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Speaker: Denise Aregba-Driollet (Bordeaux INP)
Title: Relaxation schemes for hyperbolic systems: around the BGK formalism PDF 1 ZOOM 1 PDF 2 ZOOM 2 Abstract: Beyond the modeling of certain physical phenomena, kinetic formalism is a fruitful source of inspiration for the construction of numerical schemes, particularly in fluid mechanics. The main feature is the approximation of a hyperbolic quasilinear system by a semilinear one, which avoids the use of complex and costly Riemann solvers. Reciprocally, the kinetic interpretation of a numerical scheme allows to prove useful stability properties. In this lecture I shall give some examples of such methods, first in the framework of conservation laws and then for nonconservative systems. The kinetic models are formally BGK equations where the maxwellian function contains all the nonlinear information. Stability properties will be studied and the link with relaxation methods will be done.
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Speaker: Jose A. Carrillo (University of Oxford)
Speaker: Young-Pil Choi (Yonsei University)
Title: Models for collective behavior: qualitative properties and model hierarchy PDF 1-2 ZOOM 1 ZOOM 2 PDF 3 ZOOM 3 PDF 4 ZOOM 4 Abstract: We do a state of the art in collective behavior models. The first two lectures are introductory material discussing individual based models, variations, their mean field limit and formal approximations. Qualitative properties of the particle models will also be discussed. The second two lectures are related to the most recent advances in the rigorous derivations from particles and the asymptotic limits connecting all the hierarchy of models in this active field of research: kinetic models, pressureless Euler equations with nonlocal forces, and aggregation equations.
Lecture 1. Particle models in collective behavior: a quick introduction. (JAC)
Lecture 2. Qualitative Properties: Flocks and Mills. Mean-field limit and formal hydrodynamics. (JAC)
Lecture 3. Rigorous derivation of the continuum hydrodynamic equations. (YPC)
Lecture 4. Asympotic limits connecting: kinetic, hydrodynamic and aggregation equations. (YPC)
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Speaker: François Golse (Ecole Polytechnique)
Title: Mean-field limits: a mathematical toolbox
PDF 1 ZOOM 1 PDF 2 ZOOM 2 Abstract: In these lectures, we shall present some tools for proving mean field limits in the context of classical and quantum dynamics. Some of these tools (empirical measures and Klimontovich solutions of the Vlasov equation, Wasserstein distances and so on) have been known for quite a long time. Adapting these tools to a noncommutative setting (such as the Lohe matrix model), or the quantum setting has been done only recently. The purpose of these lectures is to give an overview of recent progress on these issues, based on some joint works with I. Ben Porath, S.-Y. Ha, C. Mouhot and T. Paul.
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Speaker: Seung-Yeal Ha (Seoul National University)
Title: Higher-dimensional Kuramoto models and their relations with the Schrödinger-Lohe system
PDF 1 ZOOM 1 PDF 2 ZOOM 2 Abstract: In this lecture, we will review the state-of-the-art results on the emergent dynamics of the aggregation model on the space of rank-m tensors with the same size, and its relation with the Schr&oiml;dinger-Lohe model for quantum synchronization. The proposed aggregation model is general enough to include Lohe type synchronization models such as the Kuramoto model, the Lohe sphere model and the Lohe matrix models for the ensemble of real rank-0, rank-1 and rank-2 tensors, respectively. In this regard, we call our proposed model as the Lohe tensor model for rank-m tensors with the same size. For the proposed model, we present several sufficient frameworks leading to the collective dynamics of the Lohe tensor model in terms of system parameters and initial data, and study existence of special solutions such as completely separable solutions and quadratically separable solutions. This lecture is based on the recent joint works with Hansol Park (Seoul National Univ.) and Dohyun Kim (Sungshin Women's Univ.)
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Speaker: Jacques Schneider (University of Toulon)
Title: Relaxation models for kinetic equations PDF 1-2 ZOOM 1 ZOOM 2 Abstract: The complexity of many interaction operators occuring in kinetic theory has raised multiple interests for developping simpler operators. As soon as an entropy principle leads the solution to some equilibrium in a given regime, relaxation operators may be considered as an attractive alternative to this complexity. One of the main features of such operators is to pertain positivity of the distribution function while behaving linearly with respect to moments of this function. This has lead us to reconsider existing models (BGK, ESBGK, Shakhov and others) in a single axiomatic construction that offers the possibility to design new operators in many different context (mono and polyatomic gases, multi species reacting or not mixtures). Yet, if the principle seems quite simple at first sight, it requires deep analysis in many different area ranging from characterization of realizable moment (i.e moments of nonnegative distribution functions) via positive polynomials to variational problem and convex analysis. Our purpose is to give an insight into these numerous problems, the solution we and others have brought to them and eventually to display opened questions
Short presentations
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Speaker: Gi-Chan Bae (Seoul National University)
Title: Shakhov model near a global Maxwellian PDF ZOOM Abstract: Shakhov model is a relaxation approximation of the Boltzmann equation proposed to overcome the deficiency of the original BGK model, namely, the incorrect production of the Prandtl number. In this talk, we address the existence and asymptotic stability of the Shakhov m odel when the initial data is a small perturbation of global equilibrium. We derive a dichotomy in the coercive estimate of the linearized relaxation operator between zero and non-zero Prandtl number and observe that the linearized relaxation operator becomes more degenerate in the former case. To fill out such degeneracy and recover the full coercivity, we consider a micro-macro equation that involves non-conservative quantities. This is a joint work with Seok-Bae Yun (Sungkyunkwan University).
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Speaker: Byung-Hoon Hwang (Sungkyunkwan University)
Title: A relativistic generalization of the BGK model for gas mixtures
PDF ZOOM Abstract: The BGK model is the best-known model equation of the Boltzmann equation, which satisfies the essential features of the Boltzmann equation at a much lower numerical cost. In this talk, we introduce a relativistic generalization of the BGK model for gas mixtures derived based on the Marle's formulation of the relativistic BGK model. Here we consider an inert gas mixture satisfying the conservation laws for species number densities, global momentum, and total energy. We first present several properties that our model satisfies, and discuss the determination problem of the auxiliary temperature provided by the nonlinear relations. Then, we show that our model can recover the classical BGK model proposed by Bisi et al. [J. Phys. A: Math. Theor., 51 (2018)] in the Newtonian limit. This is based on the joint work with Seok-Bae Yun and Myeong-Su Lee (Sungkyunkwan Univ.).
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Speaker: Jinwook Jung (Seoul National University)
Title: On the large-time behavior of two-phase fluid models PDF ZOOM Abstract: In this talk, we study the large-time behavior of two--phase fluid models on the periodic domain. More precisely, we consider the pressureless Euler--Poisson/Riesz system coupled with the incompressible Navier--Stokes system through the drag force. Under suitable assumptions on the regularity of solutions, we show the fluid velocities are aligned with each other and the fluid density converges to the background state exponentially fast as time tends to infinity. This is based on the collaboration with Prof. Young-Pil Choi (Yonsei Univ.).
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Speaker: Myeongju Kang (Seoul National University)
Title: On a generalized Kuramoto model and uniform stability PDF ZOOM Abstract: We propose a generalized Kuramoto model motivated by relativistic effects and investigate uniform stability. The proposed generalized Kuramoto model incorporates relativistic Kuramoto(RK) type models which can be derived from the relativistic Cucker-Smale (RCS) on the unit sphere under suitable approximations. We present several sufficient frameworks leading to uniform stability in terms of system function, system parameters and particle network.
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Speaker: Jeongho Kim (Hanyang University)
Title: Hydrodynamic limits of the nonlinear Schr\"odinger equation with the Chern-Simons gauge fields PDF ZOOM Abstract: We present two types of the hydrodynamic limit of the nonlinear Schrödinger-Chern-Simons (SCS) system. We consider two different scalings of the SCS system and show that each SCS system asymptotically converges towards the compressible and incompressible Euler system, coupled with the Chern-Simons equations and Poisson equation respectively, as the scaled Planck constant converges to 0. Our method is based on the modulated energy estimate. In the case of compressible limit, we observe that the classical theory of relative entropy method can be applied to show the hydrodynamic limit, with the additional quantum correction term. On the other hand, for the incompressible limit, we directly estimate the modulated energy to derive the desired asymptotic convergence.
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Speaker: Woojoo Shim (Seoul National University)
Title: A mean-field limit of the Cucker-Smale model on complete Riemannian manifolds PDF ZOOM Abstract: We study a mean-field limit of the Cucker-Smale(C-S) model for flocking on complete smooth Riemannian manifolds. For this, we first formally derive the kinetic manifold C-S model on manifolds using the BBGKY hierarchy and derive several a priori estimates on emergent dynamics. Then, we present a rigorous mean-field limit from the particle model to the corresponding kinetic model by using the generalized particle-in-cell method. As a byproduct of our rigorous mean-field limit estimate, we also establish a global existence of a measure-valued solution for the derived kinetic model. Compared to the corresponding results on $\mathbb{R}^d$, our procedure requires additional assumption on holonomy and proper {\it a priori} bound on the derivative of parallel transports. As a concrete example, we verify that hyperbolic space $\mathbb{H}^d$ satisfies our proposed standing assumptions.
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Speaker: Noemi David (Sorbonne University)
Title: On the incompressible limit for tumor growth models including nutrients and convective effects PDF ZOOM Abstract: Both compressible and incompressible porous medium models are used in the literature to describe the mechanical properties of living tissues, [6, 7]. These two classes of models can be related using a stiff pressure law. In the incompressible limit, the compressible model generates a free boundary problem of Hele-Shaw type where incompressibility holds in the saturated phase, see for example [5, 4]. In this talk, we present a model including the effect of a nutrient (or possibly an external drift), [1, 3, 2]. Then, a badly coupled system of equations describes the cell population density and the nutrient concentration. For this reason, the derivation of the free boundary (incompressible) limit was an open problem, in particular a difficulty is to establish the so-called complementarity relation which allows to recover the pressure using an elliptic equation. To this end, we prove the strong compactness of the pressure gradient, blending two new techniques : an extension of the usual Aronson-B-nilan estimate in an L2-setting, also used recently for related problems, and a sharp L4-uniform bound of the pressure gradient. (Based on joint works with Benoit Perthame, Markus Schmidtchen, Xinran Ruan)
[1] N. David, B. Perthame. Free boundary limit of a tumor growth model with nutrient. Journal de Mathematiques Pures et Appliquees, 2021.
[2] N. David, X. Ruan. An asymptotic preserving scheme for a tumor growth model of porous medium type. arXiv :2105.10376, 2021.
[3] N. David, M. Schmidtchen. On the incompressible limit for a tumour growth model incorporating convective effects. arXiv :2103.02564, 2021.
[4] I. Kim, N. Pozar, B. Woodhouse. Singular limit of the porous medium equation with a drift. Adv. Math., 349, 682-732, 2019.
[5] B. Perthame, F. Quiros, J. L. Vazquez. The Hele-Shaw asymptotics for mechanical models of tumor growth. Arch. Ration. Mech. Anal., 212(1), 93-127, 2014.
[6] B. Ribba, O. Saut, T. Colin, D. Bresch, E. Grenier, J. P. Boissel. A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents. J. Theoret. Biol., 243(4), 532-541, 2006.
[7] J. A. Sherratt, M. A. J. Chaplain. A new mathematical model for avascular tumour growth. J. Math. Biol., 43(4), 291-312, 2001.
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Speaker: Théophile Dolmaire (Sorbonne University)
Title: The rigorous derivation of the Boltzmann equation: how to generalize Lanford's theorem in various domains? PDF ZOOM Abstract: It is well known that the Boltzmann equation models typically irreversible, macroscopic phenomena, although it can be formally obtained from a reversible description of the matter at a microscopic level. It is not until 1973 that this apparent paradox was properly addressed by Lanford, who provided the first rigorous derivation of the Boltzmann equation.
We will first describe the proof introduced by Lanford, in its modern version presented in GSRT[*]. This proof, providing an explicit rate of convergence, is performed in domains without boundary (namely: the whole Euclidean space, or the torus). We will then investigate how the proof of Lanford can be extended to obtain a rigorous derivation of the Boltzmann equation in a domain with boundary. We will start with the simplest case of a non-trivial domain: the half-space. In a second time, we will see how the proof dealing with the later case can be modified for the setting of particles evolving around a general, two-dimensional convex obstacle. Finally, if time allows it, we will discuss a work in progress concerning the derivation for particles contained in the disk.
GSRT[*]: From Newton to Boltzmann : hard spheres and short range potentials, I.Gallagher, L. Saint-Raymond, B. Texier, 2012.
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Speaker: Kevin Guillon (University of Bordeaux)
Title: A Fick's law recovering BGK model for a mixture of polyatomic gases PDF ZOOM Abstract: In this talk, we extend the derivation of the Fick-relaxation BGK model, to a polyatomic setting. The construction of the present model is based on the introduction of relaxation coefficients and by solving an entropy minimisation problem. The distribution functions of each species are described by adding a supplementary continous variable collecting vibrational and translational energies. Finally, by using a Chapmann-Engskog equation, we recover the Fick matrix, the volume viscosity and the shear viscosity
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Speaker: Laurent Laflêche (University of Texas)
Title: From many-body quantum dynamics with singular potentials to Vlasov equation PDF ZOOM Abstract: In this talk I will present several techniques and concepts used in the context of the mean-field and the semiclassical limit allowing to go from the quantum models to the classical mean-field equations of kinetic theory, linked to works in collaboration with Chiara Saffirio and Jacky Chong. The N-body Schrôdinger equation describes the motion of N interacting particles at the quantum scale. In the case of Fermions, when making N go to infinity in the mean-field regime, one obtains the so called Hartree-Fock equation. In parallel, one can also make the Planck constant h go to 0, leading to the Vlasov equation. In the case of singular potentials, to understand how close these equations are, one possibility is to use weak-strong uniqueness principles and understand the similarity of the models to prove propagation of a semiclassical notion of regularity independent of N and h
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Speaker: Matthieu Pauron (University of Bordeaux)
Title: A hyperbolic approach to the dead water phenomenon PDF AVI ZOOM Abstract: The dead water phenomenon is a mecanism observed when a boat moves at the surface of a stratified fluid: the perturbations created by the motion of the boat create internal waves that cause a force that opposes the motion of the boat. To model this phenomenon, we derive and study a system of nonlinear hyperbolic equations modeling the dynamics of two superposed layers of immiscible fluid of finite depth and different densities. Neglecting the waves at the surface of the fluid, one can assume that the upper surface is flat, except for a disturbance representing the bottom of the boat, and therefore moving at the speed of the boat. The resulting equations form a nonlinear hyperbolic nonconservative system. The position and speed of the boat are found by solving Newton's equation in which the force exerted by the fluid are determined by the resolution of the hyperbolic system. The full system is therefore a free boundary problem. We show that it can be reduced to an initial boundary value problem that we analyse mathematically and that also proves to be very convenient for numerical simulations.
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Speaker: Samir Salem (Ecole Polytechnique)
Title: An optimal transport approach of hypocoercivity for the 1d kinetic Fokker-Planck equation PDF ZOOM Abstract: A quadratic optimal transport metric on the set of probability measure over $\mathbb{R}^2$ is introduced. The quadratic cost is given by the euclidean norm on $\mathbb{R}^2$ associated to some well chosen symmetric positive matrix, which makes the metric equivalent to the usual Wasserstein-2 metric. The dissipation of the distance to the equilibrium along the kinetic Fokker-Planck flow, is bounded by below in terms of the distance itself. It enables to obtain some new type of trend to equilibrium estimate in Wasserstein-2 like metric, in the case of non-convex confinement potential.
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