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Mathematical Topics in Kinetic Theory
May 9 - 13, 2016 Cambridge, United Kingdom |
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Abstracts of the talks
Title: From Newton's law to the linear Boltzmann equation without cut-off
Abstract: We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling. The main difficulty in this context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford's strategy. I will present how a combination of Lanford's strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process and of new duality arguments to study the additional terms associated to the infinite range interaction, leading to some explicit weak estimates, overcomes this difficulty.
Title: From Boltzmann to incompressible Navier-Stokes on the torus in L^\infty settings
Abstract: The hydrodynamical limit from the Boltzmann equation towards the incompressible Navier-Stokes equation has been rigorously studied for years, the latest results giving the strong convergence in highly regular Sobolev spaces, namely 3+0 derivatives and an Maxwellian weight. Motivated by recent results on the Boltzmann equation in mere Lebesgue spaces with polynomial weight we would like to first get rid of the strong Maxwellian weight and second the need for derivatives in the study of the hydrodynamical limit.
In this talk I will present the non-trivial problems, as well as propose some new ideas to overcome them, arising when one wants results that are uniform in the Knudsen number, especially when one does not include x-derivatives. This is a joint work with S. Merino-Aceituno and C. Mouhot.
Title: Non-equilibrium stationary states for some kinetic systems
Abstract: We study non-equilibrium states for some simple spatially non-homogeneous kinetic equations with thermostats at two different temperatures. The thermostats act globally, and not only at the boundaries, This simplification still leaves a challenging and interesting set of problems, only some of which have been solved. For example, it is natural to ask whether all of the non-equilibrium steady states are spatially homogeneous. This is positively answered under some conditions on the strength of the thermostatic interaction. We also prove result on the rate of relaxation to these states. This is joint work with R. Esposito, J. Lebowitz, R. Marra, and Clement Mouhot.
Title: A Quantum Kac Walk and its Kinetic Limit
Abstract: We present recent results on models for quantum systems of $N$ particles undergoing random binary collisions, focusing on the rate of
convergence to equilibrium and the propagation of chaos. These questions arise from the work of Mark Kac and his investigation into the probabilistic structure underlying the Boltzmann equation. Recently, the quantum mechanical variation on Kac's question has begun to be investigated. In this case, the Kac Master equation becomes an evolution equation of Lindblad type, while the corresponding Boltzmann equation is a novel sort of non-linear evolution
equation for a density matrix. The treatment departs from the classical treatment because in quantum mechanics, conditional probability is not always well
defined. Nonetheless, a substantial quantum analog of the Kac program can be carried out, and it leads to an interesting and novel class of quantum kinetic equations. This is joint work with Eric Carlen and Michael Loss.
Title: Concentration waves of bacteria at the mesoscopic scale
Abstract: Concentration waves of swimming bacteria Escherichia coli were described in his seminal paper by Adler (Science 1966). These experiments gave rise to intensive PDE modelling and analysis, after the original model by Keller and Segel (J. Theor. Biol. 1971), and the work of Alt and co-authors in the 80`s. Together with Bournaveas, Perthame, Raoul and Schmeiser, we have revisited this old problem from the point of view of kinetic transport equations. This framework is very much adapted to the so-called run-and-tumble motion, in which any bacteria modulate the frequency of reorientation (tumble) -- and thus the duration of free runs -- depending on chemical variations in its environment.
In this talk, I will present existence results for solitary waves both at the macroscopic scale, and at the mesoscopic scale. The macroscopic problem consists of a drift-diffusion equation derived from the kinetic equation after a suitable diffusive rescaling, coupled to two reaction-diffusion equations. Mathematical difficulties arise at the mesoscopic scale, where the proof of existence of travelling waves require a refined description of spatial and velocity profiles.
I will also present numerical simulations done in collaboration with Gosse and Twarogowska, in order to illustrate some unexpected behavior of the mesoscopic problem.
Title: From particle methods to hybrid semi-lagrangian schemes
Abstract: In this talk we present two classes of particle methods with remappings, which aim at improving the accuracy of the
numerical approximations for transport problems with a minimal amount of smoothing.
Both methods use the particles pushed forward to compute local linearizations of the characteristic flow. A first
approach consists of transporting smooth particle shapes exactly along the corresponding affine flows, and then
describing the transported density as a sum of lineraly-transformed particles (LTP).
This method has good convergence properties that can be demonstrated both at the theoretical and the numerical level.
However for long remapping periods it is affected by the fact that extended particle shapes deteriorate the locality.
To avoid this weakness we have designed a second method which follows a backward semi-lagrangian approach based on the
local linearizations of the flow. The resulting scheme is more local by construction and also has enhanced convergence
properties, also validated by numerical experiments.
This is a joint work with Frédérique Charles, Antoine Le Hyaric and the Selalib group.
Title: An analog of Cercignani's conjecture for the Becker-Döring equation
Abstract: We present a recent work on the application of the entropy method to the convergence to equilibrium for subcritical solutions of the Becker-Döring equation, a well-known model for aggregation and crystallisation processes of materials. The results bear a strong resemblance to the theory of Cercignani's conjecture for the Boltzmann equation. We show that the inequalities involved can be studied by means of so-called "modified logarithmic Sobolev" inequalities in the setting of discrete Markov processes.
Title: Mathematical modeling of collective behavior
Abstract: Emergent aggregation and flocking phenomena appearing in many biological systems are simple instances of collective behavior. Recently, they have been an active research in applied mathematics, biology, engineering, and physics. In this talk, we present several di fferent types of mathematical models describing collective behaviors from microscopic to macroscopic descriptions. For those models, we discuss well-posedness, blow-up, and large-time behavior of solutions.
Title: The Boltzmann equation for a multi-species mixture close to global equilibrium
Abstract: (joint work with Marc Briant) In this talk I will present our joint work with Marc Briant on the Cauchy theory for a multi-species dilute gaseous mixture with different masses modeled by the multi-species Boltzmann equation close to equilibrium on the torus. The physically most relevant space for such a Cauchy theory is the space of density functions that only have finite mass and energy. Thus, the ultimate aim of this work is to obtain existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in $L^1_vL^\infty_x(m)$, where $m \sim (1+|v|^k)$ is a polynomial weight and $k>k_0$, recovering the optimal physical threshold of finite energy $k_0=2$ in the particular case of a multi-species hard spheres mixture with same masses. Our strategy is to combine and adapt several very recent methods, combined with new hypocoercivity estimates, in order to develop a new constructive approach that allows to deal with polynomial weights without requiring any spatial Sobolev regularity. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (\textit{e.g.} Carleman representation, Povzner inequality).
Title: Models of emergent networks
Abstract: In this talk, we will present a modelling framework for the emergence of networks and their evolution. We will provide various examples of such emergent networks: ant trails, extracellular fibers and blood capillaries. We believe this framework can apply to other types of networks in which the topology and topography of nodes and links is fuzzy and evolutive.
Title: Entropy methods for degenerate diffusions and weighted functional inequalities
Abstract: Entropy methods are well known to give sharp rates of convergence towards large time, self-similar solutions of porous medium and fast diffusion equations. These estimates have recently been reformulated using R\'enyi entropy powers and convexity properties inspired by the theory of information. In a joint work with M.J.~Esteban and M.~Loss, entropy methods and techniques related with the \emph{carr\'e du champ}, or \emph{Bakry-Emery method} have been used to produce \emph{rigidity} results in weighted elliptic PDEs related with Caffarelli-Kohn-Nirenberg inequalities and solve a longstanding conjecture on \emph{symmetry breaking}. In a recent study with M.~Bonforte, M.~Muratori and B.~Nazaret, the time-dependent version of the equations has been considered. Beyond the difficult issues related with the non-compactness of the domain and the regularity of the solutions, the degeneracy of the diffusion equation can be overcome by an artificial change of the dimension, which also results in an anisotropy of the diffusion and brings the problem back into the formalism of the R\'enyi entropy powers. The lecture will emphasize the issues of \emph{symmetry breaking}, large time asymptotics, optimality in weighted functional inequalities and optimal constants in entropy -- entropy production inequalities. Even the simple case of an evolution equation involving power law weights reveal the complex interaction of these notions.
Title: Some results on growth fragmentation equations
Abstract: I will present some recent results on growth fragmentation equations in a range of parameters where zero is the only steady state.
Abstract: I will speak about the numerical resolution of the Vlasov-Poisson system with a strong external magnetic field by Particle-In-Cell (PIC) methods. In this regime, classical PIC methods are subject to stability constraints on the time and space steps related to the small Larmor radius and plasma frequency. Here, we propose an asymptotic-preserving PIC scheme which is not subjected to these limitations. Our approach is based on first and higher order semiimplicit numerical schemes already validated on dissipative systems. Additionally, when the magnitude of the external magnetic field becomes large, this method provides a consistent PIC discretization of the guiding-center equation, that is, incompressible Euler equation in vorticity form. We propose several numerical experiments which provide a solid validation of the method and its underlying concepts.
Title: From N-Body Schrödinger to Vlasov
Abstract: It is well-known that the semiclassical limit of the N-body Schrödinger equation is the N-body Liouville equation of classical mechanics, and that it can be formulated as a small h limit of the Schrödinger equation, where h is the Planck constant. In the case of smooth interaction potentials (specifically for
Lipschitz continuous forces) it is known that the Vlasov equation, a nonlinear equation written in the single particle phase space, can be derived from the Liouville equation in the large N limit. This talk introduces new tools leading to an explicit convergence rate of the joint mean-field (large N) and semiclassical (small h) limit of the N-body Schrödinger equation. It can be regarded as a quantum analogue of Dobrushin's proof [Functional Anal. 13 (1979), 115--123] of the mean field limit in classical mechanics based on the Monge-Kantorovich distance - also known as either the Kantorovich-Rubinshtein distance or the Vasershtein distance of exponent 1
Title: The angular M1 model in a moving velocity frame for gas dynamics applications
Abstract: (joint work with Denise Aregba, Stephane Brull, Bruno Dubroca) The angular M1 model is an attractive approach to describe the transport of particles in a collisional plasma. Indeed, it is less numerically expensive than kinetic descriptions and more accurate than classical models. This model is based on a entropy minimisation closure and the angular moments equations are derived by integration on the unit sphere in the velocity space of a kinetic equation. While this procedure enables to obtain finite moments the derived model is not Galilean invariant. To overcome this major drawback and in order to simplify the collision operators, we propose to work in a moving velocity framework. Before considering complex congurations dealing with charged particles interactions, in this work, we only consider non-charged particles and present the angular M1 model expressed in the mean velocity frame for gas dynamics applications. First of all, the derivation of the model is introduced with its main properties. It is shown that the system studied is Friedrich's symmetric and that the correct conservation laws are recovered. Also, it is pointed out that in the chosen frame the mean velocity is zero. Secondly, a suitable numeri- cal scheme based on an underlying kinetic description is proposed for this model. Thirdly, first numerical results are presented. Finally, some perspectives are given.
Title: Quasineutral limit for Vlasov-Poisson systems
Abstract: We shall review some recent results on the quasineutral limit (that is the small Debye length regime) for Vlasov-Poisson systems. In particular we will present a recent work with F. Rousset that justifies the limit for initial data satisfying a pointwise Penrose stability condition.
Title: Quasineutral limit for Vlasov-Poisson via Wasserstein stability estimates
Abstract: The Debye length is the typical length of electrostatic interaction and in most physical situations is very small compared to the size of the domain. The so-called quasineutral limit consists in understanding the behavior of solutions of Vlasov-Poisson systems when the Debye length goes to zero. In this talk we present some recent results obtained in collaboration with Daniel Han-Kwan on the quasineutral limit of the Vlasov-Poisson equation for ions with massless thermalized electrons in dimension 1. Also, we will discuss the rigorous justification of the formal limit for very small but rough perturbations of analytic initial data for the Vlasov-Poisson equation in dimension 2 and 3.
Title: Harnack inequality for hypoelliptic kinetic equations with rough kernels.
Abstract: In this talk, I will present results concerned with the Hölder regularity of weak solutions of hypoelliptic equations whose coefficients are merely measurable and bounded. By combining methods developped by de Giorgi and Moser in elliptic regularity on the one hand and transfer of regularity results from the kinetic theory on the other hand, we can even prove a Harnack inequality for non-negative solutions, which is stronger than controlling the Hölder semi-norm. This talk is based on a joint work with Golse, Mouhot and Vasseur.
Title: A new flocking model through body attitude coordination
Abstract: We present a new model for multi-agent dynamics where each agent is described by its position and body attitude: agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations. Agents try to coordinate their body attitudes with the ones of their neighbours. This model is inspired by the Vicsek model. The goal of this talk will be to present this new flocking model, its relevance and the derivation of the macroscopic equations from the particle dynamics. In collaboration with Pierre Degond (Imperial College London) and Amic Frouvelle (Université Paris Dauphine).
Title: On a linear run and tumble equation
Abstract: I will present a recent result of existence of a unique positive and normalized steady state as well as its asymptotic stability for a linear runs and tumbles equation in dimension d ≥ 1. This result is picked up from a work established in collaboration with Q. Weng and improves similar results obtained by Calvez, Raoul and Schmeiser in dimension d = 1. Our analysis is based on the Krein-Rutman theory revisited by J. Scher and myself together with some new moment estimates for proving confinement mechanism as well as dispersion, multiplicator and averaging lemma arguments for proving some regularity property of suitable iterated averaging quantities.
Title: On the size of chaos in the Boltzmann-Grad limit for hard speres.
Abstract: I present a quantitative analysis of the low-density limit of a hard sphere system based on the study of a set of functions (correlation errors), measuring the deviations in time from the statistical independence of particles (propagation of chaos). In the context of the BBGKY hierarchy, a correlation error of order k measures the event where k tagged particles are connected by a chain of interactions preventing the factorization. Provided that the time is sufficiently small, such an error goes to zero with the hard spheres diameter \eps, for all k such that k< \eps^{-\alpha}, for some \alpha >0. This requires a new analysis of many recollision events, and improves previous estimates in which k can grow only a logaritmically.
Title: Hydrodynamic limit of granular gases to pressureless Euler in dimension 1
Abstract: In this joint work with P.-E. Jabin, we investigate the behavior of granular gases in the limit of small Knudsen number, that is very frequent collisions. We deal with the strongly inelastic case, in one dimension of space and velocity. We are able to prove the convergence toward the pressureless Euler system. The proof relies on dispersive relations at the kinetic level, which leads to the so-called Oleinik property at the limit.
Title: From the many-body quantum dynamics to the Vlasov equation.
Abstract: We review some results on the derivation of the Vlasov equation starting from a system of N interacting fermions, whose evolution is given by the N-particle Schrödinger equation. In order to get explicit bounds on the rate of convergence, we use the Hartree-Fock equation as a bridge between the Hamiltonian and the Vlasov dynamics. The results we present are valid both for pure and mixed states.
Title: Aggregation in kinetic transport models for chemotaxis
Abstract: Results on kinetic models for chemotaxis will be reviewed with an emphasis on questions like the description of aggregation phenomena, long time behavior, and macroscopic limits. In particular, new results on the existence of aggregated steady states and their dynamic stability will be presented.
Title: Regularity of the Boltzmann equation in bounded domains
Abstract: We consider the Boltzmann equation in a bounded domain with diffuse reflection at the boundary. The solution is known to present a singular behaviour on the grazing trajectories. In the case of a strictly convex domain, the singularities happen specifically on the (grazing) boundary: in this case, with the help of a "kinetic" distance towards the grazing boundary, we prove a result of Sobolev regularity for the solution. In the case of a non-convex domain, the singular trajectories cross the domain and the singularity propagates in the domain: in this case we obtain the BV regularity of the solution. We indicate the optimality of these results.
This is a joint work with Yan GUO, Chanwoo KIM and Daniela TONON.
Title: Convergence to equilibrium for linear and homogeneous Fokker-Planck equations
Abstract: In this work, we investigate the spectral analysis and long time asymptotic convergence of semigroups associated to discrete, fractional and classical Fokker-Planck equations in some regime where the corresponding operators are close. We successively deal with the discrete and classical Fokker-Planck models and the fractional and classical Fokker-Planck models. In each case, we present results of uniform convergence to equilibrium based on perturbation and/or enlargement arguments and obtained in collaboration with S. Mischler.
Title: Aggregation of bacteria by chemotaxis: mathematical and numerical analysis
Abstract: Chemotaxis is the phenomenon by which bacteria direct their motion in response to a chemical. At a mesoscopic scale, this motion is well described by a kinetic model modelling the run-and-tumble process. When taxis dominates the unbiaised movement, this model relaxes towards a macroscopic model describing aggregation of bacteria. This so-called aggregation equation may be considered as a transport equation whose macroscopic velocity has weak regularity, leading to finite time blow-up of weak solutions. In this talk, we will present a recent result of existence and uniqueness of weak measure solution for the aggregation equation. It allows to provide an upwind type numerical scheme able to simulate the dynamics of solution after blow up. This scheme is shown to be of order 1/2.
Title: Veiled singularities for the spherically symmetric Einstein-Vlasov system.
Abstract: It is well known that the solutions of Einstein equations combined with different matter models yield spacetimes. The best known examples of singularities in General Relativity are the black holes which are characterized by the onset of a horizon, i.e. the onset of a region from where not even light rays can connect some regions of the spacetime with regions far away from the center. Other singularities that have been studied for the Einstein equations combined with different matter models correspond to the situations usually termed as critical collapse. These solutions are associated to the transition between the situations of global existence of solutions and black holes. They are relevant because they often yield the so-called naked singularities, i.e. singular points of the spacetime which can be casually connected to regions of the spacetime arbitrarily far away from the center.
In this talk I will describe a family of solutions of the spherically symmetric Einstein-Vlasov system (or collisionless matter) exhibiting singularities different from black holes. They are obtained combining a self-similar solution with a matching procedure which allows to obtain a spacetime asymptotically similar to the Minkowski space far away from the center. The precise sense in which these solutions solve the Einstein-Vlasov equations as well as the causal structure of the spacetime will be described in detail. (Joint work with Alan D. Rendall).
Title: Variations of Kac model of a particle system
Abstract: The talk deals with variations on the work of Mark Kac concerning kinetic limits of N-particle systems. The N-particle system is in these cases represented by a a Markov jump processes in a product space E^N, where E represents the phase space of one particle. Kac's original work treats the case where the full dynamics is given by jumps involving two particles. He proved that in the limit of large N, the one-particle distribution satisfies a Boltzmann equation. The key concept in his proof is what is now called propagation of chaos.
The first example is a model for (one-dimensional) relativistic particles, and in the second example, we consider a pure jump process with jumps involving pairs of particles, that in the limit of infinitely many particles leads to the BGK-equation.
The work has been done together with Dawan Mustafa
Title: Nonlinear diffusion: From particles to thermodynamics
Abstract: In many applications, one has a 'small-scale' model which is simple but computationally intractable, and wishes to derive an effective (computable) 'large-scale' description. This talk will study this problem of scale-bridging from a thermodynamic perspective, focusing on gradient flows. We discuss the interplay between particle models and their thermodynamic description at hand of a class of nonlinear diffusion equations. It will first be shown how an underlying particle model can reveal an underlying (geo-)metric structure of the governing PDE, notably a gradient flow setting for a class of nonlinear diffusion equations. Connections to large deviation theory from probability will be made, and it will be discussed how this can link mesoscopic fluctuations and stochastic PDEs in a way that can allow to derive stochastic "corrections" for deterministic PDEs.
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