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We study the linearized Vlasov–Poisson system around suitably stable homogeneous equilibria on $${\mathbb {R}}^d\times {\mathbb {R}}^d$$ (for any $$d \ge 1$$ ) and establish dispersive $$L^\infty $$ decay estimates in the physical space.

We study the large time behaviour of small data solutions to the Vlasov–Navier–Stokes system set on Ω × R 3 , for a smooth bounded domain Ω of R 3 , with homogeneous Dirichlet boundary condition for the fluid and absorption boundary condition for the kinetic phase. We prove that the fluid velocity homogenizes to 0 while the distribution function concentrates towards a Dirac mass in velocity centred at 0, with an exponential rate. The proof, which follows the methods introduced in Han-Kwan et al (2020 Arch. Ration. Mech. Anal. 236 1273–323), requires a careful analysis of the boundary effects. We also exhibit examples of classes of initial data leading to a variety of asymptotic behaviours for the kinetic density, from total absorption to no absorption at all.

Vlasov equations can be formally derived from N-body dynamics in the mean-field limit. In some suitable singular limits, they may themselves converge to fluid dynamics equations. Motivated by this heuristic, we introduce natural scalings under which the incompressible Euler equations can be rigorously derived from N-body dynamics with repulsive Coulomb interaction. Our analysis is based on the modulated energy methods of Brenier and Serfaty., Minor typos corrected

International audience; We study the large time behavior of Fujita–Kato type solutions to the Vlasov–Navier–Stokes system set on $\T^3 \times \R^3$. Under the assumption that the initial so-called modulated energy is small enough, we prove that the distribution function converges to a Dirac mass in velocity, with exponential rate. The proof is based on the fine structure of the system and on a bootstrap analysis allowing us to get global bounds on moments.

We revisit the proof of Landau damping near stable homogenous equilibria of Vlasov-Poisson systems with screened interactions in the whole space $\mathbb{R}^d$ (for $d\geq3$) that was first established by Bedrossian, Masmoudi and Mouhot. Our proof follows a Lagrangian approach and relies on precise pointwise in time dispersive estimates in the physical space for the linearized problem that should be of independent interest. This allows to cut down the smoothness of the initial data required in Bedrossian at al. (roughly, we only need Lipschitz regularity). Moreover, the time decay estimates we prove are essentially sharp, being the same as those for free transport, up to a logarithmic correction., Comment: 25 pages, minor typos fixed

International audience; In this paper, we study the Vlasov-Maxwell system in the non-relativistic limit, that is in the regime where the speed of light is a very large parameter. We consider data lying in the vicinity of homogeneous equilibria that are stable in the sense of Penrose (for the Vlasov-Poisson system), and prove Sobolev stability estimates that are valid for times which are polynomial in terms of the speed of light and of the inverse of size of initial perturbations. We build a kind of higher-order Vlasov-Darwin approximation which allows us to reach arbitrarily large powers of the speed of light.

We prove local exact controllability in arbitrary short time of the two-dimensional incompressible Euler equation with free surface, in the case with surface tension. This proves that one can generate arbitrary small amplitude periodic gravity-capillary water waves by blowing on a localized portion of the free surface of a liquid., Comment: 71 pages

This work is concerned with the broad question of propagation of regularity for smooth solutions to nonlinear Vlasov equations. For a class of equations (that includes Vlasov–Poisson and relativistic Vlasov–Maxwell systems), we prove that higher regularity in space is propagated, locally in time, into higher regularity for the moments in velocity of the solution. This in turn can be translated into some anisotropic Sobolev higher regularity for the solution itself, which can be interpreted as a kind of weak propagation of space regularity. To this end, we adapt the methods introduced by D. Han-Kwan and F. Rousset (Ann. Sci. Ecole Norm. Sup. 49:6 (2016) 1445–1495) in the context of the quasineutral limit of the Vlasov–Poisson system.

We introduce a long wave scaling for the Vlasov-Poisson equation and derive, in the cold ions limit, the Korteweg-De Vries equation (in 1D) and the Zakharov-Kuznetsov equation (in higher dimensions, in the presence of an external magnetic field). The proofs are based on the relative entropy method.

International audience; In this paper, we study the Vlasov-Navier-Stokes system in a 2D pipe with partially absorbing boundary conditions. We show the existence of stationary states for this system near small Poiseuille flows for the fluid phase, for which the kinetic phase is not trivial. We prove the asymptotic stability of these states with respect to appropriately compactly supported perturbations. The analysis relies on geometric control conditions which help to avoid any concentration phenomenon for the kinetic phase.

Consider a system of $N$ particles interacting through Newton's second law with Coulomb interaction potential in one spatial dimension or a $\mathcal{C}^2$ smooth potential in any dimension. We prove that in the mean field limit $N \to + \infty$, the $N$ particles system displays instabilities in times of order $\log N$ for some configurations approximately distributed according to unstable homogeneous equilibria., Comment: minor typos corrected; Journal of Statistical Physics, accepted

We study the instability of solutions to the relativistic Vlasov--Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit, $\varepsilon \to 0$, with $\varepsilon$ being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution $\mu$ of Vlasov--Poisson systems in arbitrarily high Sobolev norms, but become of order one away from $\mu$ in arbitrary negative Sobolev norms within time of order $|\log \varepsilon|$. Second, we deduce the invalidity of the quasineutral limit in $L^2$ in arbitrarily short time.

We study a two-fluid description of high and low temperature components of the electron velocity distribution of an idealized tokamak plasma. We refine previous results on the laminar steady-state solution. On the one hand, we prove global stability outside a parameter set of possible linear instability. On the other hand, for a large set of parameters, we prove the primary instabilities for varying temperature difference stem from the lowest spatial harmonics. We moreover show that any codimension-one bifurcation is a supercritical Andronov-Hopf bifurcation, which yields stable periodic solutions in the form of traveling waves. In the degenerate case, where the instability region in the temperature difference is a point, we prove that the bifurcating periodic orbits form an arc of stable periodic solutions. We provide numerical simulations to illustrate and corroborate our analysis. These also suggest that the stable periodic orbit, which bifurcated from the steady-state, undergoes additional bifurcations., 25 pages

International audience; In this paper, we study the quasineutral limit of the isothermal Euler-Poisson equation for ions, in a domain with boundary. This is a follow-up to our previous work \cite{GVHKR}, devoted to no-penetration as well as subsonic outflow boundary conditions. We focus here on the case of supersonic outflow velocities. The structure of the boundary layers and the stabilization mechanism are different.

This work is devoted to the analysis of the linear Boltzmann equation on the torus, in the presence of a force deriving from a potential. The collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity. We study how the association of transport and collision phenomena can lead to convergence to equilibrium, using concepts and ideas from control theory. We prove two main classes of results. On the one hand, we show that convergence towards an equilibrium is equivalent to an almost everywhere geometric control condition. The equilibria (which are not necessarily Maxwellians with our general assumptions on the collision kernel) are described in terms of the equivalence classes of an appropriate equivalence relation involving transport and collisions. On the other hand, we characterize the exponential convergence to equilibrium in terms of the Lebeau constant, which involves some averages of the collision frequency along the flow of the transport. We also explain how to handle the case of linear Boltzmann equations posed on the phase space associated to a compact Riemannian manifold without boundary.

This work is concerned with the quasineutral limit of the one-dimensional Vlasov-Poisson equation, for initial data close to stationary homogeneous profiles. Our objective is threefold: first, we provide a proof of the fact that the formal limit does not hold for homogeneous profiles that satisfy the Penrose instability criterion. Second, we prove on the other hand that the limit is true for homogeneous profiles that satisfy some monotonicity condition, together with a symmetry condition. We handle the case of well-prepared as well as ill- prepared data. Last, we study a stationary boundary-value problem for the formal limit, the so-called quasineutral Vlasov equation. We show the existence of numerous stationary states, with a lot of freedom in the construction (compared to that of BGK waves for Vlasov-Poisson): this illustrates the degeneracy of the limit equation., 50 pages

We study the quasineutral limit of the isothermal Euler-Poisson system describing a plasma made of ions and massless electrons. The analysis is achieved in a domain of $\R^3$ and thus extends former results by Cordier and Grenier [Comm. Partial Differential Equations, 25 (2000), pp.~1099--1113], who dealt with the same problem in a one-dimensional domain without boundary., Comment: 32 pages

The purpose of this note is to extend the $L^1$ averaging lemma of Golse and Saint-Raymond \cite{GolSR} to the case of a kinetic transport equation with a force field $F(x)\in W^{1,\infty}$. To this end, we will prove a local in time mixing property for the transport equation $\partial_t f + v.\nabla_x f + F.\nabla_v f =0$., Comment: 15 pages, to be published in Kinetic and Related Models

In this paper, we study the quasineutral limit (in other words the limit when the Debye length tends to zero) of Vlasov-Poisson like equations describing the behaviour of ions in a plasma. We consider massless electrons, with a charge density following a Maxwell-Boltzmann law. For cold ions, using the relative entropy method, we derive the classical Isothermal Euler or the (inviscid) Shallow Water systems from fluid mechanics. In a second time, we study the combined quasineutral and strong magnetic field regime for such plasmas., Comment: 37 pages

This paper is concerned with the analysis of a mathematical model arising in plasma physics, more specifically in fusion research. It directly follows \cite{DHK1}, where the tri-dimensional analysis of a Vlasov-Poisson equation with finite Larmor radius scaling was led, corresponding to the case of ions with massless electrons whose density follows a linearized Maxwell-Boltzmann law. We now consider the case of electrons in a background of fixed ions, which was only sketched in \cite{DHK1}. Unfortunately, there is evidence that the formal limit is false in general. Nevertheless, we formally derive a fluid system for particular monokinetic data. We prove the local in time existence of analytic solutions and rigorously study the limit (when the Debye length vanishes) to a new anisotropic fluid system. This is achieved thanks to Cauchy-Kovalevskaya type techniques, as introduced by Caflisch \cite{Caf} and Grenier \cite{Gre1}., Comment: 28 pages

The goal of this paper is to understand from a mathematical point of view the magnetic confinement of plasmas for fusion. Following Fr\'enod and Sonnendr\"ucker \cite{FS2}, we first use two-scale convergence tools to derive a gyrokinetic system for a plasma submitted to a large magnetic field with a slowly spatially varying intensity. We formally derive from this system a simplified bi-temperature fluid system. We then investigate the behaviour of the plasma in such a regime and we prove nonlinear stability or instability depending on which side of the tokamak we are looking at. In our analysis, we will also point out that there exists a temperature gradient threshold beyond which one can expect stability, even in the "bad" side : this corresponds to the so-called H-mode., Comment: 31 pages, accepted for publication in SIAM J. Math. Anal.

Following Frenod and Sonnendrucker (SIAM J. Math. Anal. 32(6) (2001) 1227-1247), we consider the finite Larmor radius regime for a plasma submitted to a large magnetic field and take into account both the quasineutrality and the local thermodynamic equilibrium of the electrons. We then rigorously establish the asymptotic gyrokinetic limit of the rescaled and modified Vlasov-Poisson system in a three-dimensional setting with the help of an averaging lemma.