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Daniel Han-Kwan

Publications and source records attributed to Daniel Han-Kwan.

At least 19 recordsLinked to original sources
arXiv2025

Semiclassical limit of cubic nonlinear Schr\"odinger equations for mixed states

In this work, we study the semiclassical limit of cubic Nonlinear Schr\"odinger equations for mixed states. We justify the limit to a singular Vlasov equation (in which the force field is proportional to the gradient of the density), for data with finite Sobolev regularity whose velocity profiles satisfy a quantum Penrose stability condition. This latter condition is always satisfied for small data (with a smallness condition independent of the semiclassical parameter) both in the focusing and the defocusing case, and for small perturbations of a large class of physically relevant examples in the defocusing case, such as local Maxwellian-like profiles.

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arXiv2024

Nonlinear chaotic Vlasov equations

In this article, we study nonlinear Vlasov equations with a smooth interaction kernel on a compact manifold without boundary where the geodesic flow exhibits strong chaotic behavior, known as the Anosov property. We show that, for small initial data with finite regularity and supported away from the null section, there exist global solutions to the nonlinear Vlasov equations which weakly converge to an equilibrium of the free transport equation, and whose potential strongly converges to zero, both with exponential speed. Central to our approach are microlocal anisotropic Sobolev spaces, originally developed for studying Pollicott-Ruelle resonances, that we further refine to deal with the geometry of the full cotangent bundle, which paves the way to the analysis of nonlinear Vlasov equations.

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arXiv2024

Linear Landau damping for the Vlasov-Maxwell system in $\mathbb{R}^3$

In this work, we consider the relativistic Vlasov-Maxwell system, linearized around a spatially homogeneous equilibrium, set in the whole space $\mathbb{R}^3 \times \mathbb{R}^3$. The equilibrium is assumed to belong to a class of radial, smooth, rapidly decaying functions. Under appropriate conditions on the initial data, we prove algebraic decay (of dispersive nature) for the electromagnetic field. For the electric scalar potential, the leading behavior is driven by a dispersive wave packet with non-degenerate phase and compactly supported amplitude, while for the magnetic vector potential, it is driven by a wave packet whose phase behaves globally like the one of Klein-Gordon and the amplitude has unbounded support.

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arXiv2023

On well-posedness for thick spray equations

In this paper, we prove the local in time well-posedness of thick spray equations in Sobolev spaces, for initial data satisfying a Penrose-type stability condition. This system is a coupling between particles described by a kinetic equation and a surrounding fluid governed by compressible Navier-Stokes equations. In the thick spray regime, the volume fraction of the dispersed phase is not negligible compared to that of the fluid. We identify a suitable stability condition bearing on the initial conditions that provides estimates without loss, ensuring that the system is well-posed. It coincides with a Penrose condition appearing in earlier works on singular Vlasov equations. We also rely on crucial new estimates for averaging operators. Our approach allows to treat many variants of the model, such as collisions in the kinetic equation, non-barotropic fluid or density-dependent drag force.

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arXiv2022

Concentration versus absorption for the Vlasov-Navier-Stokes system on bounded domains

We study the large time behavior of small data solutions to the Vlasov-Navier-Stokes system set on $Ω\times \mathbb{R}^3$, for a smooth bounded domain $Ω$ of $\mathbb{R}^3$, 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 centered at $0$, with an exponential rate. The proof, which follows the methods introduced in [Han-Kwan - Moussa - Moyano, arXiv:1902.03864v2], requires a careful analysis of the boundary effects. We also exhibit examples of classes of initial data leading to a variety of asymptotic behaviors for the kinetic density, from total absorption to no absorption at all.

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arXiv2021

On hydrodynamic limits of the Vlasov-Navier-Stokes system

We introduce a framework to justify hydrodynamic limits of the Vlasov-Navier-Stokes system. We specifically study high friction regimes, which take into account the fact that particles of the dispersed phase are light (resp. small) compared to the fluid part, and lead to the derivation of Transport-Navier-Stokes (resp. Inhomogeneous Navier-Stokes) systems.

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arXiv2020

From Newton's second law to Euler's equations of perfect fluids

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.

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arXiv2020

Large time behavior of small data solutions to the Vlasov-Navier-Stokes system on the whole space

We study the large time behavior of small data solutions to the Vlasov-Navier-Stokes system on $\R^3 \times \R^3$. We prove that the kinetic distribution function concentrates in velocity to a Dirac mass supported at $0$, while the fluid velocity homogenizes to $0$, both at a polynomial rate. The proof is based on two steps, following the general strategy laid out in \cite{HKMM}: (1) the energy of the system decays with polynomial rate, assuming a uniform control of the kinetic density, (2) a bootstrap argument allows to obtain such a control. This last step requires a fine understanding of the structure of the so-called Brinkman force, which follows from a family of new identities for the dissipation (and higher versions of it) associated to the Vlasov-Navier-Stokes system.

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arXiv2020

The non-relativistic limit of the Vlasov-Maxwell system with uniform macroscopic bounds

We study in this paper the non-relativistic limit from Vlasov-Maxwell to Vlasov-Poisson, which corresponds to the regime where the speed of light is large compared to the typical velocities of particles. In contrast with \cite{Asano-Ukai-86-SMA}, \cite{Degond-86-MMAS}, \cite{Schaeffer-86-CMP} which handle the case of classical solutions, we consider measure-valued solutions, whose moments and electromagnetic fields are assumed to satisfy some uniform bounds. To this end, we use a functional inspired by the one introduced by Loeper in his proof of uniqueness for the Vlasov-Poisson system \cite{Loeper-2006}. We also build a special class of measure-valued solutions, that enjoy no higher regularity with respect to the momentum variable, but whose moments and electromagnetic fields satisfy all required conditions to enter our framework.

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arXiv2019

Large time behavior of the Vlasov-Navier-Stokes system on the torus

We study the large time behavior of Fujita-Kato type solutions to the Vlasov-Navier-Stokes system set on $\mathbb{T}^3 \times \mathbb{R}^3$. 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 to get global bounds on moments.

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arXiv2019

Asymptotic stability of equilibria for screened Vlasov-Poisson systems via pointwise dispersive estimates

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.

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arXiv2018

On propagation of higher space regularity for non-linear Vlasov equations

This work is concerned with the broad question of propagation of regularity for smooth solutions to non-linear Vlasov equations. For a class of equations (that includes Vlasov-Poisson and relativistic Vlasov-Maxwell), 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 in the context of the quasineutral limit of the Vlasov-Poisson system in [D. Han-Kwan and F. Rousset, Ann. Sci. École Norm. Sup., 2016].

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arXiv2017

Uniqueness of the solution to the 2D Vlasov-Navier-Stokes system

We prove a uniqueness result for weak solutions to the Vlasov-Navier-Stokes system in two dimensions, both in the whole space and in the periodic case, under a mild decay condition on the initial distribution function. The main result is achieved by combining methods from optimal transportation (introduced in this context by G. Loeper) with the use of Hardy's maximal function, in order to obtain some fine Wassestein-like estimates for the difference of two solutions of the Vlasov equation.

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arXiv2017

Long time estimates for the Vlasov-Maxwell system in the non-relativistic limit

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.

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arXiv2017

Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits

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 $μ$ of Vlasov-Poisson in arbitrarily high Sobolev norms, but become of order one away from $μ$ 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.

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arXiv2016

The Vlasov-Navier-Stokes system in a 2D pipe: existence and stability of regular equilibria

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.

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arXiv2016

Ill-posedness of the hydrostatic Euler and singular Vlasov equations

In this paper, we develop an abstract framework to establish ill-posedness in the sense of Hadamard for some nonlocal PDEs displaying unbounded unstable spectra. We apply it to prove the ill-posedness for the hydrostatic Euler equations as well as for the kinetic incompressible Euler equations and the Vlasov-Dirac-Benney system.

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