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Derivation of the Vlasov-Maxwell system from the Maxwell-Schrödinger equations with extended charges

Nikolai Leopold, Chiara Saffirio

Abstract

We consider the Maxwell-Schrödinger equations in the Coulomb gauge describing the interaction of extended fermions with their self-generated electromagnetic field. They heuristically emerge as mean-field equations from non-relativistic quantum electrodynamics in a mean-field limit of many fermions. In the semiclassical regime, we establish the convergence of the Maxwell-Schrödinger equations for extended charges towards the non-relativistic Vlasov-Maxwell dynamics and provide explicit estimates on the accuracy of the approximation. To this end, we build a well-posedness and regularity theory for the Maxwell-Schrödinger equations and for the Vlasov-Maxwell system for extended charges.

Derivation of the Vlasov-Maxwell system from the Maxwell-Schrödinger equations with extended charges

Abstract

We consider the Maxwell-Schrödinger equations in the Coulomb gauge describing the interaction of extended fermions with their self-generated electromagnetic field. They heuristically emerge as mean-field equations from non-relativistic quantum electrodynamics in a mean-field limit of many fermions. In the semiclassical regime, we establish the convergence of the Maxwell-Schrödinger equations for extended charges towards the non-relativistic Vlasov-Maxwell dynamics and provide explicit estimates on the accuracy of the approximation. To this end, we build a well-posedness and regularity theory for the Maxwell-Schrödinger equations and for the Vlasov-Maxwell system for extended charges.
Paper Structure (37 sections, 18 theorems, 305 equations)

This paper contains 37 sections, 18 theorems, 305 equations.

Table of Contents

  1. Introduction
  2. Heuristic discussion
  3. Notation
  4. Organisation of the paper
  5. Main results
  6. Comparison with the literature
  7. Strategy of the proof
  8. Preliminaries
  9. Proof of Theorem \ref{['theorem:main theorem']}
  10. Quantization of the Vlasov--Maxwell equations
  11. Grönwall estimate
  12. The term \ref{['eq: Sobolev norm estimate omega 2']}:
  13. The term \ref{['eq: Sobolev norm estimate omega 3']}:
  14. The term \ref{['eq: Sobolev norm estimate omega 4']}:
  15. The term \ref{['eq: Sobolev norm estimate omega 5']}:
  16. ...and 22 more sections

Key Result

Proposition II.1

Let $\kappa$ satisfy Assumption assumption:cutoff function. For all $(\omega_0, \alpha_0) \in \textfrak{S}_{+}^{2,1} (L^2(\mathbb{R}^3)) \times \mathfrak{h}_{1/2} \cap \dot{\mathfrak{h}}_{-1/2}$ the Cauchy problem for the Maxwell--Schrödinger system eq:Maxwell-Schroedinger equations associated with of the system are conserved, i.e.

Theorems & Definitions (38)

  • Remark II.1
  • Remark II.2
  • Proposition II.1
  • Proposition II.2
  • Theorem II.1
  • Remark II.3
  • Lemma III.1
  • proof
  • Lemma III.2
  • proof
  • ...and 28 more