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Arbitrary Polynomial Decay Rates of Neutral, Collisionless Plasmas

Grace Mattingly, Stephen Pankavich, Jonathan Ben-Artzi

TL;DR

This work analyzes the multispecies Vlasov-Poisson system in $\mathbb{R}^3$ for neutral plasmas ($\mathcal{M}=0$) and shows that, under a rapid decay assumption for the electric field, one can realize solutions with arbitrary polynomial decay rates for the charge density $\rho$ and field $E$, accompanied by a countable family of asymptotic profiles. The authors develop a polyhomogeneous expansion framework via translated distribution functions $g^\alpha=f^\alpha(t,x+vt,v)$, introducing higher-order limiting objects $F^{\alpha,\ell}_\infty$, $\rho_{\ell,\infty}$, and $E_{\ell,\infty}$, and prove convergence of these objects along with linear $L^{\infty}$ scattering for all particle distributions. The main technical engine combines Taylor expansions in the translated variables, precise control of transported moments, and Gronwall-type arguments to propagate finite-time estimates into an infinite hierarchy of asymptotic profiles. This yields a robust, constructive description of self-similar decay regimes beyond the classical dispersive rates, clarifying how charge cancellation and translated dynamics generate sharper long-time behavior in neutral plasmas with multiple species.

Abstract

A multispecies, collisionless plasma is modeled by the Vlasov-Poisson system. Assuming the plasma is neutral and the electric field decays with sufficient rapidity as $t \to\infty$, we show that solutions can be constructed with arbitrarily fast, polynomial rates of decay, depending upon the properties of the limiting spatial average and its derivatives. In doing so, we establish, for the first time, a countably infinite number of asymptotic profiles for the charge density, electric field, and their derivatives, each of which is necessarily realized by a sufficiently smooth solution and exceeds the established dispersive decay rates. Finally, in each case we establish a linear $L^\infty$ scattering result for every particle distribution function, namely we show that they converge as $t \to \infty$ along the transported spatial characteristics at increasingly faster rates.

Arbitrary Polynomial Decay Rates of Neutral, Collisionless Plasmas

TL;DR

This work analyzes the multispecies Vlasov-Poisson system in for neutral plasmas () and shows that, under a rapid decay assumption for the electric field, one can realize solutions with arbitrary polynomial decay rates for the charge density and field , accompanied by a countable family of asymptotic profiles. The authors develop a polyhomogeneous expansion framework via translated distribution functions , introducing higher-order limiting objects , , and , and prove convergence of these objects along with linear scattering for all particle distributions. The main technical engine combines Taylor expansions in the translated variables, precise control of transported moments, and Gronwall-type arguments to propagate finite-time estimates into an infinite hierarchy of asymptotic profiles. This yields a robust, constructive description of self-similar decay regimes beyond the classical dispersive rates, clarifying how charge cancellation and translated dynamics generate sharper long-time behavior in neutral plasmas with multiple species.

Abstract

A multispecies, collisionless plasma is modeled by the Vlasov-Poisson system. Assuming the plasma is neutral and the electric field decays with sufficient rapidity as , we show that solutions can be constructed with arbitrarily fast, polynomial rates of decay, depending upon the properties of the limiting spatial average and its derivatives. In doing so, we establish, for the first time, a countably infinite number of asymptotic profiles for the charge density, electric field, and their derivatives, each of which is necessarily realized by a sufficiently smooth solution and exceeds the established dispersive decay rates. Finally, in each case we establish a linear scattering result for every particle distribution function, namely we show that they converge as along the transported spatial characteristics at increasingly faster rates.

Paper Structure

This paper contains 18 sections, 16 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Review of Previous Results
  4. Modified Scattering
  5. Polyhomogeneous Expansions
  6. Notation
  7. Main Results
  8. Strategy of the Proof
  9. Preliminary Lemmas
  10. Further Notation and Preliminary Estimates
  11. Preliminary Bounds on the Particle Density and the Field
  12. Bounds on Derivatives of $F$
  13. Improved Bounds on Gradients of the Particle Density and the Field
  14. Improved Bounds on the Particle Density
  15. Proofs of Theorems
  16. ...and 3 more sections

Key Result

Theorem 1.1

Consider a solution $f^\alpha\in C^2((0,\infty)\times\mathbb{R}^6)$ of VP with initial data $f^\alpha_0 \in C_c^2(\mathbb{R}^6)$. Assume that Assumption holds. Then, we have the following:

Theorems & Definitions (38)

  • Theorem 1.1: Pankavich2022
  • Theorem 1.2: Pankavich2022
  • Theorem 1.3
  • Remark 1.1
  • Theorem 1.4
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • ...and 28 more