Singularly perturbed resistive-viscous models in magnetohydrodynamics

Abstract

We consider singularly perturbed resistive-viscous MHD equations of the form u' = f(u,v,lambda), epsivv' = g(u,v,lambda), where ' stands for derivative with respect to thetas = x - st, s is the wave speed, 0 < epsiv Lt 1 and lambda is a parameter. Such systems of singlularly perturbed MHD equations include the MHD models of intermediate shocks when the resistivity eta and viscosity mu and/or nu are present and one of the viscosity parameters plays the role of small epsiv. The u = [By,Bz], two components of the magnetic induction vector (Bx = const) and v is the velocity. When epsiv rarr 0 we obtain a system of differential-algebraic equations (DAEs) rather than singularly perturbed ODEs. The former have singularities which typically behave as impasse points, singular pseudo nodes, saddles, foci points, or singularity induced bifurcation (SIB) points. The pseudo equilibrium and SIB points allow for smooth transitions between the plus (supersonic) and minus (subsonic) Rie- mann sheets with either one or two analytic trajectories crossing the singularity (sonic) curve and other trajectories of lower smoothness. In the paper we analyze the singularly perturbed MHD equations in the context of their relations to DAEs and the recent developments in the qualitative analysis of systems with folded pseudo equilibrium points.