Convergence from two-fluid incompressible Navier-Stokes-Maxwell system with Ohm's law to solenoidal Ohm's law: Classical solutions

The asymptotics from the two-fluid incompressible Navier-Stokes-Maxwell system with Ohm's law to solenoidal Ohm's law was pointed out in Arsénio-Saint-Raymond's work [3]. We justify rigorously this limit in the context of global-in-time classical solutions. The key is to derive the global-in-time uniform in ε energy estimate of the rescaled system with Ohm's law by employing the decay properties of both the electric field Eε and the wave equation with linear damping of the divergence free magnetic field Bε, then take the limit as ε→0 to obtain the solutions of the system with solenoidal Ohm's law.