Long-Time Dynamics of the 3D Vlasov-Maxwell System with Boundaries

Abstract

We construct global-in-time classical solutions to the nonlinear Vlasov--Maxwell system in a three-dimensional half-space beyond the vacuum scattering regime in a physical setting of the solar wind model. Our approach combines the construction of stationary solutions to the associated boundary-value problem with a proof of their asymptotic dynamical stability in $L^\infty$ under small perturbations, providing a new framework for understanding long-time wave-particle interactions in the presence of boundaries and interacting magnetic fields. A key difficulty is that the transport dynamics remains coupled on arbitrarily long time scales to a wave component whose sharp $t^{-1}$ pointwise decay is neither integrable in time nor improved by the boundary. We resolve this obstruction by establishing a new decay mechanism that converts the spatial localization of particle--field interactions into temporal decay through the light-cone geometry of wave propagation. This enables us to close the nonlinear field--particle feedback and establish sharp $t^{-1}$ decay for both the particle distribution and the electromagnetic fields. To the best of our knowledge, this work presents the first construction of asymptotically stable non-vacuum steady states under general perturbations in the full three-dimensional nonlinear Vlasov--Maxwell system.