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Webinar Kinetic and fluid equations for collective behavior

• Twice a month on Monday: a 40 mn talk
• Summer French time: 10:00-11:00;   Korean time: 17:00-18:00
       UTC offset France = +1; UTC offset Korea = +9
       Winter time: +0 additional offset in France (2023/10/29 → 2024/03/30)
       Summer time: +1 additional offset in France (2024/03/31 → 2024/10/27)
• Zoom link: ask the organizers

• 2024/06/24: Pierre Gervais (University of Lille)
  PDF Title: Hydrodynamic limit of elastic kinetic equations by a spectral approach
  
  Abstract: Among the 23 problems listed by D. Hilbert during the International Congress of Mathematicians in 1900, the 6th one concerns the derivation of macroscopic descriptions of fluids from their microscopic descriptions. One possible strategy involves going through an intermediate level of description called mesoscopic, or kinetic, such as the Boltzmann or Landau models. This is referred to as the problem of hydrodynamic limits.
  In the early 1990s, C. Bardos, F. Golse, and D. Levermore proved that one could formally derive the Navier-Stokes equations from kinetic equations conserving mass, velocity, and energy, and dissipating entropy, and the specific cases of the Boltzmann and Landau equations were gradually and independently addressed over the following three decades, despite their common structure.
  The work on hydrodynamic limits is partly constrained by tools dating back to the early days of Boltzmann theory in the 1960s, allowing only for solutions satisfying a very restrictive integrability assumption, but also by results established using non-constructive arguments. In the case of Cauchy theories of kinetic equations, these restrictions have been lifted thanks to modern tools of "enlargement theory" and hypocoercivity methods developed from the 2000s onwards, notably by C. Mouhot, S. Mischler, and M. Gualdani.
  In this talk, I present a collaboration with Bertrand Lods in which we have, on the one hand, considered the question of hydrodynamic limit for a kinetic equation under generic assumptions close to those of Bardos-Golse-Levermore, thus unifying previous results, and, on the other hand, modernized the necessary spectral study using the new theories of enlargement and hypocoercivity, thus providing the first fully quantitative results of hydrodynamic limits.
  
  
• 2024/06/10: Gyounghun Ko from (POSTECH and CM2LA)
  PDF ZOOM Title: Dynamical Billiard and a long-time behavior of the Boltzmann equation in general 3D toroidal domains
  
  Abstract: In this talk, we consider the Boltzmann equation in general 3D toroidal domains with a specular reflection boundary condition. So far, it is a well-known open problem to obtain the low-regularity solution for the Boltzmann equation in general non-convex domains because there are grazing cases, such as inflection grazing. Thus, it is important to analyze trajectories which cause grazing. We will provide new analysis to handle these trajectories in general 3D toroidal domains. This is a joint work with C.Kim and D. Lee.
  
  
• 2024/05/27: Erwan Faou (University of Rennes)
  ZOOM Title: linear damping around inhomogeneous stationary states of the Vlasov-HMF model
  
  Abstract: We will consider the dynamics of perturbations around an inhomogeneous stationary state of the Vlasov-HMF (Hamiltonian Mean-Field) model, satisfying a linearized stability criterion. Such stationary states are closely related to the dynamics of the pendulum system. We consider solutions of the linearized equation around the steady state, and prove the algebraic decay in time of the Fourier modes of their density. We prove moreover that these solutions exhibit a scattering behavior to a modified state, implying a linear damping effect with an algebraic rate of damping. This is a joint work with F. Rousset and R. Horsin.
  
  
• 2024/05/13: Dr. Hongxu Chen (Chinese University of Hong Kong)
  PDF ZOOM Title: Macroscopic estimate of Boltzmann equation with mixed boundary
  
  Abstract: In this talk, I will discuss the Boltzmann equation in a bounded domain. In the first part, we consider a mixed boundary condition where a portion of the boundary follows the pure specular boundary condition, while the remaining portion follows the pure diffuse boundary condition. We investigate the dynamical stability of the nonlinear problem by employing the $L^2$-$L^\infty$ framework. In the second part, we consider the pure specular boundary condition. We establish an $L^6$ bound for the macroscopic component of the linear problem. The $L^6$ estimate is crucial in the future study of the hydrodynamic limit under the specular boundary condition. These are joint work with Renjun Duan and Chanwoo Kim.
  
  
• 2024/04/22: Andrea Tosin (Politecnico di Torino)
  PDF Title: Network-based kinetic models: Emergence of a statistical description of the graph topology
  
  Abstract: In this talk, we present a novel approach that employs kinetic equations to describe the aggregate dynamics emerging from graph-mediated pairwise interactions in multi-agent systems. We formally show that for large graphs and specific classes of interactions a statistical description of the graph topology, given in terms of the degree distribution embedded in a Boltzmann-type kinetic equation, is sufficient to capture the aggregate trends of networked interacting systems. This proves the validity of a commonly accepted heuristic assumption in statistically structured graph models, namely that the so-called connectivity of the agents is the only relevant parameter to be retained in an aggregate description of the graph topology.
  
  
• 2024/04/08: I-Kun Chen (National Taiwan University)
  PDF ZOOM Title: Geometric effects on $W^{1, p}$ regularity of the stationary linearized Boltzmann equation
  
  Abstract: We study the incoming boundary value problem for the stationary linearized Boltzmann equation in bounded convex domains. The geometry of the domain has a dramatic effect on the space of solutions. We prove the existence of solutions in $W^{1,p}$ spaces for $1\leq p<2$ for small domains. In contrast, if we further assume the positivity of the Gaussian curvature on the boundary, we prove the existence of solutions in $W^{1, p}$ spaces for $1 \leq p < 3$ provided that the diameter of the domain is small enough. In both cases, we provide counterexamples in the hard sphere model; a bounded convex domain with a flat boundary for $p = 2$, and a small ball for $p = 3$. This talk is based on a joint work with Chun-Hsiung Hsia, Daisuke Kawagoe, and, Jhe-Kuan Su.
  
  
• 2024/03/25: Milana Pavić-Čolić (University of Novi Sad)
  PDF ZOOM Title: Homogeneous polyatomic Boltzmann flow: wellposedness and integrability
  
  Abstract: This talk will focus on the space homogeneous Boltzmann equation modelling a single polyatomic gas based on the continuous approach for particle’s internal energy. We will present existence and uniqueness theory under an extended Grad-type assumption on the collision kernel, which comprises hard potentials for both the relative speed and internal energy with the rate in the interval $(0, 2]$, multiplied by an integrable angular part. The Cauchy problem is resolved by means of an abstract ordinary differential equation (ODE) theory in Banach spaces for the initial data with finite and strictly positive mass and energy, and additionally finite 2+ moment. Moreover, we will explore Lebesgue's integrability propagation using entropy-based estimates. The differential inequality approach allows to prove the propagation property of the polynomially weighted $L^p$ norms, $p \in (1,\infty)$, associated to the solution of the Boltzmann equation. The case $p = \infty$ is found as a limit of the case $p < \infty$. The extension to multi-component mixtures will be discussed. This is a joint work with Ricardo Alonso and Irene M. Gamba.
  
  
• 2024/03/11: Prof. Junha Kim (Ajou University)
  PDF ZOOM Title: On the wellposedness of $\alpha$-SQG equation
  
  Abstract: In this talk, we consider the $\alpha$-SQG equation in a half-plane, where $\alpha=0$ and $\alpha=1$ correspond to the 2D Euler and SQG equations respectively. We prove the local wellposedness of $\alpha$-SQG in an anisotropic Lipschitz space and the instantaneous blow-up of solutions in Hölder spaces. This is a joint work with In-Jee Jeong(SNU) and Yao Yao(NUS).
  
  
• 2023/12/18: Sara Merino Aceituno (Universität Wien)
  ZOOM Title: Well-posedness, large-particle limit, phase transitions and open questions for the Vicsek model
  
  Abstract: The Vicsek model is a well studied model for collective dynamics where particles move at a constant speed while trying to align their orientations, up to some noise. The particle description consists of a coupled system of stochastic differential equations which in the large particle limits leads, formally, to a transport-type equation. Different modelling choices give rise to various challenges in proving the well-posedness of the equations and the large-particle derivation. (Based mostly on a joint work with Marc Briant and Antoine Diez.)
  
  
• 2023/12/04: Dr. Kunlun Qi (Univ of Minnesota)
  PDF ZOOM Title: On the kinetic description of the objective molecular dynamics
  
  Abstract: In this talk, I'll introduce a new multiscale hierarchy framework for objective molecular dynamics (OMD), reduced-order molecular dynamics with a certain symmetry that connects it to the statistical kinetic equation, and the macroscopic hydrodynamic model. In the mesoscopic regime, we exploit two interaction scalings that lead, respectively, to either a mean-field type or to a Boltzmann-type equation. It turns out that, under the special symmetry of OMD, the mean-field scaling results in vastly simplified dynamics that extinguish the underlying molecular interaction rule, whereas the Boltzmann scaling yields a meaningful reduced model called the homo-energetic Boltzmann equation. At the macroscopic level, we derive the corresponding Euler and Navier-Stokes systems by conducting a detailed asymptotic analysis. If time allows, some future work in both analytic and numerical fields will be discussed. The talk is based on the joint work with Li Wang (UMN) and Richard D. James (UMN).
  
  
• 2023/11/20: Vincent Calvez (CNRS & Université de Bretagne Occidentale)
  ZOOM Title: Modeling crowds of bacteria
  
  Abstract: I will present a joint work with Jean-Baptiste Saulnier (Marseille), Michèle Romanos (Lyon) and Tam Mignot (Marseille). We revisited the modeling of the so-called rippling phase of the soil bacteria Myxococcus xanthus lifecycle. The emergence of counter-propagating waves has attracted great attention from the mathematical and biophysical communities in the past two decades. Combining recent biological insights, high-resolution microscopy, trajectory analysis, kinetic modeling and numerical simulations, we proposed an integrated scenario for the collective behavior of this fascinating micro-organism, in which traffic congestion plays a key role.
  
  
• 2023/11/06: Seung-Yeon Cho (Gyeongsang National University)
  PDF ZOOM Title: A conservative semi-Lagrangian scheme for the Boltzmann equation
  
  Abstract: In this talk, I will introduce a high order conservative semi-Lagrangian scheme for the Boltzmann equation of rarefied gas flow. In order to construct an efficient method, we use semi- Lagrangain approach for treating convection term and the well-known fast spectral method to compute the collision operator. Moreover, we have adopt a conservative reconstruction and L2-minimization techniques to prevent the loss of conservation. Our high order methods require less computational cost than splitting methods of same order. Several numerical results will be presented to demonstrate the performance of the proposed scheme. This talk is based on the joint work with G. Russo and S. Boscarino.
  
  
• 2023/10/23: Mattia Zanella (University of Pavia)
  PDF ZOOM Title: Stochastic Galerkin particle methods for kinetic equations of plasmas with uncertain data
  
  Abstract: Kinetic equations play a leading role in the modelling of large systems of interacting particles with a recognized effectiveness in describing real world phenomena ranging from plasma physics to multi-agent dynamics. The derivation of these models has often to deal with limited data availability. Hence, to produce realistic descriptions of collisional systems, it is of paramount importance to embed uncertain quantities as a structural feature and to understand their propagation across scales. In this talk, we focus on a class of methods that guarantees the preservation of main physical properties of kinetic models with uncertainties. In contrast to the direct application of classical uncertainty quantification methods, which may lead to the loss of structural properties, we discuss the construction of particle methods that are able to achieve high accuracy in the random space without losing nonnegativity of the solution and hyperbolicity of hydrodynamic models.
  
  
• 2023/10/16: Prof. Jinmyoung Seok (Seoul National University)
  PDF ZOOM Title: Semiclassical limit of fermion stars
  
  Abstract: In this talk, we discuss the consistency of the super-cooled fermion stars and the Chandrasekhar's fluid model of white dwarfs in the semi-classical limit. The former is realized as a minimizer of the relativistic Hartree-Fock energy subject to the mass and a spectral bound. Similarly, the latter is realized as a minimizer of the relativistic Vlasov-Poisson energy subject to the mass and a uniform bound. The Lieb-Thirring inequality and relativistic Weyl's law play prominent roles for the proof.
  
  
• 2023/09/25: Nastassia Pouradier Duteil (Inria Paris)
  PDF ZOOM Title: Title: Graph Limit for Interacting Particle Systems on Weighted Random Graphs
  
  Abstract: We study the large-population limit of interacting particle systems posed on weighted random graphs. In that aim, we introduce a general framework for the construction of weighted random graphs, generalizing the concept of graphons. We prove that as the number of particles tends to infinity, the finite-dimensional particle system converges in probability to the solution of a deterministic graph-limit equation, in which the graphon prescribing the interaction is given by the first moment of the weighted random graph law. We also study interacting particle systems posed on switching weighted random graphs, which are obtained by resetting the weighted random graph at regular time intervals. We show that these systems converge to the same graph-limit equation, in which the interaction is prescribed by a constant-in-time graphon.
  
  
• 2023/09/18: In-Jee Jeong (Seoul National University)
  PDF ZOOM Title: Twisting in Hamiltonian Flows
  
  Abstract: We prove that the twisting in Hamiltonian flows on annular domains, which can be quantified by the differential winding of particles around the center of the annulus, is stable to perturbations. In fact, it is possible to prove the stability of the whole of the lifted dynamics to non-autonomous perturbations, though single particle paths are generically unstable. These all-time stability facts are used to establish a number of results related to the long-time behavior of fluids and kinetic equations. (Joint work with T. Drivas and T. Elgindi)
  
  
• 2023/06/30: Emmanuel Trélat (Sorbonne Université)
  PDF Title: Exponential convergence towards consensus for non-symmetric linear first-order systems in finite and infinite dimensions
  
  Abstract: I will first recall some results on how to achieve consensus for well known classes of systems, like the celebrated Cucker-Smale or Hegselmann-Krause models. When the systems are symmetric, convergence to consensus is classically established by proving, for instance, that the usual variance is an exponentially decreasing Lyapunov function: this is a "$L^2$ theory". When the systems are not symmetric, no $L^2$ theory existed until now and convergence was proved by means of a "$L^\infty$ theory".
  In this talk I will show how to develop a $L^2$ theory by designing an adequately weighted variance, and how to obtain the sharp rate of exponential convergence to consensus for general finite and infinite-dimensional linear first-order consensus systems. If time allows, I will show applications in which one is interested in controlling vote behaviors in an opinion model.
  This is a work in collaboration with Laurent Boudin and Francesco Salvarani.
  
  
• 2023/06/23: Dr. Gyuyoung Hwang (Seoul National University)
  PDF ZOOM Title: Asymptotic analysis of the Schrödinger-Lohe model for quantum synchronization
  
  Abstract: The Schrödinger-Lohe model is a system of coupled nonlinear Schrödinger equations designed for the feedback control of the quantum system. This model derives the quantum particles to exhibit the collective behavior called synchronization. In this thesis, we study the asymptotic behavior of the Schrödinger-Lohe model in two different perspectives : discrete and semiclassical. We begin with concerning the semi-discrete Schrödinger-Lohe model. Namely, we discretize the space domain into lattices to consider the interaction of wave functions on the infinite graph. Next, we study the semiclassical analysis of the Schrödinger-Lohe system. That is to say, we investigate the passage from the quantum to the classical system. Due to the structure of our model, we use the Wigner matrix method to study the behavior of the correlations of the wave functions as the Planck constant converges to zero.
  
  
• 2023/06/09: Alain Blaustein (University of Toulouse)
  PDF ZOOM Title: An asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck equation
  
  Abstract: We will propose a numerical method for the Vlasov-Poisson-Fokker-Planck model and prove quantitative results ensuring that it is Asymptotic-Preserving in both the macroscopic and the long time regime in a linearized framework. We will also focus on two simulations in which we will observe plasma echoes and transition phase between macroscopic and long time behavior.
  
  
• 2023/05/26: Dr. Junhyuk Byeon (Institute of Natural Sciences, Seoul National University)
  PDF ZOOM Title: Asymptotic tracking of a point cloud moving on Riemannian manifolds
  
  Abstract: We present two flocking type models for the asymptotic tracking of a point cloud moving on complete, connected, and smooth Riemannian manifolds. For each model, we provide a sufficient framework in terms of a moving target point cloud, system parameters, and initial data. In the proposed framework, we show asymptotic flocking, collision avoidance, and asymptotic tracking to a given point cloud.
  
  
• 2025/05/12: Raphael Winter (University of Vienna)
  PDF ZOOM Title: The Vicsek-BGK equation in collective dynamics
  
  Abstract: The Vicsek-BGK equation describes the collective motion of agents with local alignment. It is known that the spatially homogeneous model undergoes a phase transition from disoriented motion to collective motion. In this contribution we give a prove the onset of a phase transition in the spatially inhomogeneous case. Joint work with Sara Merino Aceituno and Christian Schmeiser.
  
  
• 2023/04/28: Myeongju Kang (Seoul National Unioversity)
  ZOOM Title: Continuum limit of Lohe model on some class of Lie groups
  
  Abstract: The Lohe model is a generalized high dimensional Kuramoto-type model whose oscillators lie on a special class of matrix Lie group. The continuum Lohe model is a time-evolutionary integro-differential equation which governs the Lohe phase field, and solution to the Lohe model becomes a simple function valued solution to the continuum Lohe model. In this talk, we study the emergent dynamics and global well-posedness of the continuum Lohe model which can be obtained by a continuum limit of the Lohe model. We first construct a local solution to the continuum Lohe model. Then, we find an invariant set to extend our local solution to a global one. Lastly, we show that sequence of simple functions obtained from the Lohe model converges to a solution of the continuum Lohe model in supreme norm sense. This talk is based on the joint work with Hangjun Cho and Seung-Yeal Ha.
  
  
• 2023/04/14: Marzia Bisi (University of Parma)
  PDF ZOOM Title: A mixed Boltzmann-BGK model for inert gas mixtures
  
  Abstract: We present a mixed Boltzmann-BGK model for mixtures of monatomic gases, where some collisions are described by Boltzmann operators and the others by suitable binary BGK terms. For the most general form of this hybrid kinetic description, where each different kind of intra-species or inter-species interactions may be described by a Boltzmann or by a BGK operator, we are able to prove that the model guarantees conservations of global momentum and energy, positivity of all temperatures, as well as the validity of Boltzmann H-theorem, allowing to conclude that the unique admissible equilibrium state corresponds to the expected Maxwellian distributions with all species sharing a common mean velocity and a common temperature. We also investigate in more detail the model with all intra-species collisions described by Boltzmann operators, and all inter-species interactions modelled by BGK terms. This option allows to derive consistent hydrodynamic equations not only in the classical collision dominated regime, but also in situations with intra-species collisions playing the dominant role, as in suitable mixtures with very disparate particle masses.
  
  
• 2023/03/24: Prof. Kung-Chien Wu (Department of Math. National Cheng Kung University)
  PDF ZOOM Title: Space-Velocity Bridge is Falling Down- Fractional Mixture Lemma
  
  Abstract: In this talk, we consider the Boltzmann equation with angular-cutoff for very soft potential case $-3 < \nu \leq -2$. We prove a regularization mechanism that transfers the microscopic velocity regularity to macroscopic space regularity in the fractional sense. The result extends the smoothing effect results of Liu-Yu (see ``mixture lemma'' in Comm Pure Appl Math 57:1543-1608, 2004), and of Gualdani-Mischler-Mouhot (see ``iterated averaging lemma'' in Mém Soc Math Fr 153, 2017), both established for the hard sphere case. A precise pointwise estimate of the fractional derivative of collision kernel, and a connection between velocity derivative and space derivative in the fractional sense are exploited to overcome the high singularity for very soft potential case. As an application of fractional regularization estimates, we prove the global well-posedness and large time behavior of the solution for non-smooth initial perturbation.
  
  
• 2023/03/10: Pierre Degond (University of Toulouse 1)
  PDF ZOOM Title: Topological states in a swarmalator model
  
  Abstract: Swarmalators are agents that combine the features of swarming particles and oscillators hence the name, contraction of ``swarmer'' and ``oscillator''. Each particle is endowed with a phase which modulates its interaction force with the other particles. In return, relative positions modulate phase synchronization between interacting particles. In the talk, I will present a model where there is no force reciprocity: when a particle attracts another one, the latter repels the former. This results in a pursuit behavior. I will derive a hydrodynamic model and show that it has explicit special solutions enjoying non-trivial topology quantified by a phase index. I will present a theoretical and numerical study of these solutions. This is joint work with Antoine Diez (Kyoto University) and Adam Walczak (Imperial College London).
  
  
• 2022/12/16: Vincent Giovangigli (Ecole Polytechnique)
  PDF ZOOM Title: Kinetic derivation and existence of strong solutions for diffuse interface fluid models
  
  Abstract: The diffuse interface fluid equations---also termed capillary fluids---are first derived from the kinetic theory of dense gases. These equations involve van der Waals' gradient energy, Korteweg's tensor, Dunn and Serrin's heat flux as well as viscous and heat diffusion fluxes. We next investigate existence of strong solutions to the resulting compressible nonisothermal diffuse interface fluid model. The density gradient is added as an extra variable and the augmented system of equations is recast into a normal form with symmetric hyperbolic first order terms, symmetric dissipative second order terms and antisymmetric capillary second order terms. New a priori estimates are obtained for such augmented systems of equations in normal form. Using the augmented system in normal form and a priori estimates, local existence of strong solutions is established in an Hilbertian framework.
  
  
• 2022/12/09: Yonggeun Cho (Jeonbuk National University)
  PDF ZOOM Title: Global well-posedness of Hartree type Dirac equations at critical regularity
  
  Abstract: In this talk I will introduce a recent result on the global well-posedness of classical Dirac equation with Hartree type nonlinearity in $\mathbb R^{1+3}$. The equation is essentially $L^2$-critical. A standard argument is to utilise spinorial null structure inside the equations. However, the null structure is not enough to attain the global well-posedness at critical regularity. I will impose an extra regularity assumption with respect to the angular variable to prove global well-posedness and scattering of Dirac equations for small $L^2_x$-data with additional angular regularity. This talk is based on the joint works with S. Hong and T. Ozawa, and S. Hong and K. Lee.
  
  
• 2022/11/25: Daniel Han-Kwan (Ecole polytechnique)
  PDF ZOOM Title: On the hydrodynamic description of aerosols
  
  Abstract: We are interested in the dynamics of an aerosol, that is a cloud of fine particles immersed in an ambient fluid. Starting with a description by the Vlasov-Navier-Stokes system, we will show that in a certain high friction limit, the aerosol can be well described by the inhomogeneous incompressible Navier-Stokes system. This is a joint work with David Michel.
  
  
• 2022/11/18: Doctor Gi-Chan Bae (Seoul National University)
  PDF ZOOM Title: The quantum Boltzmann and BGK model near a global equilibrium
  
  Abstract: This talk considers the existence and asymptotic behavior of two quantum kinetic equations, the quantum BGK model and the relativistic quantum Boltzmann equation. More precisely, we establish the existence of unique classical solutions and their exponentially fast stabilization when the initial data starts sufficiently close to a global quantum equilibrium based on the nonlinear energy method. The two models have different difficulties. For the quantum BGK model, the difficulty is to extract dissipation from the highly nonlinear quantum local equilibrium. For the relativistic quantum Boltzmann equation, we should control the nonlinear part on the energy-momentum 4-vector space to close the argument on $L^2$ space.
  
  
• 2022/10/28: Marwa Shahine (University of Bordeaux)
  PDF ZOOM Title: Fredholm Property of the Linearized Boltzmann Operator for a Mixture of Polyatomic Gases
  
  Abstract: In this talk, we consider the Boltzmann equation that models a mixture of polyatomic gases assuming the internal energy to be continuous. Under some convenient assumptions on the collision cross-section, we prove that the linearized Boltzmann operator L is a Fredholm operator. For this, we write L as a perturbation of the collision frequency multiplication operator. We prove that the collision frequency is coercive and that the perturbation operator is Hilbert-Schmidt integral operator.
  
  
• 2022/10/14: Dr. Jae Yong Lee (KIAS, Korea)
  PDF ZOOM Title: Deep learning approach for solving kinetic equations
  
  Abstract: Recently, deep learning-based methods have been developed to solve PDEs with many advantages. In this talk, I introduce our recent results on the deep neural network solutions to the kinetic equation. We study Vlasov-Poisson-Fokker-Planck equation and its diffusion limit via the deep learning approach. Also, we propose a new framework to approximate the solution to Fokker-Planck-Landau equation which has a nonlinearity and a high dimensionality of variables.
  
  
• 2022/09/23: Francis Filbet (Université Paul Sabatier - Toulouse)
  PDF ZOOM Title: On the stability of Hermite spectral methods for the Vlasov-Poisson system and Fokker-Planck equation
  
  Abstract: We study a class of spatial discretizations for the Vlasov-Poisson system and Fokker-Planck equation written as an hyperbolic system using Hermite polynomials. To obtain L^2 stability properties, we introduce a new $L^2$ weighted space, with a time dependent weight. For the Hermite spectral form of the Vlasov-Poisson system, we prove conservation of mass, momentum and total energy, as well as global stability for the weighted $L^2$ norm. These properties are then discussed for several spatial discretizations. For the Fokker-Planck equation, this approach allows to investigate the long time behavior and the asymptotic limit of the discrete model.
  
  
• 2022/06/24: Gael Raoul (Ecole Polytechnique)
  PDF ZOOM Title: Wasserstein estimates and convergence to equilibrium for an evolutionary biology model
  
  Abstract: Titre : We are interested in the dynamics of a population structured by a phenotypic trait. Individuals reproduce sexually, which is represented by a non-linear integral operator close to an inelastic Boltzmann operator. This operator is combined to a multiplicative operator representing selection. When the strength of selection is small, we show that the dynamics of the population is governed by a simple macroscopic differential equation, and that solutions converge exponentially to a steady-state. The analysis is based on Wasserstein distance inequalities.
  
  
• 2022/06/10: Dr. Jinwook Jung (Department of Statistics, Seoul National University)
  ZOOM Title: Large time behavior of solutions to the pressureless Euler-Navier-Stokes system in the whole space
  
  Abstract: In this talk, we present a refined framework for the large time behavior estimates for the pressureless Euler-Navier-Stokes system. Specifically, under a suitable assumption on the density of the pressureless Euler fluid flow, we show that the decay rate of the higher-order derivatives of fluid velocities, whose order is smaller than the dimension, is faster than that of lower-order derivatives. As a byproduct, we establish the global-in-time existence and uniqueness of classical solutions to our main system in the two-dimensional case. This talk is based on the joint work with Young-Pil Choi (Yonsei University).
  
  
• 2022/05/20: Filippo Santambrogio (Université Claude-Bernard Lyon 1)
  PDF ZOOMTitle: The Fokker-Planck equation as a gradient-flow in the Wasserstein space : estimates in the time-discrete scheme
  
  Abstract: From the work by Jordan, Kinderlehrer and Otto it is known that some parabolic PDEs have a variational structure (of steepest descent type) in the Wassestein space of probability densities endowed with a distance coming from optimal transport. The most typical example is the Fokker-Planck equation $\partial_t \rho = \Delta \rho + \nabla\cdot (\rho \nabla V)$, associated with the energy $F(\rho):=\int \rho\log\rho+\rho V$. Due to this variational structure, a very natural time-discretization scheme can be built, known as the JKO scheme : at each time step the sum of $F$ plus a suitable transport cost from the previous density is minimized, thus obtaining a recursive sequence of densities which approximate the solution of the PDE. It is interesting to see which bounds and regularity properties known to be satisfied by the solutions of the continuous-time equation are also satisfied in the discrete scheme. In the present talk, after recalling the main ingredients to understand the JKO scheme, I will give $L^\infty$ estimates on the solution and on its gradient, based on some easy manipulations on the Monge-Ampère equation.
  
  
• 2022/05/13: Prof. Chanwoo Kim (Univ. of Wisconsin)
  PDF ZOOM Title: Vorticity Convergence from Boltzmann to 2D incompressible Euler equations below Yudovich class
  
  Abstract: It has been an open problem to prove the convergence of solutions of the Boltzmann equations to non-Lipschitz solutions of the incompressible Euler equations. We settle this question affirmatively for Lagrangian solutions of the 2D incompressible Euler equation when the vorticity belongs to $L^p$ for any $p\geq 1$.
  
  
• 2022/04/22: Bertrand Lods (Università degli studi di Torino)
  PDF ZOOM Title: Hydrodynamic limit for granular gases: from Boltzmann equation to some modified Navier-Stokes-Fourier system
  
  Abstract: In this talk, we aim to present recent results about the rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres with small inelasticity. The hydrodynamic system that we obtain is an incompressible Navier-Stokes-Fourier system with self-consistent forcing terms and, to our knowledge, it is the first hydrodynamic system that properly describes rapid granular flows consistent with the kinetic formulation in physical dimension d=3. For that purpose, one of the main mathematical difficulty is to understand the relation between the restitution coefficient, which quantifies the energy loss at the microscopic level, and the Knudsen number. This is achieved by identifying the correct nearly elastic regime to capture nontrivial hydrodynamic behavior. The talk is based on a joint work with Ricardo Alonso (Texas A&M University at Qatar) and Isabelle Tristani (ENS Paris, Université PSL).
  
  
• 2022/04/08: Dongnam Ko (Catholic University of Korea)
  PDF ZOOM Title: Convergence of the discrete consensus-based optimization algorithm with heterogeneous noises
  
  Abstract: We present stochastic convergence analysis of the discrete consensus-based optimization (CBO) algorithm with random batch interactions and heterogeneous external noises, which guarantees the termination of the CBO algorithm. Despite successful performance in many practical simulations and remarkable analysis on kinetic level, the termination of the CBO algorithm was not rigorously investigated in such a generality as a discrete-time numerical algorithm. For this, we generalize the CBO algorithm with an abstract representative point, and then derive stochastic convergence of the individuals to a common point in mean-square and almost-sure sense under small noise assumption.
  
  
• 2022/03/25: Anne Nouri (Aix-Marseille University)
  PDF ZOOM Title: Discrete velocity Boltzmann equations in the plane: stationary solutions
  
  Abstract: Existence of stationary mild solutions for discrete velocity Boltzmann equations in the plane with no pair of colinear interacting velocities and given ingoing boundary values will be proven. A key property is $L^1$ compactness of integrated collision frequency for a sequence of approximations. This replaces the $L^1$ compactness of velocity averages in the continuous velocity case, not available when the velocities are discrete. This a joint work with Leif Arkeryd.
  
  
• 2022/03/18: Insuk Seo (Seoul National University)
  PDF ZOOM Title: Metastability of random processes and solution of resolvent equations
  
  Abstract: A random process exhibits the so-called metastability when the associated potential function has multiple local minima. The quantitative analysis of the metastability is based on the careful investigation of the generator (which is usually a second-order elliptic operator) associated with the random process. Recently, it has been observed that the analysis of solutions of a class of resolvent equations associated with the generator of the random process contain all the essential information regarding the quantitative feature of the metastability. In this lecture, we explain this relation and its application to the small random perturbations of dynamical system studied originally by Freidlin and Wentzell in 70s. The lecture is self-contained and will not assume any prior knowledge regarding the study of metastability. This presentation is based on the work arXiv:2102.00998 and an ongoing work with Claudio Landim and Jungkyoung Lee.
  
  
• 2021/12/17: Yann Brenier (Ecole Normale Supérieure)
  PDF ZOOM Title: Phase-space and optimal-transport formulations of Einstein's gravitation in vacuum
  
  Abstract: Einstein's theory of gravitation, at least in vacuum, can be treated as a kind of quadratic matrix-valued optimal transportation problem. This goes through a kind of "kinetic" or "phase-space" formulation of the concept of Ricci curvature
  
  
• 2021/12/10: Woojoo Shim (Seoul National University)
  PDF ZOOM Title: Cucker-Smale inspired deterministic Mean Field Game with velocity interactions
  
  Abstract: In this talk, I will present a mean field game model for agents moving in a given domain and choosing their trajectories so as to minimize a cost including a penalization on the difference between their own velocity and that of the other agents they meet. For the proposed model, we discuss the existence of an equilibrium in a Lagrangian setting by using its variational structure, and then study its properties including regularity.
  
  
• 2021/11/26: Teddy Pichard (Ecole polytechnique)
  PDF ZOOM Title: Some recent advances in the theory of moment model
  
  Abstract: In kinetic theory, the method of moments is used to reduce a scalar transport equation depending on multiple variables (typically Boltzmann equation involves seven variables) into a system involving less variables (typically Euler equations). In this talk, I will present some recent constructions of moment closures in velocity. I will focus on three approaches that will be compared: 1-a family of methods based on quadrature formulae (QMOM methods); 2-a family of models based on the entropy dissipation of the kinetic equation ; 3-a family of methods based on the study of the set of admissible moments and exploiting projection techniques on it.
  
  
• 2021/11/12: Jin Woo Jang (Department of Math. Pohang University of Science and Technology)
  PDF ZOOM Title: LTE and Non-LTE Solutions in Gases Interacting with Radiation
  
  Abstract: The goal of this talk is to discuss a class of kinetic equations describing radiative transfer in gases which include also the interaction of gas molecules with themselves. We first introduce a system of kinetic PDEs that describes the dynamics of gas molecules coupled with an equation for photons radiative transfer. We then discuss several scaling limits and introduce some Euler-like systems coupled with radiation as an aftermath of specific scaling limits. We consider scaling limits in which local thermal equilibrium (LTE) holds, as well as situations in which this assumption fails (non-LTE). We understand that the structure of the equations describing the gas-radiation system is very different in the LTE and non-LTE cases. We prove the existence of stationary solutions to the resulting limit models in the LTE case. Lastly, we will also prove the non-existence of stationary solutions with zero velocities in a non-LTE situation. This is a joint work with Juan J. L. Velazquez at Bonn.
  
  
• 2021/10/29: Amic Frouvelle (University of Dauphine)
  PDF ZOOM Title: Body-attitude alignment : phase transition, link with suspensions of rodlike polymers and quaternions
  
  Abstract: We present a model of alignment of individuals based on body attitude (birds aligning their heading and wings directions for instance). The kinetic model in consideration is a Fokker-Planck model for which the velocity variable is a rotation matrix of dimension 3. We present an interesting link between this model and a generalization of the Maier-Saupe model for alignment of diluted rodlike polymers in dimension 4, due to the fact that a rotation can be represented by a unit quaternion (or its opposite, which relates to the fact that a rodlike polymer is unoriented). We obtain the phase diagram of this model : when the alignment strength is low, the uniform distribution is the only equilibria, when the strength is sufficiently large, there exists a unique family of stable (concentrated) distributions, and in between, we have stability of both non-aligned and aligned states.
  
  This comes from works in collaboration with Pierre Degond, Antoine Diez, Sara Merino-Aceituno and Ariane Trescases.
  
  
• 2021/10/15: Renjun Duan (The Chinese University of Hong Kong)
  PDF ZOOM Title: The Boltzmann equation for plane Couette flow
  
  Abstract: In the talk I will report a recent work in collaboration with Shuangqian Liu and Tong Yang on a study of the Couette flow for a rarefied gas between two parallel infinite plates moving relative to each other. We reformulate it as the boundary value problem on the Boltzmann equation with a shear force subject to the homogeneous non-moving diffuse reflection boundary, and then establish the existence and large time asymptotic stability of stationary solutions for any small enough shear strength.
  
  
• 2021/09/24: Laurent Desvillettes (Université de Paris)
  PDF ZOOM Title: Some new use of the duality methods for parabolic systems
  
  Abstract: Duality methods for L^2-estimates of solutions of reaction-diffusion systems were introduced by Pierre and Schmitt. They can also be used to obtain estimates for coagulation-fragmentation-diffusion systems, or cross diffusion systems. We present in this talk a new application of those methods for a system coming out of the modeling of cells/chemical species including chemotaxis terms.
  
  
• 2021/09/10: Jeongho Kim (Hanyang University)
  PDF ZOOM Title: Hydrodynamic limits of the Schrodinger equation with gauge fields
  
  Abstract: In this talk, we present the hydrodynamic limits of the Schrodinger equation, affected by different gauge fields. Precisely, we first present the hydrodynamic limit of the Schrodinger equation with the Chern-Simons gauge fields (Chern-Simons-Schrodinger equation), toward to the Euler-Chern-Simons equation on the two-dimensional state space. Then, we consider the hydrodynamic limit of the Schrodinger equation with the Maxwell gauge fields (Maxwell-Schrodinger equation), toward to the Euler-Maxwell equation on the three-dimensional state space. Both estimate use the estimate on the modulated energy functionals.
  
  

• 2021/06/25: Isabelle Tristani (ENS Ulm Paris)
  PDF ZOOM Title: Incompressible Navier-Stokes limit of the Boltzmann equation
  
  Abstract: In this talk, we are interested in the link between strong solutions of the Boltzmann and the Navier-Stokes equations. The problem of justifying the connection between mesoscopic and macroscopic equations has been extensively studied. Here, we propose an approach that intertwines fluid mechanics and kinetic estimates. It enables us to prove convergence of smooth solutions of the Boltzmann equation to solutions to the fluid dynamics equations when the Knudsen number goes to zero. We do not require any smallness at initial time, and our result is valid for any initial data (well prepared or not) in the case of the whole space. We also prove that the time of existence of the solution to the Boltzmann equation is bounded from below by the existence time of the fluid equation as soon as the Knudsen number is small enough. This is a joint work with Isabelle Gallagher
  
  
• 2021/06/11: Dohyun Kim (Sungshin Women's Univ.)
  PDF ZOOM Title: Asymptotic emergent dynamics of the Schrödinger-Lohe model
  
  Abstract: In this talk, we introduce a coupled system of nonlinear Schrödinger equations, so-called the Schrödinger-Lohe (S-L) model as a phenomenological model for quantum synchronization. Then, we briefly review recent progress on the S-L model from the perspective of asymptotic emergent dynamics. For the analytic results, the two-point correlation function defined as the inner product of two wavefunctions is mainly used. On the other hand for the numerical result, we adopt the time splitting spectral method together with the Crank-Nicolson method to discretize the S-L model.
  
  
• 2021/05/28: Benoit Perthame (Sorbone Université)
  PDF ZOOM Title: From voltage-conductance kinetic models to integrate and fire equation for neural assemblies
  
  Abstract: The voltage-conductance systems for neural networks has been introduced by biophysicists for modeling the visual cortex. In terms of mathematical structure, it can be compared to a kinetic equations with a macroscopic limit which turns out to be the Integrate and Fire equation. This talk is devoted to a mathematical description of the slow-fast limit of the kinetic type equation to an I\&F equation. After proving the weak convergence of the voltage-conductance kinetic problem to potential only I\&F equation, we prove strong a priori bounds and we study the main qualitative properties of the solution of the I\&F model, with respect to the strength of interconnections of the network. In particular, we obtain asymptotic convergence to a unique stationary state for weak connectivity regimes. For intermediate connectivities, we prove linear instability and numerically exhibit periodic solutions. These results about the I\&F model suggest that the more complex voltage-conductance kinetic equation shares some similar dynamics in the correct range of connectivity.
  
  
• 2021/05/14: Donghyun Lee (Postech)
  PDF ZOOM Title: Large amplitude solution of the Boltzmann equation
  
  Abstract: We study well-posedness theory the Boltzmann equation in low regularity L^{\infty} space. After low regularity approach with L^2-L^\infty bootstrap argument was introduced, the method was widely used to solve many boundary condition problems in small perturbation framework. Moreover, these results have extended into more general L^\infty solution whose amplitude can be arbitrary large. We mainly discuss about recent developments in large amplitude Boltzmann theory including boundary condition problems. This is a joint work with R.Duan(CUHK) and G. Ko(POSTECH).
  
  
• 2021/04/23: Clément Mouhot (University of Cambridge)
  ZOOM Title: Quantitative De Giorgi Methods in Kinetic Theory
  
  Abstract: We consider hypoelliptic equations of kinetic Fokker-Planck type, also sometimes called of Kolmogorov or Langevin type, with rough coefficients in the diffusion matrix. We present novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities (which imply Holder continuity with quantitative estimates). This is a joint work with Jessica Guerand.
  
  
• 2021/04/09: Young-Pil Choi (Yonsei Univ.)
  ZOOM Title: Quantified overdamped limit for Vlasov-Fokker-Planck equations with singular interaction forces
  
  Abstract: In this talk, I will discuss a quantified overdamped limit for kinetic Vlasov-Fokker-Planck equations with nonlocal interaction forces. We provide explicit bounds on the error between solutions of that kinetic equation and the limiting equation, which is a diffusive model with nonlocal velocity fields often referred to as aggregation-diffusion equation or McKean-Vlasov equation. Our strategy only requires weak integrability of the interaction potentials, thus in particular it includes the quantified overdamped limit of the kinetic Vlasov-Poisson-Fokker-Planck system to the aggregation-diffusion equation with either repulsive electrostatic or attractive gravitational interactions.
  
  
• 2021/03/26: Helge Dietert (Université Paris Diderot)
  ZOOM Title: Hypocoercivity with spatial weight
  
  Abstract: We will study linear kinetic equations for a density in phase-space evolving through Hamiltonian transport and a linear collision operator in the velocity variable. In this talk, we study the effect of a spatial weight for the collision operator which vanishes in part of the domain. In particular, I will present a quantitative proof for exponential relaxation under a geometric control condition.
  
  
• 2021/03/12: Jihoon Lee (Dept. of Math, Chung-Ang Univ.)
  PDF ZOOM Title: Decay estimates of solutions to the fluid equations with rotation or stratification
  
  Abstract: In this talk, we consider the incompressible fluid equations with rotation or stratification. First, we consider three dimensional incompressible Navier-Stokes equations with fractional dissipation and Coriolis force. We find Coriolis force gives extra temporal decay of the solutions under some conditions on the initial data. Next, we consider the three-dimensional damped Boussinesq equations with strong stratification. We find the global-in-time existence of solutions under some conditions of the initial data and the temporal decay of solutions.
  This is based on the joint work with Jaewook Ahn(Dongguk Univ.) and Junha Kim(Chung-Ang Univ.).
  
  
• 2021/02/26: François Golse (Ecole Polytechnique)
  PDF ZOOM Title: Half-space problem for the Boltzmann equation with phase transition at the boundary
  
  Abstract: Y. Sone, K. Aoki and their group have studied numerically the existence of a solution to the steady half-space problem for a rarefied gas whose behavior is described by the Boltzmann equation (with slab symmetry). The gas is assumed to fill a half-space on top of a liquid which is its condensed phase, and the velocity distribution function of molecules entering the half-space from the condensed phase is the centered Maxwelllian parametrized by the temperature at the gas-liquid interface, and the saturating vapor pressure at this temperature. The state at infinity (i.e. far from the interface) is (another) Maxwellian. In a remarkable paper T.-P. Liu and S.-H. Yu [Arch. Rational Mech. Anal. 209 (2013), 869-997] have proposed a complete method for handling this kind of problem. The purpose of this talk is to present an alternative, self-contained proof of one of results in the work of Liu and Yu, specifically the existence and uniqueness of solutions that are decaying as the distance to the interface goes to infinity, uniformly in the Mach number of the Maxwellian at infinity. The proof uses a variant of the generalized eigenvalue problem studied by Nicolaenko in his work on the shock profile for the Boltzmann equation, and the Ukai-Yang-Yu penalization method for half-space problems in kinetic theory (suitably modified).
  [Work with N. Bernhoff.]
  
  
• 2021/02/08: Seok-Bae Yun (Sunkyunkwan University (SKKU))
  PDF Title: Ellipsoidal BGK model of the Boltzmann equation with the correct Prandtl number
  
  Ellipsoidal BGK model (ES-BGK) is a generalized version of the Boltzmann-BGK model where the local Maxwellian in the relaxation operator of the BGK model is extended to an ellipsoidal Gaussian with a Prandtl parameter ν, so that the correct Prandtl number can be computed in the Navier-Stokes limit. In this talk, we review some of the recent results on ES-BGK model such as the existence (stationary or non-stationary) theory and the entropy-entropy production estimates. A dichotomy is observed between −1/2 < v < 1 and ν=-1/2. In the former case, an equivalence relation between the local temperature and the temperature tensor enables one to apply theories developed for the original BGK model in a modified form. In the critical case (ν=-1/2), where the correct Prandtl number is achieved, such equivalence break down, and the structure of the flow has to be incorporated to estimate the temperature tensor from below. This is from joint works with Stephane Brull and Doheon Kim.