José Alfredo Cañizo
(by M. C. Escher)
I am currently a Juan de la Cierva researcher at the Department of Mathematics of the Universitat Autònoma de Barcelona. I work mainly on existence theory and asymptotic behavior of kinetic equations, especially for coagulation and fragmentation models.
Address
My office is -164 at the C1 tower of the Faculty of Science at the campus of Bellaterra. Notice that's minus 164 (floor -1; luckily not floor -164).
The address is:
Departament de Matemàtiques
Facultat de Ciències
Universitat Autònoma de Barcelona
08193 Bellaterra (Barcelona)
Spain
Tel. +34 93 581 31 04
Fax +34 93 581 27 90
Email: canizo@mat.uab.cat
Publications
The following are research papers. I also have some notes on several math-related topics.-
With L. Desvillettes and K. Fellner, Absence of Gelation for Models of Coagulation-Fragmentation with Degenerate Diffusion, (Preprint, Sep. 2009)• PDF
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.
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With J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, (Preprint, July 2009)
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.
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With L. Desvillettes and K. Fellner, Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion, Accepted in Annales de l'Institut Henri Poincare, (Sep. 2009)
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, Accepted in SIAM Journal on Mathematical Analysis, (Sep. 2009)
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.
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With S. Mischler, Regularity, asymptotic behavior and partial uniqueness for Smoluchowski's coagulation equation, (Preprint, Feb. 2008)
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.
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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.
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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.
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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.
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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:- Research group in partial differential equations and applications (GREDPA).
- ANR Project SPINADA: "Systèmes de Particules en Interactions Non Réversibles - Approches Déterministes et Aléatoires".
- Research Group on PDE's in Kinetic Theory, Quantum-Kinetics, Fluid Mechanics and Developmental Biology.
Upcoming conferences and stays
- SIAM Conference on Analysis of Partial Differential Equations, Miami, December 7-10, 2009.
- Emerging Topics in Dynamical Systems and Partial Differential Equations DSPDEs'10, Barcelona, May 31 - June 4, 2010.
- Porto Ercole Summer School on Mathematical Models and Methods in Kinetic Theory, Porto Ercole, Italy, Jun 13 - 19, 2010.
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