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

We construct local (in time) strong solutions in $H^s(\mathbb{R}^3)$, $s>3/2$ and global weak solutions with finite energy for the Pauli-Darwin equation and the Pauli-Poisswell equation. These are the first rigorous results on these two models, which couple a 2-spinor with an electromagnetic field. The proofs rely on energy estimates for the strong solutions and compactness with an appropriate regularization for the weak solutions.