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Propagation of Velocity Moments for the Magnetized Vlasov-Poisson System with Space-Time Dependent Magnetic Fields

Immanuel Ben Porat, Antoine Gagnebin, Mikaela Iacobelli, Jonathan Junné

TL;DR

This work analyzes the propagation of velocity moments for the 2D magnetized Vlasov-Poisson system and the 3D magnetized screened Vlasov-Poisson equation with a space-time dependent external magnetic field. It develops a moment-based framework combining Eulerian and velocity moments, kinetic interpolation inequalities, and the no-work property of magnetic forces to obtain global moment propagation in 2D and a two-stage strategy (short-time for $n=3,4$ and long-time via $L_4$) in 3D, along with propagation of regularity. It also improves stability estimates in the kinetic Wasserstein distance, achieving an optimal $\sqrt{\log}$ rate by leveraging the no-work identity, and thereby establishing global classical solutions under explicit decay and smallness conditions. The results bridge the non-magnetized theory and magnetized dynamics, providing rigorous control of moments, regularity, and stability for plasmas with non-uniform magnetic fields.

Abstract

We prove that polynomial velocity moments of solutions to the 2D magnetized Vlasov-Poisson system and the 3D magnetized screened Vlasov-Poisson equation remain finite for all times, provided they are finite initially, even when the external magnetic field $B=B(t,x)$ is space-time dependent. We deduce propagation of regularity, thereby implying the existence of global classical solutions. Moreover, we prove optimal stability estimates in the kinetic-Wasserstein distance on par with the unmagnetised case.

Propagation of Velocity Moments for the Magnetized Vlasov-Poisson System with Space-Time Dependent Magnetic Fields

TL;DR

This work analyzes the propagation of velocity moments for the 2D magnetized Vlasov-Poisson system and the 3D magnetized screened Vlasov-Poisson equation with a space-time dependent external magnetic field. It develops a moment-based framework combining Eulerian and velocity moments, kinetic interpolation inequalities, and the no-work property of magnetic forces to obtain global moment propagation in 2D and a two-stage strategy (short-time for and long-time via ) in 3D, along with propagation of regularity. It also improves stability estimates in the kinetic Wasserstein distance, achieving an optimal rate by leveraging the no-work identity, and thereby establishing global classical solutions under explicit decay and smallness conditions. The results bridge the non-magnetized theory and magnetized dynamics, providing rigorous control of moments, regularity, and stability for plasmas with non-uniform magnetic fields.

Abstract

We prove that polynomial velocity moments of solutions to the 2D magnetized Vlasov-Poisson system and the 3D magnetized screened Vlasov-Poisson equation remain finite for all times, provided they are finite initially, even when the external magnetic field is space-time dependent. We deduce propagation of regularity, thereby implying the existence of global classical solutions. Moreover, we prove optimal stability estimates in the kinetic-Wasserstein distance on par with the unmagnetised case.
Paper Structure (10 sections, 19 theorems, 225 equations)

This paper contains 10 sections, 19 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Propagation of moments
  3. Preliminaries
  4. The case $d=2$
  5. The case $d=3$
  6. Short time propagation of moments of order $n=3,4$
  7. Long time propagation of moments of order $4$
  8. Long time estimate for moments of arbitrary order
  9. Propagation of regularity
  10. Improved stability estimate

Key Result

Theorem 1.1

Assume that $K$ is the kernel as in Coulomb def. Let $0\leq f^{\mathrm{in}}\in L^{1}(\mathbb{R}^{d}\times\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d})$ and suppose there exist $\mathbf{M}^{\mathrm{in}}_{k}>0$ for $k=0,\dots,n$ such that Let $f$ be a classical solution to MAGNETIC Vlasov Intro on $[0,T]\times\mathbb{R}^{d}$ with initial data $f^{\mathrm{in}}$. Then,

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 22 more