The semiclassical limit from the Pauli-Poisswell/Darwin to the Euler-Poisswell/Darwin system by WKB methods

TL;DR

The paper analyzes the semiclassical limit $oldsymbol{ abla} o0$ for the self-consistent Pauli-Poisswell and Pauli-Darwin equations, establishing a local well-posedness framework and a blow-up dichotomy for the associated Euler-type limit systems. It develops two complementary routes: a WKB approach yielding local-in-time convergence to the Euler-Poisswell/Darwin equations, and a Wigner-transform approach yielding a monokinetic, Vlasov-type limit to Vlasov-Poisswell/Darwin with convergence of macroscopic densities. Key contributions include rigorous a priori energy estimates, elliptic bounds for the self-consistent fields, and a detailed semiclassical limit that ties the quantum spin-magnetic dynamics to classical fluid/kinetic models. The work clarifies the hierarchical relationship among fully relativistic Euler-Maxwell, semi-relativistic Euler-Darwin/Poisswell, and nonrelativistic Euler-Poisson/Vlasov-Poisson systems, and provides a robust mathematical foundation for the short-time behavior of spinful self-consistent plasmas in the semiclassical regime.

Abstract

The self-consistent Pauli-Poisswell and Pauli-Darwin equations for 2-spinors are $O(1/c)$ (where $c$ denotes the speed of light) semi-relativistic approximations of the Dirac-Maxwell equation for 4-spinors coupled to the self-consistent electromagnetic fields generated by the charge and current densities of a fast moving electric charge. They consist of a vector-valued magnetic Schrödinger equation with the Stern-Gerlach term which couples spin and magnetic field, coupled to 1+3 Poisson equations as the magnetostatic approximation of Maxwell's equations. The Pauli-Poisswell and Pauli-Dariwn euqations are $O(1/c)$ models keeping both relativistic effects magnetism and spin, both of which are absent in the non-relativistic Schrödinger-Poisson equation and inconsistent in the magnetic Schrödinger-Maxwell equation. We prove the local in time semiclassical limit $\hbar \rightarrow 0$ to the Euler-Poisswell equation and Euler-Darwin equations based on WKB analysis and energy estimates. Moreover we obtain weak convergence of the monokinetic Wigner transform to the monokinetic scalar Wigner measure solving the Vlasov-Poisswell and Vlasov-Darwin equations and strong convergence of the macroscopic densities. We introduce the Euler-Poisswell/Darwin equation and prove local wellposedness and a blow up alternative.

Theorems & Definitions (26)

• Remark 1
• Remark 2
• Theorem 1.a
• Theorem 1.b
• Theorem 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Lemma 1