Abstracts

Anton Arnold (Vienna University of Technology, Austria)

Title: Hypocoercivity and hypocontractivity concepts for linear dynamical systems

Abstract: Evolution equations with a system matrix or generator that is not coercive are called hypocoercive, if they still exhibit uniform exponential decay towards the steady state. Typical examples are kinetic Fokker-Planck equations, BGK equations, or even ODE systems. The "index of hypocoercivity" describes the interplay between between the dissipative and conservative part of their generator. As a first main result we shall show that this index characterizes the polynomial decay of the propagator norm for short time. Discrete time analogues of the above systems (obtained via the mid-point rule) are contractive, but typically not strictly contractive. For this setting we introduce "hypocontractivity" and an "index of hypocontractivity" and discuss their close connection to the continuous time evolution equations.

This talk is based on joint work with F. Achleitner, E. Carlen, and V. Mehrmann.

References:

* F. Achleitner, A. Arnold, V. Mehrmann: Hypocoercivity and hypocontractivity concepts for linear dynamical systems, submitted 2022.

* F. Achleitner, A. Arnold, E. Carlen: The hypocoercivity index for the short time behavior of ODEs, preprint 2022.

* A. Arnold, C. Schmeiser, B. Signorello. Propagator norm and sharp decay estimates for Fokker-Planck equations with linear drift, Comm. Math. Sc. 2022.

_________________________________________________________________________

Jose A. Carrillo (University of Oxford, U.K.)

Title: Nonlocal Aggregation-Diffusion Equations: entropies, gradient flows, phase transitions and applications

Abstract: This talk will be devoted to an overview of recent results understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics and financial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear diffusion Keller-Segel type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential. The analysis is based on linear stability in the right functional space associated to the regularity of the problem at hand. While in the case of linear diffusion, one can work in the L2 framework, nonlinear diffusion needs the stronger Linfty topology to proceed with the analysis based on Crandall-Rabinowitz bifurcation analysis applied to the variation of the entropy functional. Explicit examples show that the global bifurcation branches can be very complicated. Stability of the solutions will be discussed based on numerical simulations with fully explicit energy decaying finite volume schemes specifically tailored to the gradient flow structure of these problems. The theoretical analysis of the asymptotic stability of the different branches of solutions is a challenging open problem. 

This overview talk is based on several works in collaboration with R. Bailo, A. Barbaro, J. A. Canizo, X. Chen, P. Degond, R. Gvalani, J. Hu, G. Pavliotis, A. Schlichting, Q. Wang, Z. Wang, and L. Zhang. This research has been funded by EPSRC EP/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363. 

_________________________________________________________________________

Pierre Degond (Institut de Mathématiques de Toulouse CNRS & Université Paul Sabatier, France)

Title: Moment method for rod-like polymers

Abstract: There are two methods to derive hydrodynamic equations from kinetic equations when the Knudsen number tends to zero: the Hilbert method and the moment method. While the former only requires mild properties on the linearized collision operator, the latter requires specific conservation relations to hold. For classical models, the two methods are equivalent (at least formally), as the required properties of the linearized collision operator can be related to conservations. However, there are cases where conservation relations are lacking and only (so far) the Hilbert method is applicable. In this talk, I will focus on one of these examples, the kinetic model of rod-like polymers, or Doi model. I will show that suitable generalized conservation relations (aka generalized collision invariants) hold and make the moment method applicable in spite of the lack of conservation relations in the strict sense. It could lead to a better understanding of the structural properties of the Doi model and open the way to rigourous convergence proofs that would require less regularity than the Hilbert method. 

_________________________________________________________________________

Laurent Desvillettes (Université Paris Diderot, France)

Title: Linear stability of thick sprays

Abstract: Thick spray equations consist of a coupling between a Vlasov equation (for droplets) and an equation of fluid mechanics (for a surrounding gas), which involves both the drag force and the volume fraction occupied by each phase. In a work in collaboration with C. Buet and B. Després, we study the linear stability of homogeneous solutions, and compare it to related models where this stability is known to hold, or not to hold. 

_________________________________________________________________________

Emanuele Dolera (University of Pavia, Italy) 

Title: Asymptotics of weighted Poincaré-Wirtinger constants in Hilbert spaces

Abstract: The abstract is available here.

_________________________________________________________________________

Irene Gamba (University of Texas at Austin, U.S.A.)

Title: Weak turbulence for electron flows by quasilinear particle systems for unmagnetized and magnetized regimes

Abstract: We will discuss recent development in the numerical simulations and analytical estimates towards the understanding stability properties for weak perturbations dynamics from collisionless statistical equilibrium states. We show that under well prepared data, a stable non-equilibrium state may emerge in a  model given by a  mean field system of quasilinear diffusion model for electron particles in momenta space coupled to an spectral energy density wave perturbed by a small deviations from a dominant statistical equilibrium state.

_________________________________________________________________________

Amit Einav (Durham University, U.K.)

Title: The entropic journey of Kac’s model

Abstract: Kac’s model, introduced in 1956 by Marc Kac for the purpose of giving a probabilistic justification to Boltzmann’s equation, is one of the first mean field limit model of its kind. Moreover, in creating his model Kac has introduced the fundamental notion of chaoticity – a correlation assumption pertaining to asymptotic independence – a notion all mean field limit models to date use. The relative simplicity of Kac’s model, especially in contrast to the highly non-linear Boltzmann equation, has inspired Kac to try and investigate properties of his model that could somehow be “pushed down” to the Boltzmann equation via the mean field limit process. In particular, Kac was interested in the study of convergence to equilibrium.

In this talk, in loving memory of Maria Conceição Carvalho, we will review the investigation of the interplay of the convergence to equilibrium in Kac’s model and in Boltzmann’s equation, mainly through the lenses of the so-called Entropy Method. We will be mostly motivated by our own contribution to this study, culminating in our work with Eric Carlen and Maria Conceição Carvalho.

___________________________________________________________________________

Luís Simão Ferreira (CMAFcIO, University of Lisbon, Portugal)

Title: Spectral gaps and entropy production for Kac's walk by means of coupling

Abstract: Kac's model for particle collisions has been a central topic in kinetic theory ever since it was introduced in 1954, mainly due to its connection with classical PDEs in thermodynamics. In particular, this class of models describes the dynamics of a system of N particles undergoing energy and momentum preserving collisions as a stochastic jump process on a sphere, such that, by adequately re-scaling, the limiting one-particle marginal evolves under the spatially homogeneous nonlinear Boltzmann equation. Even though new technical difficulties arise from considering many bodies, this description allows us to make use of probabilistic tools, such as coupling methods, to study properties related to relaxation rates and approach to equilibrium. In this talk, we will explore how new results regarding lower bounds on spectral gaps and entropy production estimates for Maxwellian molecules may be obtained by combining these techniques with analytic methods.

___________________________________________________________________________

Francois Golse (Ecole Polytechnique, Paris, France)

Title: Observability for Quantum Dynamics with Optimal Transport

Abstract: We explain how the quantum analogue of the Wasserstein distance defined in [F. Golse, T. Paul, Archive Rational Mech. Anal. 223 (2017) 57-94] can be used to prove an observation inequality for quantum dynamics provided that classical particle trajectories with initial data in a compact phase-space subset enter the observation region in finite time (Bardos-Lebeau-Rauch geometric condition). Two advantages of this approach are (1) that it requires only the force field to be Lipschitz continuous, and (2) the observability constants are explicit in terms of the Bardos-Lebeau-Rauch condition. 

(Work in collaboration with T. Paul)

_________________________________________________________________________

Michael Loss (Georgia Institute of Technology, Atlanta, U.S.A.)

Title: Which magnetic fields support a zero mode?

Abstract: I present some results concerning the size of magnetic fields that support zero modes for the three dimensional Dirac equation and related problems for spinor equations. Critical quantities measuring this size are the 3/2 norm of the magnetic field B and the 3 norm of the vector potential A. 

The point is that the spinor structure enters the analysis in a crucial way. This is joint work with Rupert Frank at LMU Munich.

___________________________________________________________________________

Rossana Marra (University of Rome Tor Vergata, Italy)

Title: Stationary non-equilibrium states in kinetic theory

Abstract: I will review some results on the construction of stationary non equilibrium solutions for the Boltzmann equation, in a general domain in contact with a slightly non-homogeneous thermal reservoir, both for finite and small Knudsen number. I will describe different approaches and different techniques developed. The main focus will be on stationary solutions close to hydrodynamics.

___________________________________________________________________________

R. Vilela Mendes (CMAFcIO, University of Lisbon, Portugal)

Title: On the support of Lévy processes

Abstract: Characterization of the distributional support of paths of Lévy processes is an important issue for the construction of sparse statistical models, for the existence of generalized solutions to stochastic PDE's driven by Lévy white noise as well as for theories of integration in infinite dimensions. After a review of the conditions that insure support in S', a family of Lévy processes without support in S' is studied and its support characterized.

___________________________________________________________________________

Gabriel Nahum (IST, University of Lisbon, Portugal)

Title: A microscopic model for the transition from slow to fast diffusion

Abstract: In this talk we present a nearest neighbour interacting particle system of exclusion and gradient types, which illustrates the transition from constant to slow or fast diffusion. More precisely, the hydrodynamic limit of this microscopic system in the diffusive space-time scaling is a parabolic diffusion equation with diffusion coefficient following a power law, whose exponent can be adjusted to observe the transition from the Porous Media Equation to the Fast Diffusion Equation. The model's construction is based on the generalised binomial theorem, and as a consequence it interpolates continuously the Porous Media Model and the Symmetric Simple Exclusion Process, while going further into a Fast Diffusion Model up to some degree.

___________________________________________________________________________

Amy Novick-Cohen (Technion IIT, Israel)

Title: Surface diffusion, and surface diffusion coupled with mean curvature motion

Abstract: Surface diffusion as well as mean curvature motion constitute geometric motions relevant to modelling various phenomena, such as hillock and island

formation, as well as agglomeration which are seen in thin poly-crystalline film dynamics. We explain the physical context and then focus on certain composite

axi-symmetric geometries, where the steady states may be described by piecing together Delaunay surfaces, and the related evolutionary issues (existence, stability) are pertinent to solid state wetting and dewetting. The notion of an effect radius is introduced in this context.

Lecture in part reflects work with A. Zigelman,  D. Goldberg,  K. Golubkov

___________________________________________________________________________

Stefano Olla (Université Paris-Dauphine-PSL, France, and GSSI-L'Aquila, Italy)

Title: Heat flow in a system in contact with a thermal bath subjected to periodic forcing

Abstract: We investigate the properties of a harmonic chain in contact at its left end with a thermal bath and subjected at its right end to a periodic force. The particles are also subjected to a random velocity reversal action. The latter gives the system a finite heat conductivity. We prove the approach of the system to a time-periodic state and compute the heat current, equal to the time-averaged work done on the system, in that state. Rescaling space, time, and the strength of the force leads to a continuum heat equation with Neumann boundary condition on the right, fixing the amount of heat current flowing into the system, and Dirichlet condition on the left, fixing the temperature in accord to the one of the thermal bath.

Works in collaboration with Tomasz Komorowski (Polish Academy of Science) and Joel Lebowitz (Rutgers).

___________________________________________________________________________

Mario Pulvirenti (Università di Roma La Sapienza, Italy)

Title: On the BGK model and particle approximations

Abstract: I consider the well known BGK kinetic equation and suitable stochastic particle systems approaching its solutions under various scaling limits. I discuss results and conjectures. 

This research is in collaboration with P. Buttà, M. Hauray.

___________________________________________________________________________

Valeria Ricci (Università degli Studi di Palermo, Italy)

Title: Collective Modes Associated with Rarefied Populations of Heavy Nuclei

Abstract: We deal with plasmas in which low density populations of heavy nuclei (impurities) is present, with cyclotron frequencies different from that of the main population of ions. Modes at the cyclotron frequency of the impurities are found to be substained by the impurity density gradient: these are nearly electrostatic in the short transverse wavelenghts while they involve significant magnetic field fluctuations in the relatively long wavelength limit. Relevant transfer phase velocities are in the direction of the local impurity diamagnetic velocity and produce a transport of this population across the magnetic field. Growth rate are found that depend on the pre-existing thermal energy transport processes.

This work is a cooperation with B.Coppi and B. Basu in the frame of the Ignitor project.

[1] B.Coppi, S.Cowley et Al. Phys.Fluids 29, 4060 (1986)

[2] C.Mazzotta et al., Paper IAEA-CN-899 MFE Fusione Energy Conference (I.A.E.A., Vienna, 2021)

[3] B.Coppi, H.Furth, M.Rosenbluth and R.Sagdeev Phys.Rev.Lett.17.377 (1966)

___________________________________________________________________________

Francesco Salvarani (Università di Pavia, Italy and ALDV-DVCR Paris, France)

Title: Kinetic description of polyatomic gases undergoing resonant collisions

Abstract: This talk is devoted to the study of a kinetic model describing a polyatomic gas in which the microscopic internal and kinetic energies are separately conserved during a collision process (resonant collisions). This behaviour has been observed in some physical phenomena, for example in the collisions between selectively excited CO$_2$ molecules. After describing the model itself, we prove the related $H$-theorem and show that, at the equilibrium, two temperatures are expected. We moreover prove a compactness property of the corresponding linearized Boltzmann operator. The peculiar structure of resonant collision rules allows to tensorize the problem and separately treat the internal energy contributions. We also propose a geometric variant of Grad's proof of the compactness property in the monatomic case.  These results have been obtained in collaboration with T. Borsoni, L. Boudin and A. Rossi. 

___________________________________________________________________________

Gunter Schütz (CAMGSD, University of Lisbon, Portugal)

Title: Dynamical universality classes: Recent results and open questions

Abstract: Universality asserts that, especially near phase transitions, the macroscopic properties of a physical system do not depend on its details such as the precise form of microscopic interactions. We show that the two best-known examples of dynamical universality classes, the diffusive and Kardar-Parisi-Zhang-classes, are only part of an infinite discrete family. The members of this family have dynamical exponents which surprisingly can be expressed by the Kepler ratio of consecutive Fibonacci numbers. This strongly indicates the existence of a simpler but still unknown underlying mechanism that determines the different classes.

___________________________________________________________________________

Bernt Wennberg (University of Gothenburg, Sweden)

Title: The Lorentz gas in a nearly periodic distribution of scatterers

Abstract: Let $\chi_{\epsilon}\subset\mathbb{R}^2$ be a random point set such that then number of points in a set $A$, $\chi_{\epsilon}(A)$ satisfies $\epsilon \chi_{\epsilon}(A) \rightarrow 1$ when $\epsilon\rightarrow 0$, and consider the motion of a point particle moving in the region obtained by putting a circular obstacle of radius $\epsilon$ at each point of $\chi_{\epsilon}$, and being specularly reflected when hitting an obstacle. This is the Boltzmann Grad limit of the Lorentz gas in $\mathbb{R}^2$. It is known since the works of Gallavotti that when $\chi_{\epsilon}$ is a Poisson distribution with intensity $1/\epsilon$  a density of point particles evolving according to this process satisfies a linear Boltzmann equation in the limit $\epsilon\rightarrow 0$, but it is also known that if $\chi_{\epsilon}$ is a periodic set, such as $\sqrt{\epsilon} \mathbb{\Z}^2$, then the limiting density does not satisfy the Boltzmann equation. We construct a random pointset $\chi_{\epsilon}$ such that $\epsilon{-1/2}\chi_{\epsilon}$ converges to $\mathbb{Z}^2$ when $\epsilon$ converges to zero, but yet the limiting point particle distribution satisfies the same linear Boltzmann equation as when $\chi_{\epsilon}$ is a Poisson distribution.