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 E^ε and the wave equation with linear damping of the divergence free magnetic field B^ε, then take the limit as ε → 0 to obtain the solutions of the system with solenoidal Ohm's law.