I am a lecturer at the School of Mathematics of the University of Birmingham. I work mainly on existence theory and asymptotic behavior of kinetic equations, especially for coagulation and fragmentation models.
My office is 215 in the Watson building of the Edgbaston Campus.
The address is:
School of Mathematics
Watson Building
University of Birmingham
Edgbaston
Birmingham B15 2TT
UK
Tel. +44 (0) 121 414 6586
Email: j.a.canizo@bham.ac.uk
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.
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.
We present a refined duality estimate for parabolic equations. This estimate entails new results for systems of reaction-diffusion equations, including smoothness and exponential convergence towards equilibrium for equations with quadratic right-hand sides in two dimensions. For general systems in any space dimension, we obtain smooth solutions of reaction-diffusion systems coming out of reversible chemistry under an assumption that the diffusion coefficients are sufficiently close one to another.
We prove that any subcritical solution to the Becker-Döring equations converges exponentially fast to the unique steady state with same mass. Our convergence result is quantitative and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, for which several bounds are provided. This improves the known convergence result by Jabin & Niethammer (see ref [14]). Our approach is based on a careful spectral analysis of the linearized Becker-Döring equation (which is new to our knowledge) in both a Hilbert setting and in certain weighted $\ell^1$ spaces. This spectral analysis is then combined with uniform exponential moment bounds of solutions in order to obtain a convergence result for the nonlinear equation.
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.
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.
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 L^{1} 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.
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 L^{1} norm towards the fundamental solution of the heat equation.
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 ∈ (α_{0},1), with some constructive α_{0} ∈ [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 F_{α} are also derived; in particular, we prove that F_{α} is bounded from above and from below by two explicit universal (i.e. independent of α) Maxwellian distributions.
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.
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 L^{2} spaces of the operator defining the time evolution. In the present work we prove that there is also a spectral gap in weighted L^{1} spaces, thus extending exponential convergence to a larger set of initial conditions. The main tool is an extension result in ref. [4].
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.
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.
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.
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.
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.
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 L^{2} 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.
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 L^{2} 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.
We consider Smoluchowski's equation with a homogeneous kernel of the form a(x,y) = x^{α} y^{β} + y^{α} x^{β}, 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.
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.
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 ρ_{s} such that solutions with an initial density ρ_{0} ≤ ρ_{s} converge strongly to the equilibrium with density ρ_{0}, and solutions with initial density ρ_{0} > ρ_{s} converge (in a weak sense) to the equilibrium with density ρ_{s}. 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.
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.
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.
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