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Local Wellposedness and Global Weak Solutions of the Pauli-Darwin/Poisswell Equations

Pierre Germain, Norbert J. Mauser, Jakob Möller

TL;DR

The authors establish the first rigorous local well-posedness results in $H^s(\mathbb{R}^3)$ with $s>\tfrac{3}{2}$ and global finite-energy weak solutions for the Pauli-Darwin and Pauli-Poisswell semi-relativistic Pauli systems coupled to self-consistent Maxwell-type fields. They develop a parabolic regularization that yields energy dissipation, derive robust elliptic bounds for the potentials $V$ and $A$, and employ compactness (Aubin–Lions) to pass to the limit, obtaining global weak solutions in the energy space. The work contextualizes the models within semi-relativistic quantum electrodynamics and contrasts them with the broader Schrödinger–Maxwell literature, highlighting spin coupling via the Pauli operator and the Leray projection’s role in enforcing divergence-free currents. Together, these results advance the rigorous mathematical theory of first-order semi-relativistic quantum models with self-consistent electromagnetic fields and lay groundwork for future research at lower regularity and in broader gauge settings.

Abstract

We construct local (in time) strong solutions in {$H^s(\mathbb{R}^3)$, $s>3/2$} and global weak solutions with finite energy for both the Pauli-Darwin and the Pauli-Poisswell systems. These are the first rigorous results on local and global wellposedness for these nonlinear first-order semi-relativistic quantum models for fast moving electrons. The Pauli equation is essentially a vector-valued magnetic Schrödinger equation for a 2-spinor with an additional Stern-Gerlach term coupling spin and magnetic field, keeping terms up to first order in $1/c$, where $c$ denotes the speed of light. The self-consistent electromagnetic field is computed from the charge density and current density by semi-relativistic approximations of the Maxwell equations: the Poisswell equation at $O(1/c)$ and the Darwin equation at $O(1/c^2)$.\\ We present the physics and asymptotic relations and provide proofs that rely on energy estimates for the strong solutions and compactness with an appropriate regularization for the weak solutions.

Local Wellposedness and Global Weak Solutions of the Pauli-Darwin/Poisswell Equations

TL;DR

The authors establish the first rigorous local well-posedness results in with and global finite-energy weak solutions for the Pauli-Darwin and Pauli-Poisswell semi-relativistic Pauli systems coupled to self-consistent Maxwell-type fields. They develop a parabolic regularization that yields energy dissipation, derive robust elliptic bounds for the potentials and , and employ compactness (Aubin–Lions) to pass to the limit, obtaining global weak solutions in the energy space. The work contextualizes the models within semi-relativistic quantum electrodynamics and contrasts them with the broader Schrödinger–Maxwell literature, highlighting spin coupling via the Pauli operator and the Leray projection’s role in enforcing divergence-free currents. Together, these results advance the rigorous mathematical theory of first-order semi-relativistic quantum models with self-consistent electromagnetic fields and lay groundwork for future research at lower regularity and in broader gauge settings.

Abstract

We construct local (in time) strong solutions in {, } and global weak solutions with finite energy for both the Pauli-Darwin and the Pauli-Poisswell systems. These are the first rigorous results on local and global wellposedness for these nonlinear first-order semi-relativistic quantum models for fast moving electrons. The Pauli equation is essentially a vector-valued magnetic Schrödinger equation for a 2-spinor with an additional Stern-Gerlach term coupling spin and magnetic field, keeping terms up to first order in , where denotes the speed of light. The self-consistent electromagnetic field is computed from the charge density and current density by semi-relativistic approximations of the Maxwell equations: the Poisswell equation at and the Darwin equation at .\\ We present the physics and asymptotic relations and provide proofs that rely on energy estimates for the strong solutions and compactness with an appropriate regularization for the weak solutions.

Paper Structure

This paper contains 25 sections, 11 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. The equations
  3. Main results
  4. The Schrödinger-Maxwell equation
  5. Organization of the article
  6. The Physics
  7. The equations including the physical dimensions
  8. Semi-relativistic approximation
  9. From Maxwell to Darwin and Poisswell
  10. Notations and formulas
  11. Inequalities
  12. Fourier multipliers and Sobolev spaces
  13. Linear and multilinear operations
  14. Useful identities
  15. The energy structure
  16. ...and 10 more sections

Key Result

Theorem 1

Let $s>3/2$ and $u_{0} \in H^s(\mathbb{R}_x^3)$. Then there exists $T>0$ such that the Pauli-Darwin equation eq:PD_Pauli-eq:PD_data and the Pauli-Poisswell equation eq:PPw_Pauli-eq:PPW_data have unique solutions $u^{\text{D}},u^{\text{PW}} \in C([0,T],H^s(\mathbb{R}_x^3))$ with initial data $u_{0}$.

Theorems & Definitions (26)

  • Theorem 1: Local wellposedness
  • Remark 1
  • Theorem 2: Global weak solutions
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 16 more