by showing a spectral gap in weighted L2 spaces of the operator defining the time evolution. In the present work we prove that there is also a spectral gap in weighted L1 spaces, thus extending exponential convergence to a larger set of initial conditions. The main tool is an extension result in ref. [4].

• With M. J. Cáceres and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, Journal de Mathémathiques Pures et Appliquées, Vol. 96, No. 4, pp. 334-362 (2011)

We study the asymptotic behavior of linear evolution equations of the type ∂t g = Dg + Lg - λ g, where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator Dg + Lg. In the case Dg = -∂x g, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg = -x ∂x g, it is known that λ = 2 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation ∂t f = Lf. By meansof entropy-entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In both cases mentioned above we show these conditions are met for a wide range of fragmentation coefficients, so the exponential convergence holds.

• With F. Bolley and J. A. Carrillo, Stochastic Mean-Field Limit: Non-Lipschitz Forces & Swarming, Mathematical Models and Methods in Applied Sciences, Vol. 21, No. 11, pp. 2179-2210 (2011)

We consider general stochastic systems of interacting particles with noise which are relevant as models for the collective behavior of animals, and rigorously prove that in the mean-field limit the system is close to the solution of a kinetic PDE. Our aim is to include models widely studied in the literature such as the Cucker-Smale model, adding noise to the behavior of individuals. The difficulty, as compared to the classical case of globally Lipschitz potentials, is that in several models the interaction potential between particles is only locally Lipschitz, the local Lipschitz constant growing to infinity with the size of the region considered. With this in mind, we present an extension of the classical theory for globally Lipschitz interactions, which works for only locally Lipschitz ones.

• With L. Desvillettes and K. Fellner, Absence of Gelation for Models of Coagulation-Fragmentation with Degenerate Diffusion, Nuovo Cimento C, Vol. 33, No. 1, pp. 79-86 (2010)

We show in this work that gelation does not occur for a class of discrete coagulation-fragmentation models with size-dependent diffusion. We do not assume here that the diffusion rates of clusters are bounded below. The proof uses a duality argument first devised for reaction-diffusion systems with a finite number of equations due to M. Pierre.

• With J. A. Carrillo and J. Rosado, Collective Behavior of Animals: Swarming and Complex Patterns, ARBOR, Vol. 186, No. 746, pp. 1035-1049 (November-December 2010)

In this short note we review some of the individual based models of the collective motion of agents, called swarming. These models based on ODEs exhibit a complex rich asymptotic behavior in terms of patterns, that we show numerically. Moreover, we comment on how these particle models are connected to partial differential equations to describe the evolution of densities of individuals in a continuum manner. The mathematical questions behind the stability issues of these PDE models are questions of actual interest in mathematical biology research.

• With J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, Vol. 21, No. 3, pp. 515-539 (March 2011)

We present existence, uniqueness and continuous dependence results for some kinetic equations motivated by models for the collective behavior of large groups of individuals. Models of this kind have been recently proposed to study the behavior of large groups of animals, such as flocks of birds, swarms, or schools of fish. Our aim is to give a well-posedness theory for general models which possibly include a variety of effects: an interaction through a potential, such as a short-range repulsion and long-range attraction; a velocity-averaging effect where individuals try to adapt their own velocity to that of other individuals in their surroundings; and self-propulsion effects, which take into account effects on one individual that are independent of the others. We develop our theory in a space of measures, using mass transportation distances. As consequences of our theory we show also the convergence of particle systems to their corresponding kinetic equations, and the local-in-time convergence to the hydrodynamic limit for one of the models.

• With L. Desvillettes and K. Fellner, Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, Vol. 27, No. 2, pp. 639-654 (April 2010)

We present a new a-priori estimate for discrete coagulation-fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a-priori estimate provides a global L2 bound on the mass density and was previously used, for instance, in the context of reaction-diffusion equations.

In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case.

• With S. Mischler and C. Mouhot, Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients, SIAM Journal on Mathematical Analysis, Vol. 41, No. 6, pp. 2283-2314 (2010)

We show that solutions to Smoluchowski's equation with a constant coagulation kernel and an initial datum with some regularity and exponentially decaying tail converge exponentially fast to a self-similar profile. This convergence holds in a weighted Sobolev norm which implies the L2 convergence of derivatives up to a certain order k depending on the regularity of the initial condition. We prove these results through the study of the linearized coagulation equation in self-similar variables, for which we show a spectral gap in a scale of weighted Sobolev spaces. We also take advantage of the fact that the Laplace or Fourier transforms of this equation can be explicitly solved in this case.

• With S. Mischler, Regularity, local behavior and partial uniqueness of self-similar profiles for Smoluchowski's coagulation equation, Revista Matemática Iberoamericana, Vol. 27, No. 3, pp. 803-839 (2011)

We consider Smoluchowski's equation with a homogeneous kernel of the form a(x,y) = xα yβ + yα xβ, with -1 < α ≤ β < 1, and -1 < α + β < 1. We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at y = 0 in the case α < 0. We also give some partial uniqueness results for self-similar profiles: in the case α = 0 we prove that two profiles with the same mass and moment of order α+β are necessarily equal, while in the case α < 0 we prove that two profiles with the same moments of order α and β, and which are asymptotic at y=0, are equal. Our methods include a new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.

• Convergence to equilibrium for the discrete coagulation-fragmentation equations with detailed balance, Journal of Statistical Physics, Vol. 129, No. 1, pp. 1-26 (13 October 2007)

Under the condition of detailed balance and some additional restrictions on the size of the coefficients, we identify the equilibrium distribution to which solutions of the discrete coagulation-fragmentation system of equations converge for large times, thus showing that there is a critical mass which marks a change in the behavior of the solutions. This was previously known only for particular cases as the generalized Becker-Döring equations. Our proof is based on an inequality between the entropy and the entropy production which also gives some information on the rate of convergence to equilibrium for solutions under the critical mass.

• Asymptotic behavior of solutions to the generalized Becker-Döring equations for general initial data, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 461, No. 2064, pp. 3731-3745 (2005)

We prove the following asymptotic behavior for solutions to the generalized Becker-Döring system for general initial data: under a detailed balance assumption and in situations where density is conserved in time, there is a critical density ρs such that solutions with an initial density ρ0 ≤ ρs converge strongly to the equilibrium with density ρ0, and solutions with initial density ρ0 > ρs converge (in a weak sense) to the equilibrium with density ρs. This extends the previous knowledge that this behavior happens under more restrictive conditions on the initial data. The main tool is a new estimate on the tail of solutions with density below the critical density.

• With J. L. López and J. Nieto, Global L1 theory and regularity for the 3D nonlinear Wigner-Poisson-Fokker-Planck system, Journal of Differential Equations, Vol. 198, No. 2, pp. 356-373 (2004)

A global existence, uniqueness and regularity theorem is proved for the simplest Markovian Wigner–Poisson–Fokker–Planck model incorporating friction and dissipation mechanisms. The proof relies on Green function and energy estimates under mild formulation, making essential use of the Husimi function and the elliptic regularization of the Fokker–Planck operator.

• With J. C. Neu and L. L. Bonilla, Three Eras of Micellization, Phys. Rev. E, Vol. 66, No. 061406, (2002)

Micellization is the precipitation of lipids from aqueous solution into aggregates with a broad distribution of aggregation number. Three eras of micellization are characterized in a simple kinetic model of Becker-Döring type. The model asigns the same constant energy to the (k-1) monomer-monomer bonds in a linear chain of k particles. The number of monomers decreases sharply and many clusters of small size are produced during the first era. During the second era, nucleii are increasing steadily in size until their distribution becomes a self-similar solution of the diffusion equation. Lastly, when the average size of the nucleii becomes comparable to its equilibrium value, a simple mean-field Fokker-Planck equation describes the final era until the equilibrium distribution is reached.

## Ph.D. Thesis

Some problems related to the study of interaction kernels: coagulation, fragmentation and diffusion in kinetic and quantum equations. Advisor: Juan Soler. Universidad de Granada (June 2006) [PDF (1.4 Mb)], [frontpage image (2.3 Mb)], [at the University of Granada's library]

## Research groups

Some research groups of which I am or have been part of:

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