Incompressible Euler and E-MHD as scaling limits of the Vlasov-Maxwell system

Abstract

We consider two different asymptotic limits of the Vlasov-Maxwell system describing a quasineutral plasma with a uniform ionic background. In the first case, as the magnetic field is preserved in the limiting process, we obtain the so-called electron magnetohydrodynamics equations. In the second case, we obtain the incompressible Euler equations with no more magnetic field left.

Combined relaxation and non-relativistic limit of non-isentropic Euler–Maxwell equations

“…Here, the non-dimensionless parameters γ > 0 and τ > 0 can be chosen independently on each other, according to the desired scaling. [4] γ is proportional to 1 c , where c = ( 0 ν 0 ) − 1 2 is the speed of light, with 0 and ν 0 being the vacuum permittivity and permeability. τ is the momentum relaxation time for some physical flow.…”

Convergence of Compressible Euler–Maxwell Equations to Incompressible Euler Equations

“…However, we keep them in the system because this redundancy may be lost in the asymptotic limit. The non-dimensionless parameters and can be chosen independently on each other, according to the desired scaling, see Brenier et al (2003). Physically, and are proportional to the Debye length and 1 c , where c = 0 0 − 1 2 is the speed of light, with 0 and 0 being the vacuum permittivity and permeability.…”

The combined non-relativistic and quasi-neutral limit of two-fluid Euler–Maxwell equations

“…We refer to [2] for a similar choice of the scaling in the Vlasov-Maxwell equations. When the ions are non-moving and become a uniform background with a given stationary density, by letting n i = b, u i = 0 and deleting the Euler equations for ions, a one-fluid Euler-Maxwell model is formally derived.…”