skip to main content

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


This content will become publicly available on June 1, 2025

Title: Nonlinear Landau Damping for the Vlasov–Poisson System in $$\mathbb {R}^3$$: The Poisson Equilibrium
We prove asymptotic stability of the Poisson homogeneous equilibrium among solu- tions of the Vlasov–Poisson system in the Euclidean space R3. More precisely, we show that small, smooth, and localized perturbations of the Poisson equilibrium lead to global solutions of the Vlasov–Poisson system, which scatter to linear solutions at a polynomial rate as t → ∞. The Euclidean problem we consider here differs signif- icantly from the classical work on Landau damping in the periodic setting, in several ways. Most importantly, the linearized problem cannot satisfy a “Penrose condition”. As a result, our system contains resonances (small divisors) and the electric field is a superposition of an electrostatic component and a larger oscillatory component, both with polynomially decaying rates.  more » « less
Award ID(s):
2154162
PAR ID:
10504458
Author(s) / Creator(s):
; ; ;
Publisher / Repository:
Springer
Date Published:
Journal Name:
Annals of PDE
Volume:
10
Issue:
1
ISSN:
2524-5317
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ; ( , Peking Mathematical Journal)
    null (Ed.)
    Abstract: We construct (modified) scattering operators for the Vlasov–Poisson system in three dimensions, mapping small asymptotic dynamics as t→ - ∞ to asymptotic dynamics as t → + ∞. The main novelty is the construction of modified wave operators, but we also obtain a new simple proof of modified scattering. Our analysis is guided by the Hamiltonian structure of the Vlasov–Poisson system. Via a pseudo-conformal inversion, we recast the question of asymptotic behavior in terms of local in time dynamics of a new equation with singular coefficients which is approximately integrated using a generating function. 
    more » « less
  2. ; ; ; ( , International Mathematics Research Notices)
    Abstract We prove small data modified scattering for the Vlasov–Poisson system in dimension $d=3$, using a method inspired from dispersive analysis. In particular, we identify a simple asymptotic dynamics related to the scattering mass. 
    more » « less
  3. ; ( , Communications in Mathematical Physics)
    null (Ed.)
    Abstract We consider the Vlasov–Poisson system with repulsive interactions. For initial data a small, radial, absolutely continuous perturbation of a point charge, we show that the solution is global and disperses to infinity via a modified scattering along trajectories of the linearized flow. This is done by an exact integration of the linearized equation, followed by the analysis of the perturbed Hamiltonian equation in action-angle coordinates. 
    more » « less
  4. ; ; ( , Inverse Problems)
    Abstract

    We study the inverse problem of recovering the doping profile in the stationary Vlasov–Poisson equation, given the knowledge of the incoming and outgoing measurements at the boundary of the domain. This problem arises from identifying impurities in the semiconductor manufacturing. Our result states that, under suitable assumptions, the doping profile can be uniquely determined through an asymptotic formula of the electric field that it generates.

     
    more » « less
  5. ; ( , Physical Review E)
    We present a method for solving the linearized Vlasov-Poisson equation, based on analyticity properties of the equilibrium and initial condition through Cauchy-type integrals, that produces algebraic expressions for the distribution and field, i.e., the solution is expressed without integrals. Standard extant approaches involve deformations of the Bromwich contour that give erroneous results for certain physically reasonable configurations or eigenfunction expansions that are misleading as to the temporal structure of the solution. Our method is more transparent, lacks these defects, and predicts previously unrecognized behavior. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email