arXiv subjects

José A. Cañizo

Publications and source records attributed to José A. Cañizo.

At least 19 recordsLinked to original sources

arXiv2026

Self-similarity and diffusive limits for linear kinetic equations: a Wild sum approach

We prove that linear collisional kinetic equations in the whole space without confinement mechanism display a long-time self-similar behaviour. This drastically improves the recently known results (decay estimates) about the solutions in such a context, providing the first result regarding this self-similar behaviour. As a consequence, we also establish a uniform-in-time convergence of the suitably rescaled solutions to their diffusion limit, which is also new. The class of equations considered includes some BGK type equations, some kinetic nonlocal Fokker--Planck-type equations and some kinetic (possibly fractional) Fokker--Planck equations, for which we are able to write explicitly solutions through a Wild sum (or Dyson series) or we can manage some accurate computations on the Fourier side.

math.AP↗

arXiv2026

A uniform-in-time nonlocal approximation of the standard Fokker-Planck equation

We study a nonlocal approximation of the Fokker-Planck equation in which we can estimate the speed of convergence to equilibrium in a way which does not degenerate as we approach the local limit of the equation. This uniform estimate cannot be easily obtained with standard inequalities or entropy methods, but can be obtained through the use of Harris's theorem, finding interesting links to quantitative versions of the central limit theorem in probability. As a consequence one can prove that solutions of this nonlocal approximation converge to solutions of the usual Fokker-Planck equation uniformly in time-hence we show the approximation is asymptotic-preserving in this sense. The associated equilibrium has some interesting tail and regularity properties, which we also study.

math.AP↗

arXiv2025

Large-time estimates for the Dirichlet heat equation in exterior domains

We give large-time asymptotic estimates, both in uniform and $L^1$ norms, for solutions of the Dirichlet heat equation in the complement of a bounded open set of $\mathbb{R}^d$ satisfying certain technical assumptions. We always assume that the initial datum has suitable finite moments (often, finite first moment). All estimates include an explicit rate of approach to the asymptotic profiles at the different scales natural to the problem, in analogy with the Gaussian behaviour of the heat equation in the full space. As a consequence we obtain by an approximation procedure the asymptotic profile, with rates, for the Dirichlet heat kernel in these exterior domains. The estimates on the rates are new even when the domain is the complement of the unit ball in $\mathbb{R}^d$, except for previous results by Uchiyama in dimension 2, which we are able to improve in some scales. We obtain that the heat kernel behaves asymptotically as the heat kernel in the full space, with a factor that takes into account the shape of the domain through a harmonic profile, and a second factor which accounts for the loss of mass through the boundary. The main ideas we use come from entropy methods in PDE and probability, whose application seems to be new in the context of diffusion problems in exterior domains.

math.AP↗

arXiv2024

Sequence of pseudoequilibria describes the long-time behavior of the nonlinear noisy leaky integrate-and-fire model with large delay

There is a wide range of mathematical models that describe populations of large numbers of neurons. In this article, we focus on nonlinear noisy leaky integrate-and-fire (NNLIF) models that describe neuronal activity at the level of the membrane potential. We introduce a sequence of states, which we call pseudoequilibria, and give evidence of their defining role in the behavior of the NNLIF system when a significant synaptic delay is considered. The advantage is that these states are determined solely by the system's parameters and are derived from a sequence of firing rates that result from solving a recurrence equation. We propose a strategy to show convergence to an equilibrium for a weakly connected system with large transmission delay, based on following the sequence of pseudoequilibria. Unlike direct entropy dissipation methods, this technique allows us to see how a large delay favors convergence. We present a detailed numerical study to support our results. This study helps us understand, among other phenomena, the appearance of periodic solutions in strongly inhibitory networks.

math.AP↗

arXiv2024

Discrete minimizers of the interaction energy in collective behavior: a brief numerical and analytic review

We consider minimizers of the N-particle interaction potential energy and briefly review numerical methods used to calculate them. We consider simple pair potentials which are repulsive at short distances and attractive at long distances, focusing on examples which are sums of two powers. The range of powers we look at includes the well-known case of the Lennard-Jones potential, but we are also interested in less singular potentials which are relevant in collective behavior models. We report on results using the software GMIN developed by Wales and collaborators for problems in chemistry. For all cases, this algorithm gives good candidates for the minimizers for relatively low values of the particle number N. This is well-known for potentials similar to Lennard-Jones, but not for the range which is of interest in collective behavior. Standard minimization procedures have been used in the literature in this range, but they are likely to yield stationary states which are not minimizers. We illustrate numerically some properties of the minimizers in 2D, such as lattice structure, Wulff shapes, and the continuous large-N limit for locally integrable (that is, less singular) potentials.

math.NA↗

arXiv2024

On the asymptotic behavior of the NNLIF neuron model for general connectivity strength

We prove new results on the asymptotic behavior of the nonlinear integrate-and-fire neuron model. Among them, we give a criterion for the linearized stability or instability of equilibria, without restriction on the connectivity parameter, which provides a proof of stability or instability in some cases. In all cases, this criterion can be checked numerically, allowing us to give a full picture of the stable and unstable equilibria depending on the connectivity parameter and transmission delay. We also give further spectral results on the associated linear equation, and use them to give improved results on the nonlinear stability of equilibria for weak connectivity, and on the link between linearized and nonlinear stability.

math.AP↗

arXiv2022

One-dimensional inelastic Boltzmann equation: Regularity \& uniqueness of self-similar profiles for moderately hard potentials

We prove uniqueness of self-similar profiles for the one-dimensional inelastic Boltzmann equation with moderately hard potentials, that is with collision kernel of the form | $\bullet$ | $γ$ for $γ$ > 0 small enough (explicitly quantified). Our result provides the first uniqueness statement for self-similar profiles of inelastic Boltzmann models allowing for strong inelasticity besides the explicitly solvable case of Maxwell interactions (corresponding to $γ$ = 0). Our approach relies on a perturbation argument from the corresponding Maxwell model through a careful study of the associated linearised operator. In particular, a part of the paper is devoted to the trend to equilibrium for the Maxwell model in suitable weighted Sobolev spaces, an extension of results which are known to hold in weaker topologies. Our results can be seen as a first step towards a full proof, in the one-dimensional setting, of a conjecture in Ernst \& Brito (2002) regarding the determination of the long-time behaviour of solutions to inelastic Boltzmann equation.

math.AP↗

arXiv2021

Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups

We provide simple and constructive proofs of Harris-type theorems on the existence and uniqueness of an equilibrium and the speed of equilibration of discrete-time and continuous-time stochastic semigroups. Our results apply both to cases where the relaxation speed is exponential (also called geometric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive estimates in the subgeometric case and discrete-time statements which seem both to be new. The method of proof also differs from previous works, based on semigroup and interpolation arguments, valid for both geometric and subgeometric cases with essentially the same ideas. In particular, we present very simple new proofs of the geometric case.

math.PR↗

arXiv2021

Spectral gap for the growth-fragmentation equation via Harris's Theorem

We study the long-time behaviour of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show the existence of a spectral gap under conditions that generalise those in the literature by using a method based on Harris's theorem, a result coming from the study of equilibration of Markov processes. The difficulty posed by the non-conservativeness of the equation is overcome by performing an $h$-transform, after solving the dual Perron eigenvalue problem. The existence of the direct Perron eigenvector is then a consequence of our methods, which prove exponential contraction of the evolution equation. Moreover the rate of convergence is explicitly quantifiable in terms of the dual eigenfunction and the coefficients of the equation.

math.AP↗

arXiv2020

Contractivity for Smoluchowski's coagulation equation with solvable kernels

We show that the Smoluchowski coagulation equation with the solvable kernels $K(x,y)$ equal to $2$, $x+y$ or $xy$ is contractive in suitable Laplace norms. In particular, this proves exponential convergence to a self-similar profile in these norms. These results are parallel to similar properties of Maxwell models for Boltzmann-type equations, and extend already existing results on exponential convergence to self-similarity for Smoluchowski's coagulation equation.

math.AP↗

arXiv2019

The scaling hypothesis for Smoluchowski's coagulation equation with bounded perturbations of the constant kernel

We consider Smoluchowski's coagulation equation with a kernel of the form $K = 2 + εW$, where $W$ is a bounded kernel of homogeneity zero. For small $ε$, we prove that solutions approach a universal, unique self-similar profile for large times, at almost the same speed as the constant kernel case (the speed is exponential when self-similar variables are considered). All the constants we use can be explicitly estimated. Our method is a constructive perturbation analysis of the equation, based on spectral results on the linearisation of the constant kernel case. To our knowledge, this is the first time the scaling hypothesis can be fully proved for a family of kernels which are not explicitly solvable.

math.AP↗

arXiv2019

Hypocoercivity of linear kinetic equations via Harris's Theorem

We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $(x,v) \in \mathbb{T}^d \times \mathbb{R}^d$ or on the whole space $(x,v) \in \mathbb{R}^d \times \mathbb{R}^d$ with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively $L^1$ or weighted $L^1$ norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem.

math.AP↗

arXiv2018

Asymptotic behaviour of neuron population models structured by elapsed-time

We study two population models describing the dynamics of interacting neurons, initially proposed by Pakdaman, Perthame, and Salort (2010, 2014). In the first model, the structuring variable $s$ represents the time elapsed since its last discharge, while in the second one neurons exhibit a fatigue property and the structuring variable is a generic "state". We prove existence of solutions and steady states in the space of finite, nonnegative measures. Furthermore, we show that solutions converge to the equilibrium exponentially in time in the case of weak nonlinearity (i.e., weak connectivity). The main innovation is the use of Doeblin's theorem from probability in order to show the existence of a spectral gap property in the linear (no-connectivity) setting. Relaxation to the steady state for the nonlinear models is then proved by a constructive perturbation argument.

math.AP↗

arXiv2018

Exponential equilibration of genetic circuits using entropy methods

We analyse a continuum model for genetic circuits based on a partial integro-differential equation initially proposed in Friedman, Cai \& Xie (2006) as an approximation of a chemical master equation. We use entropy methods to show exponentially fast convergence to equilibrium for this model with explicit bounds. The asymptotic equilibration for the multidimensional case of more than one gene is also obtained under suitable assumptions on the equilibrium stationary states. The asymptotic equilibration property for networks involving one and more than one gene is investigated via numerical simulations.

math.AP↗

arXiv2017

Uniform moment propagation for the Becker-Döring equation

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.

math-ph↗

arXiv2017

Close-to-equilibrium behaviour of quadratic reaction-diffusion systems with detailed balance

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 $L^p$ norms for $p > 1$, which up to our knowledge is new in dimensions 3 and 4.

math.AP↗

arXiv2015

Measure solutions for some models in population dynamics

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.

math.AP↗

arXiv2015

Phase Transitions in a kinetic flocking model of Cucker-Smale type

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.

math.AP↗