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

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 Schr\"odinger 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.