José A. Cañizo

A WELL-POSEDNESS THEORY IN MEASURES FOR SOME KINETIC MODELS OF COLLECTIVE MOTION

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 hy...

Stochastic Mean-Field Limit: Non-Lipschitz Forces & Swarming

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.

Mean-field limit for the stochastic Vicsek model

Abstract We consider the continuous version of the Vicsek model with noise, proposed as a model for collective behaviour 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 R d . As part of the study we give existence and uniqueness results for both the particle system and the PDE.

Improved Duality Estimates and Applications to Reaction-Diffusion Equations

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.

Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates

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

Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations

0

Measure Solutions for Some Models in Population Dynamics

and

A New Approach to the Creation and Propagation of Exponential Moments in the Boltzmann Equation

+\infty

Exponential convergence to equilibrium for subcritical solutions of the Becker–Döring equations

. Using these estimates we prove a spectral gap result by following the technique in [Caceres, Canizo, 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.

Rate of convergence to self-similarity for the fragmentation equation in L1 spaces

Abstract We study the asymptotic behavior of linear evolution equations of the type ∂ t g = D g + L g − λ g , where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator D g + L g . In the case D g = − ∂ x g , this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case D g = − ∂ x ( x g ) , it is known that λ = 1 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation ∂ t f = L f . By means of 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 other words, the linear operator has a spectral gap in the natural L 2 space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part.