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A new framework for particle-wave interaction

Toan T. Nguyen

Abstract

In plasma physics, collisionless charged particles are transported following the dynamics of a meanfield Vlasov equation with a self-consistent electric field generated by the charge density. Due to the long range interaction between particles, the generating electric field oscillates and disperses like a Klein-Gordon dispersive wave, known in the physical literature as plasma oscillations or Langmuir's oscillatory waves. The oscillatory electric field then in turn drives particles. Despite its great physical importance, the question of whether such a nonlinear particle-wave interaction would remain regular globally and be damped in the large time has been an outstanding open problem. In this paper, we propose a new framework to resolve this exact nonlinear interaction. Specifically, we employ the framework to establish the large time behavior and scattering of solutions to the nonlinear Vlasov-Klein-Gordon system in the small initial data regime. The novelty of this work is to provide a detailed physical space description of particles moving in an oscillatory field and to resolve oscillations for the electric field generated by the collective interacting particles. This appears to be the first such a result analyzing oscillations in the physical phase space $\mathbb{R}^3_x\times \mathbb{R}_v^3$.

A new framework for particle-wave interaction

Abstract

In plasma physics, collisionless charged particles are transported following the dynamics of a meanfield Vlasov equation with a self-consistent electric field generated by the charge density. Due to the long range interaction between particles, the generating electric field oscillates and disperses like a Klein-Gordon dispersive wave, known in the physical literature as plasma oscillations or Langmuir's oscillatory waves. The oscillatory electric field then in turn drives particles. Despite its great physical importance, the question of whether such a nonlinear particle-wave interaction would remain regular globally and be damped in the large time has been an outstanding open problem. In this paper, we propose a new framework to resolve this exact nonlinear interaction. Specifically, we employ the framework to establish the large time behavior and scattering of solutions to the nonlinear Vlasov-Klein-Gordon system in the small initial data regime. The novelty of this work is to provide a detailed physical space description of particles moving in an oscillatory field and to resolve oscillations for the electric field generated by the collective interacting particles. This appears to be the first such a result analyzing oscillations in the physical phase space .

Paper Structure

This paper contains 21 sections, 16 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Nonlinear framework
  4. Characteristics
  5. Nonlinear electric field
  6. Nonlinear iterative scheme
  7. The bootstrap argument
  8. Characteristics
  9. Introduction
  10. Collective electric fields
  11. Velocity description
  12. Characteristic description
  13. Derivative bounds
  14. Decay for higher derivatives
  15. Boundedness for top derivatives
  16. ...and 6 more sections

Key Result

Theorem 1.1

Fix $\alpha_0\ge 8$, and let $f_0(x,v), \phi_0(x), \phi_1(x)$ be initial data of the relativistic Vlasov-Klein-Gordon system VKG1-VKG3. Suppose that $f_0(x,v)$ is compactly supported in $v$, and in addition, Then, for sufficiently small $\epsilon_0$, the solution $(f(t,x,v), E(t,x))$ to the relativistic Vlasov-Klein-Gordon system VKG1-VKG3 exists globally in time, and the nonlinear electric field

Theorems & Definitions (32)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Corollary 2.2: Nonlinear electric field
  • proof
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Proposition 3.3
  • ...and 22 more