Well-posedness and stability of the Navier-Stokes-Maxwell equations

The paper is devoted to studying the well-posedness and stability of the generalized Navier-Stokes-Maxwell (NSM) equations with the standard Ohm's law in ℝ^d for d ∈{2,3}. More precisely, the global well-posedness is established in case of fractional Laplacian velocity (-Δ)^α v with α = d/2 for suitable data. In addition, the local well-posedness in the inviscid case is also provided for sufficient smooth data, which allows us to study the inviscid limit of associated positive viscosity solutions in the case α = 1, where an explicit bound on the difference is given. On the other hand, in the case α = 0 the stability near a magnetohydrostatic equilibrium with a constant (or equivalently bounded) magnetic field is also obtained in which nonhomogeneous Sobolev norms of the velocity and electric fields, and the L^∞ norm of the magnetic field converge to zero as time goes to infinity with an implicit rate. In this velocity damping case, the situation is different both in case of the two and a half, and three-dimensional magnetohydrodynamics (MHD) system, where an explicit rate of convergence in infinite time is computed for both the velocity and magnetic fields in nonhomogeneous Sobolev norms. Therefore, there is a gap between NSM and MHD in terms of the norm convergence of the magnetic field and the rate of decaying in time, even the latter equations can be proved as a limiting system of the former one in the sense of distributions as the speed of light tends to infinity.