Mini-courses:
Anna De Masi
Title: Particle systems and free boundary problems
Abstract: I will consider stochastic interacting particle systems described by diffusive equations in the (space-time) hydrodynamic scaling limit. In the first lecture I will concentrate on what happens when there are phase transitions. In particular I will present recent results on the stationary non equilibrium measures. In the hydrodynamic limit such states exhibit a sharp interface which separates the different phases. In a second part I will discuss the microscopic nature of boundary conditions.
Yann Brenier
Title: Convex minimization techniques for initial value problems and
generalized mean-field games
Abstract: We consider systems of particles mostly at the macroscopical level of fluid equations of hyperbolic type (such as the Euler equations of isentropic gases or the inviscid Burgers equation) or parabolic type (such as the porous medium equations). For all these models, we show how to formulate the initial value problem as a convex space-time minimization problem, taking advantage of their weak formulations. This looks strange since convex space-time minimization problems presumably lead to elliptic space-time equations (such as the Cauchy-Riemann equations) that should correspond to ill-posed initial value problems. We show how to face this apparent contradiction. In some cases (Burgers, porous medium equation) we can recover the exact solutions for arbitrarily long time intervals. In the case of Euler equations, the recovery is obtained only for short time intervals and is also related to the method of convex integration, through a suitable concept of sub-solutions. We will also show how the resulting minimization problems look very much like generalized mean-field games, involving fields of symmetric nonnegative matrices rather than density fields.
Talks:
Nathalie Ayi
Title: Analysis of an asymptotic preserving scheme for stochastic linear kinetic equations in the diffusion limit
Abstract: We present an asymptotic preserving scheme based on a micro-macro decomposition for stochastic linear transport equations in kinetic and diffusive regimes. We perform a mathematical analysis and prove that the scheme is uniformly stable with respect to the mean free path of the particles. Several numerical tests validating our scheme will be presented.
Giada Basile
Title: Large deviations of a simplified Kac model’
Abstract: I will discuss the large deviation asymptotics of a Kac random walk with bounded velocities, and I will show how the large deviation functional is related to an entropy dissipation inequality for the Boltzmann-Kac equation.
Joint work with L. Bertini, D. Benedetto and C. Orrieri.
Gioia Carinci
Title: Sticky Brownian motion as limit of the Inclusion process
Abstract: In this talk I will show how a system of sticky brownian particles emerges as the scaling limit of symmetric random walkers with inclusion interaction. The convergence holds in a suitable condensation regime where the attractive part of the interaction has a prominent relevance. By using the self-duality property we will see, moreover, how the information provided by the two-particles dynamics is sufficient to characterize the time-dependent variance of the density fluctuation field of the inclusion process in the condensation regime.
Joint works with M. Ayala, C. Giardina, F. Redig.
Eric Carlen
Title: Spectral Gaps for Reversible Markov Processes with Chaotic Invariant Measures
Abstract: We present a method for obtaining quantitative spectral gaps for families of Markov jump process with non-uniform rates whose sequence of invariant measures is chaotic in the sense of Mark Kac. The main example is the Kac process for physical 3-dimensional hard sphere collisions, but we present a general framework. This is based on joint work with M. Carvalho and M. Loss.
Maria Conceição Carvalho
Title: The Entropy Problem for the Kac Model
Abstract: The Entropy Problem for the Kac Model has been actively studied in recent years. We present some new results and discuss some of the remaining problems.
This is joint work with E. Carlen and A. Einav.
Laurent Desvillettes
Title: A rigorous proof of the H-theorem for collision kernels appearing in weak turbulence theory
Abstract: Collision kernels appearing in weak turbulence theory and in semiconductor models differ somewhat from those found in the usual Boltzmann operator for rarefied gases (or even from the relativistic Boltzmann kernel). The proof that equilibria are (modified) Maxwellian functions for those kernels necessitates a specific treatment which uses ideas coming out of the theory of Cercignani's conjecture for the Landau equation from plasma physics.
Clément Erignoux
Title: Understanding the phenomenology of active matter with hydrodynamic limits
Abstract: (J.W. With Mourtaza Kourbane-Houssene, Julien Tailleur and Thierry Bodineau) Motility Induced Phase Separation (MIPS) and alignment phase transitions are two of the phenomena characteristic of active matter which have been the most widely studied by the physics community. However, although some progress has been achieved for mean-field models on the mathematical front, there is still a lack of mathematical understanding of many active matter models, which are by essence driven out of equilibrium by energy consumption at a microscopic model.
In this talk, I will give an brief description of active matter and of the two phenomena cited above, and will show how the theory of hydrodynamic limits for particle lattice gases can, under the right scaling, allow to compute exact phase diagrams for stochastic models with purely microscopic interactions and prove the emergence of MIPS and alignment phase transition.
François Golse
Title: Mean-field and synchronization for the Lohe matrix lattice
Abstract: The Lohe matrix lattice is a continuous-time dynamical system describing the collective dynamics of group elements in the unitary group tangent bundle, and it has been introduced as a toy model of a nonabelian generalization of the Kuramoto phase lattice. In the absence of couplings, it reduces to the finite-dimensional decoupled free Schrödinger equations with constant Hamiltonians. We study a rigorous mean-field limit of the Lohe matrix lattice resulting in a Vlasov type equation for the probability density function on the corresponding phase space. Then we discuss the emergent synchronous dynamics of the kinetic Lohe equation in terms of the initial data and the coupling strength. (Work in collaboration with S.-Y. Ha)
Daniel Han Kwan
Title: Large time behavior of the Vlasov-Navier-Stokes system
Abstract: I will report on recent works with O. Glass, A. Moussa and I. Moyano about the large time behavior of the Vlasov-Navier-Stokes system in different geometric settings.
François Huveneers
Title: Two surprises with disordered chains of oscillators
Abstract: I will discuss two examples with disordered chains of oscillators, each showing some folk belief to be wrong. First, I will derive Euler’s equations from an unpinned harmonic chain with random masses. This is an integrable Hamiltonian system, and the result shows thus that the derivation of Euler’s equations does not rest primarily on ergodicity, but rather on the separation of scales. Second, I will consider a pinned disordered harmonic chain and show that the thermal conductivity vanishes exactly, even if a non-quadratic potential is added on some small enough proportion of the atoms. This is a purely one-dimensional phenomenon, due to the presence of bottlenecks in the chain. This system originated as a toy model to understand the Griffiths effects that are presumably responsible for anomalous transport just above the many-body localization transition in quantum chains.
From joint work with Cédric Bernardin and Stefano Olla.
Ning Jiang
Title: Incompressible Navier-Stokes-Maxwell limit from Vlasov-Boltzmann-Maxwell system
Abstract: Recently, D. Arsenio and L. Saint-Raymond justified the limits from Vlasov-Boltzmann-Maxwell system to incompressible Navier-Stokes-Maxwell equations in the context of renormalized solutions. In this work, we work in classical solutions and derive the uniform estimate uniformly in Knudsen number. This is a joint work with Y.L. Luo and T. F. Zhang.
Jani Lukkarinen
Title: Multi-state condensation in Berlin-Kac spherical models
Abstract: The Berlin-Kac spherical model is a family of probability measures for classical lattice fields similar to equilibrium states of free bosonic quantum field theory. In particular, both models are known to exhibit Bose-Einstein condensation in three dimensions. We consider here supercritical densities for general lattice energy functions, focusing on systems with a degenerate condensate. We allow for continuum energy functions with degenerate minima and for degenerate states occurring on certain finite lattices when the minimum of the continuum function cannot be reached. We prove that the original Berlin-Kac measure may be replaced by a measure where the condensate and normal fluid degrees of freedom become independent random variables, and the normal fluid part converges to the critical Gaussian free field. The proof can be found in [arXiv.org:1806.01806] and it is based on a construction of a suitable coupling between the two measures, proving that their Wasserstein distance is small enough for the error in any finite moments of the field to vanish as the lattice size is increased to infinity.
Stefano Olla
Title: Hydrodynamic fluctuations in Euler scaling and beyond, with boundary tension.
Abstract: We consider the equilibrium dynamics of a chain of anharmonic oscillators (FPU type) perturbed by a noise conserving volume, momentum and energy, and with a constant tension force applied on the boundaries. We prove that, after hyperbolic scaling of space and time, the fluctuation fields of the conserved quantities (volume stretch, momentum and energy) evolve deterministically following
linearized Euler equations with boundary conditions. This deterministic evolution is still valid beyondhyperbolic time scale, but well shorter that the time scale where superdiffusion of the heat mode should appear. The proof of the linearization (so called Boltzmann-Gibbs principle) relies on a uniform lower bound on the spectral gap of the generator of the noise.
Work in collaboration with Lu Xu.
Milana Pavic Colic
Title: The Cauchy problem for spatially homogeneous Boltzmann system of Monatomic Gas Mixtures
Abstract: We solve the Cauchy problem for the full non-linear homogeneous Boltzmann system of equations describing multi-component monatomic gas mixtures for binary interactions in three dimensions. More precisely, we show existence and uniqueness of the vector value solution by means of an existence theorem for ODE systems in Banach spaces under the transition probability rates assumption corresponding to hard potentials rates $\gamma \in (0,1]$, with a bounded angular section modeled by a bounded function of the scattering angle. The initial data for the vector-valued solutions needs to be a vector of non-negative measures with finite total number density, momentum, and strictly positive energy, as well as to have finite $2+2\gamma$ scalar polynomial moment. The existence and uniqueness proof relies on a new angular averaging lemma adjusted to vector values solution that yields a Povzner estimate with explicit constants that decay with the order of the corresponding scalar polynomial moment. In addition, such initial data yields a global generation of such scalar polynomial moments at any order as well as their summability of moments to obtain estimates for corresponding scalar exponentially decaying high energy tails, referred as to scalar exponential moments associated to the system solution. Such scalar polynomial and exponential moments propagate as well.
Nicolas Perkowski cancelled
Title: Controlled distributions in infinite dimensions: The Burgers generator
Abstract: I will present a probabilistic theory for a class of Burgers type singular SPDEs. We construct a domain of controlled (and non-smooth) test functions for the infinitesimal generator and use it to show existence for the Kolmogorov backward equation via energy estimates and compactness. We also formulate a martingale problem and obtain the existence of solutions by approximation. Uniqueness for backward equation and martingale problem then follows by duality. This generalizes the uniqueness result for the Goncalves-Jara-Gubinelli energy solutions to a wider class of equations, including fractional and multi-component Burgers equations. Joint work with Massimiliano Gubineli.
Ellen Saada
Title: Zero-range process in random environment
Abstract: In this talk I will consider a zero-range process with site disorder. This one-dimensional, nearest-neighbor, attractive dynamics with a bounded jump rate, exhibits a phase transition: there are no invariant measures above some critical density. In collaboration with C. Bahadoran, T. Mountford and K. Ravishankar, we have first obtained necessary and sufficient conditions for weak convergence to the critical invariant measure. We have then derived the hydrodynamical behavior of the system, and finally, we have proven local equilibrium results, and a dynamical loss of mass.
Marco Sammartino
Title: Vortex Layers of small thickness
Abstract: In this talk we shall consider a 2D incompressible non viscous flow with an initial datum with vorticity concentrated close to a curve y = phi(x) and exponentially decaying away from it. We shall suppose the vorticity intensity to be O(1/epsilon) while the exponential decay occurs on a scale O(epsilon).
We shall prove that, if the initial data are analytic, the solution of the above problem will preserve
the vortex layer structure for a time that does not depend on epsilon.
Moreover the dynamics of the layer is well approximated by the motion predicted by the Birkhoff-Rott equation for a vortex sheet of equivalent vorticity intensity.
The possibility of extending the above result to the Navier-Stokes solutions at small viscosity will be discussed.
This is joint work with R.Caflisch and M.C.Lombardo.
Marielle Simon
Title: Hydrodynamic limit for an activated exclusion process
Abstract: In this talk, we present a microscopic model in the family of conserved lattice gases. Its stochastic short-range interaction exhibits a continuous phase transition to an absorbing state at a critical value of the particle density. We prove that, in the active phase (i.e. for initial profiles smooth enough and uniformly larger than the critical density 1/2), the macroscopic behavior of this microscopic dynamics, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to the class of fast diffusion equations. The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time. Joint work with O. Blondel, C. Erignoux and M. Sasada.
Dimitri Tsagkarodgiannis
Title: Nonequilibrium fluctuations for current reservoirs
Abstract: Stationary non-equilibrium states are characterized by the presence of steady currents flowing through the system as a response to external forces. We model this process considering the simple exclusion process in one space dimension with appropriate boundary mechanisms which create particles on the one side and kill particles on the other. The system is designed to model Fick's law which relates the current to the density gradient. In this talk, we focus on the fluctuations around the hydrodynamic limit of the system. The main technical difficulty lies on controlling the correlations induced by the boundary action. This is work in progress jointly with Panagiota Birmpa and Patricia Gonçalves.
Ewelina Zatorska
Title: On the Existence of Solutions to the Two-Fluids Systems
Abstract: In this talk, I will present the recent developments in the topic of existence of solutions to the two-fluid systems. I will discuss the application of approach developed by P.-L. Lions and E. Feireisl and explain the limitations of this technique in the context of multi-component flow models. A particular example of such a model is two-fluids Stokes system with single velocity field and two densities, and with an algebraic pressure law closure. The existence result that uses the compactness criterion introduced for the Navier-Stokes system by D. Bresch and P.-E. Jabin will be presented. I will also mention an innovative construction of solutions relying on the G. Crippa and C. DeLellis stability estimates for the transport equation.
This talk is based on a joint result with D. Bresch and P.B. Mucha