Global finite-energy weak solutions to the Navier--Stokes--Maxwell in 1D

We construct global finite-energy weak solutions for a 1D barotropic Navier–Stokes–Maxwell system with displacement current and the algebraic Ohm law. The result holds for every $\gamma >1$ and arbitrary finite-energy initial data, allowing vacuum and requiring only $L^2$ initial electromagnetic fields. A key ingredient is a weak-to-strong compactness principle for the Maxwell–Ohm subsystem: weak convergence of the velocity coefficients in $L_t^2{H}_x^1$, together with strong convergence of the initial fields, yields strong electromagnetic trajectories in $C_t{L}_x^2$. This permits identification of the weak current and the distributional Lorentz force without strong convergence of the velocity. The electromagnetic closure is combined with the classical artificial-viscosity and artificial-pressure construction for compressible flow. A density-primitive effective-flux argument identifies the physical pressure and eliminates the physical and artificial pressure-concentration defects.