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Modified Scattering of Solutions to the Relativistic Vlasov-Maxwell System Inside the Light Cone

Stephen Pankavich, Jonathan Ben-Artzi

Abstract

We consider the relativistic Vlasov-Maxwell system in three dimensions and study the limiting asymptotic behavior as $t \to \infty$ of solutions launched by small, compactly supported initial data. In particular, we prove that such solutions scatter to a modification of the free-streaming asymptotic profile. More specifically, we show that the spatial average of the particle distribution function converges to a smooth, compactly-supported limit and establish the precise, self-similar asymptotic behavior of the electric and magnetic fields, as well as, the macroscopic densities and their derivatives in terms of this limiting function. Upon constructing the limiting fields, a modified $L^\infty$ scattering result for the particle distribution function along the associated trajectories of free transport corrected by the limiting Lorentz force is then obtained. When the limiting charge density does not vanish, our estimates are sharp up to a logarithmic correction. However, when this quantity is identically zero in the limit, the limiting current density and fields may also vanish, which gives rise to decay rates that are faster than those attributed to the dispersive mechanisms in the system.

Modified Scattering of Solutions to the Relativistic Vlasov-Maxwell System Inside the Light Cone

Abstract

We consider the relativistic Vlasov-Maxwell system in three dimensions and study the limiting asymptotic behavior as of solutions launched by small, compactly supported initial data. In particular, we prove that such solutions scatter to a modification of the free-streaming asymptotic profile. More specifically, we show that the spatial average of the particle distribution function converges to a smooth, compactly-supported limit and establish the precise, self-similar asymptotic behavior of the electric and magnetic fields, as well as, the macroscopic densities and their derivatives in terms of this limiting function. Upon constructing the limiting fields, a modified scattering result for the particle distribution function along the associated trajectories of free transport corrected by the limiting Lorentz force is then obtained. When the limiting charge density does not vanish, our estimates are sharp up to a logarithmic correction. However, when this quantity is identically zero in the limit, the limiting current density and fields may also vanish, which gives rise to decay rates that are faster than those attributed to the dispersive mechanisms in the system.
Paper Structure (13 sections, 26 theorems, 426 equations, 2 figures)

This paper contains 13 sections, 26 theorems, 426 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Idea of the Proof
  4. Preliminary Lemmas
  5. Lower Regularity Estimates
  6. Characteristics
  7. Properties of the Translated Distribution Function
  8. Higher Regularity Estimates
  9. Field Limits and Modified Scattering
  10. Convergence of the Electric and Magnetic Fields
  11. Modified Scattering
  12. Proofs of Theorems
  13. The Non-Relativistic Vlasov-Maxwell System

Key Result

Theorem 1.1

For any $L > 0$ there exist $\epsilon_0 > 0$ and $\beta > 0$ with the following property. Let $f^\alpha_0 \in C^2$, $\alpha=1,\dots,N$, be non-negative functions supported on $\overline{\Gamma}_L \times \overline{\Gamma}_L$. Let $E_0, B_0 \in C^3$ be supported on $\overline{\Gamma}_L$ satisfying the then there exists a unique classical solution of (Vlasov) for all $x, p \in \mathbb{R}^3$ and $t \g

Figures (2)

  • Figure 1: Throughout this article, $p\in\mathbb{R}^3$ shall denote the momentum variable and $q=v(p)=p/p_0\in\Gamma_1$ shall denote the associated relativistic velocity.
  • Figure 2: Comparison of the sets enclosed by $L+\zeta t$ and $\gamma t$ where $0<\zeta<\gamma<1$.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 2.1: Glassey-Strauss GS
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • ...and 45 more