Talks (titles and abstracts)
Claude Bardos: About the quasilinear approximation for Vlasov equation and related equations
The quasilinear approximation for solutions \(f(t,x,v)\) of the Vlasov is a non linear diffusion equation of the form \(\delta_t F(t,v) -\nabla_v (D(\nabla G) \nabla_v G) =0\) with \(D(\nabla G(t,v)\) a diffusion coefficient which depends implicitly on \(\nabla_v G(t,v)\,. \) It is used in some circonstances to describe the total spatial (over the spatial domain \(\Omega_x\)) density of the solution of the Vlasov equation \(F(t,v)=\int_{\Omega_x} f(t,v,x)dx\,\).
As such it shares some similarity with the Balescu Lenard or the Fokker Planck equation but also some differences in its range of applications and in the way it can be derived.
In particular its main goal is to provide a link between linearly unstable regime and the Landau Damping.
This is following an ongoing project with Nicolas Besse, what I intend to discuss in this talk.
Jacob Bedrossian: Landau damping and related effects in kinetic models of plasma physics
In this talk I will attempt to overview past and recent progress in understanding asymptotic stability in collisionless or weakly collisional plasmas. Results such as Landau damping, plasma echoes, and collision-enhanced dissipation will be discussed followed by discussions of some open problems.
Yann Brenier: When Einstein's equations meet Kinetic Theory and Fluid Mechanics
Einstein's equations in vacuum can be recovered from a variational principle strikingly similar to the one needed to get the isothermal Euler equations with a correspondance between the cosmological constant and the speed of sound. This requires the introduction of a suitable phase space, just as in kinetic theory, to express the Einstein equations as a kind of generalized, matrix-valued and kinetic, version of the isothermal Euler equations.
José A. Carrillo: The Landau Equation as a Gradient Flow
The Landau equation is one of the most important partial differential equations in kinetic theory. It gives a description of colliding particles in plasma physics, and it can be formally derived as a limit of the Boltzmann equation when grazing collisions are dominant. The purpose of this talk is to propose a new perspective inspired from gradient flows for weak solutions of the spatially homogeneous Landau equation, which is in analogy with the relationship of the heat equation and the 2-Wasserstein metric gradient flow of the Boltzmann entropy. From the analytical viewpoint, we use the theory of metric measure spaces for the Landau equation by introducing a bespoke Landau distance. We show for a regularized version of the Landau equation that we can construct gradient flow solutions, curves of maximal slope, via the corresponding variational scheme. The main results obtained for the Landau equation shows that H-solutions with certain apriori bounds are equivalent to gradient flow solutions of the Landau equation, and that the grazing collision limit from Boltzmann to Landau can be obtained via a Gamma-convergence approach. From the numerical viewpoint, we aim at using this interpretation to derive a structure preserving deterministic particle method to solve efficiently the spatially homogeneous and inhomogenous Landau equation. This talk is based on a summary of works in collaboration with R. Bailo, M. Delgadino, L. Desvillettes, J. Hu, L. Wang, and J. Wu.
Peter Constantin: On the Nernst-Planck-Navier-Stokes system
The NPNS system models the macroscopic evolution of ionic concentrations in electrically conducting fluids. I will present some background and recent results.
Michele Coti Zelati: Orientation mixing in active suspensions
We study a popular kinetic model introduced by Saintillan and Shelley for the dynamics of suspensions of active elongated particles. We focus on the linear analysis of incoherence, that is on the linearized equation around the uniform distribution, in the regime of parameters corresponding to spectral (neutral) stability. We show that in the absence of rotational diffusion, the suspension experiences a mixing phenomenon similar to Landau damping. We show that this phenomenon persists for small rotational diffusion, and is combined with an enhanced dissipation at time scale at a faster time scale than the diffusive one.
Pierre Degond: Geometry and topology in collective dynamics models
Collective dynamics arises in systems of self-propelled particles and plays an important role in life sciences, from collectively migrating cells in an embryo to flocking birds or schooling fish. It has stimulated intense mathematical research in the last decade. Many different models have been proposed but most of them rely on point particles. In practice, particles often have more complex geometrical structures. Here, we will consider particles as rigid bodies whose body attitude is described by an orthonormal frame. Particles tend to align their frame with those of their neighbours. A hydrodynamic model will be derived when the number of particles is large. It will be used to exhibit solutions having non-trivial topology. We will investigate whether topology provides enhanced stability against perturbations, as observed in other systems such as topological insulators. This talk is based on recent results issued from collaborations with Antoine Diez, Amic Frouvelle, Sara Merino-Aceituno, Mingye Na and Ariane Trescases.
Laurent Desvillettes: Coupling kinetic equations and fluid mechanics equations: recent results on sprays
Sprays are complex multiphase flows in which a disperse phase of small volume interacts with an underlying gas. They can be modeled in different ways, including a coupling between a Vlasov-like equation and an Euler or Navier-Stokes system. This coupling can be done through a drag force (thin sprays) or through a drag force and a link between the volume fraction of each phase (thick sprays). We explain the state of the existing theory for existence and regularity for those couplings, with some emphasis on the difficult case of thick sprays.
Irene Gamba: Weak turbulence models for electron plasma flows by quasilinear particle systems
We discuss perturbation models associated to the evolution of a Maxwellian equilibria with a high energy anisotropy model by Vlasov Maxwell systems when that are expected to result in long time stability away for the dominant Maxwellian steady state.
These models are discussed in the unmagnetized and strongly magnetize framework. Recent developments from numerical simulations and analytical estimates help the understanding of conservation, entropy decay to strongly anisotropic states and overall stability properties for these type of weak perturbations dynamics from collisionless statistical equilibrium states.
In particular, under well prepared data, stable non-equilibrium state may emerge being as a solution of a steady mean field system of quasilinear diffusion model for electron particles in momenta space coupled to an spectral energy density wave or plasmon.
This is work in collaboration with Kun Huang and William Porteous, Michael Abdelmalik and Boris Breizman.
François Golse: Local Regularity for the Landau Equation (with Coulomb Interaction Potential)
The Landau equation is the kinetic equation describing collisions between charged particles interacting via the Coulomb potential. It has been proposed in 1936 by Landau as a substitute for the Boltzmann equation, since the Boltzmann collision integral cannot be defined when the particle interact via the Coulomb potential. To this date, whether there is finite time blow-up or global existence of classical solutions of the Cauchy problem for the space homogeneous Landau equation with Coulomb interaction potential remains an outstanding open problem in the mathematical analysis of kinetic models. This talk discusses the regularity of axisymmetric or radial weak solutions of the Landau equation. (with Cyril Imbert and Alexis Vasseur)
Thierry Goudon: Where does friction or dissipation come from?
It is quite intuitive to think of friction/dissipation as the result of the interactions between a particle and its complex environment. The formalization of this idea dates back to Caldeira-Leggett in the 80's, and it leads to couple particle's motion to a vibrational field, through momentum and energy exchanges. The starting point of this research, performed with S. De Bievre, R. Alonso, A. Vavasseur, L. Vivion, S. Rota Nodari, is a model introduced by S. De Bievre and L. Bruneau where a single classical particle interacts with a transverse wave equation. This model is revisited by considering several classical or quantum particles. We are particularly interested in asymptotic and stability issues.
Pierre-Emmanuel Jabin: A new approach to the mean-field limit of Vlasov-Poisson-Fokker-Planck
We introduce a novel approach to the mean-field limit of stochastic systems of interacting particles, leading to the first ever derivation of the mean-field limit to the Vlasov-Poisson-Fokker-Planck system for plasmas in dimension 2 and in higher dimensions with stronger restrictions on the initial distribution of particles. The method is broadly compatible with second order systems that lead to kinetic equations and it relies on novel estimates on the BBGKY hierarchy. By taking advantage of the diffusion in velocity, those estimates bound weighted L^p norms of the marginals or observables of the system, uniformly in the number of particles. This allows to treat very singular interaction kernels between the particles, including repulsive Poisson interactions.
This is a joint work with D. Bresch and J. Soler.
Juhi Jang: Gravitational collapse of gaseous stars
A classical model of a self-gravitating Newtonian star is given by the gravitational Euler-Poisson system, while a relativistic star is modeled by the Einstein-Euler system. I will review some recent progress on the local and global dynamics of Newtonian star solutions, and discuss mathematical construction of gravitational collapse of Newtonian and relativistic stars. For Newtonian stars, two distinct collapses will be presented: dust-like collapse and self-similar collapse, known as Larson-Penston solution for the isothermal stars and Yahil solution for polytropic stars, which show the existence of smooth initial data that lead to finite time collapse, characterized by the blow-up of the star density. If time permits, I will also discuss the relativistic analogue of Larson-Penston solutions and formation of naked singularities for the Einstein-Euler system.
Nader Masmoudi: tba
Clément Mouhot: Quantitative Geometric Control in Kinetic Theory
We consider a large class of linear kinetic equations combining transport and a linear collision operator on the kinetic variable, where the collision kernel is allowed to vanish on part of the spatial domain. We prove quantitative estimates of relaxation to equilibrium (spectral gap) under a geometric control condition similar to that used for wave equations. The proof is based on a novel elementary method based on trajectories and quantitative divergence inequalities, which sheds new light on the initial-boundary-value problem as well. This is a joint work with F. Hérau, H. Hutridurga and H. Dietert.
Toan Nguyen: Landau damping and plasma echoes
After recalling the classical notion of Landau damping and plasma echoes, the talk is to give an overview of the recent advances on the linear and nonlinear asymptotic stability of general spatially homogenous equilibria for the Vlasov-Poisson models. This is based on a joint program with Daniel Han-Kwan, Frederic Rousset, Emmanuel Grenier, and Igor Rodnianski.
Benoît Pausader: Stability of a point charge for the repulsive Vlasov-Poisson system
We consider solutions of the repulsive Vlasov-Poisson systems which are a combination of a point charge and a small density with respect to Liouville measure cloud, and we show that these solutions exists globally, that the electric field decay at an optimal rate and that the particle distribution converges along a modified scattering dynamics. This follows by a Lagrangian study of the linearized equation, which is integrated by means of an asymptotic action-angle coordinate transformation, and an Eulerian study of the nonlinear dynamic. This is joint work with Klaus Widmayer and Jiaqi Yang.
Nataša Pavlović: Two tales of a rigorous Derivation of the Hamiltonian Structure
Many mathematical works have focused on understanding the manner in which the dynamics of a nonlinear equation arises as an effective equation. By effective equation, we mean that solutions of the nonlinear equation approximate solutions to an underlying physical equation in some topology in a particular asymptotic regime.
- For example, the cubic nonlinear Schrodinger equation (NLS) is an effective equation for a system of N bosons interacting pairwise via a delta or approximate delta potential. In this talk, we will advance a new perspective on deriving an effective equation, which focuses on structure. In particular, we will show how the Hamiltonian structure for the cubic NLS in any dimension arises from corresponding structure at the N-particle level.
- On the other hand, the Vlasov equation in any spatial dimension has long been known to be an infinite-dimensional Hamiltonian system whose bracket structure is of Lie-Poisson type. In parallel, it is classical that the Vlasov equation is a mean-field limit for a pairwise interacting Newtonian system. Motivated by this knowledge, we provide a rigorous derivation of the Hamiltonian structure of the Vlasov equation, both the Hamiltonian functional and Poisson bracket, directly from the many-body problem. This work settles a question of Marsden, Morrison, and Weinstein on providing a "statistical basis" for the bracket structure of the Vlasov equation.
The talk is based on joint works with Dana Mendelson, Joseph Miller, Andrea Nahmod, Matthew Rosenzweig and Gigliola Staffilani.
Giuseppe Toscani: tba
Sijue Wu: The quartic integrability and long time existence of steep water waves in 2d
It is known since the work of Dyachenko & Zakharov in 1994 that for the weakly nonlinear 2d infinite depth water waves, there are no 3-wave interactions and all of the 4-wave interaction coefficients vanish on the resonant manifold. In this talk I will present a recent result that proves this partial integrability from a different angle. We construct a sequence of energy functionals \(\mathfrak E_j(t)\), directly in the physical space, that involves material derivatives of order \(j\) of the solutions for the 2d water wave equation, so that \(\frac d{dt} \mathfrak E_j(t)\) is quintic or higher order.
We show that if some scaling invariant norm, and a norm involving one spacial derivative above the scaling of the initial data are of size no more than \(\varepsilon\), then the lifespan of the solution for the 2d water wave equation is at least of order \(O(\varepsilon^{-3})\), and the solution remains as regular as the initial data during this time. If only the scaling invariant norm of the data is of size \(\varepsilon\), then the lifespan of the solution is at least of order \(O(\varepsilon^{-5/2})\).
Our long time existence results do not impose size restrictions on the slope of the initial interface and the magnitude of the initial velocity, they allow the interface to have arbitrary large steepnesses and initial velocities to have arbitrary large magnitudes.