This paper proves that a one-dimensional model of a viscous, electrically conducting fluid coupled to the full Maxwell equations has global “finite-energy weak” solutions. The model keeps the displacement current (so the electromagnetic part behaves like waves) and uses a simple algebraic Ohm law for the electric current. The existence result holds for every pressure exponent γ>1 and for any finite-energy initial data. The allowed initial data can include vacuum regions and only require the initial electric and magnetic fields to be square integrable (L^2).

The main technical hurdle is that the electromagnetic force on the fluid — the Lorentz force — is a product of quantities that are only controlled weakly by the basic energy estimates. Weak control by itself does not let one pass to limits in that product. The authors overcome this by keeping the Maxwell–Ohm (electromagnetic plus Ohm’s law) subsystem exact throughout the approximation. They prove a new weak-to-strong compactness statement: if the velocity coefficients converge weakly in time-space H^1 in velocity and the initial electromagnetic fields converge strongly, then the evolving electric and magnetic fields converge strongly in time-continuous L^2. That strong convergence of the fields gives a well-defined limit for the current and for the Lorentz force, even though the velocity does not converge strongly.

At a higher level, the construction combines this Maxwell closure with a classical compressible-fluid approximation method. The fluid part uses artificial viscosity and artificial pressure to build approximate solutions, together with a density-primitive effective-flux identity. Testing the momentum equation with a specially chosen density primitive cancels problematic terms and recovers extra compactness that isolates the physical pressure. The full argument then removes the artificial regularization and shows that any remaining pressure concentration defects vanish.