EXISTENCE AND STABILITY OF VISCOUS SHOCK PROFILES FOR 2-D ISENTROPIC MHD WITH INFINITE ELECTRICAL RESISTIVITY

For the two-dimensional Navier-Stokes equations of isentropic magnetohydrodynamics(MHD) with γ-law gas equation of state,γ≥ 1,and infinite electrical resistivity,we carry out a global analysis categorizing all possible viscous shock profiles.Precisely,we show that the phase portrait of the traveling-wave ODE generically consists of either two rest points connected by a viscous Lax profile,or else four rest points,two saddles and two nodes.In the latter configuration,which rest points are connected by profiles depends on the ratio of viscosities,and can involve Lax,overcornpressive,or undercornpressive shock profiles.Considered as three-dimensional solutions,undercompressive shocks are Lax-type(Alfven) waves.For the rnonatornic and diatornic cases γ=5/3 and γ= 7/5,with standard viscosity ratio for a nonmagnetic gas,we find numerically that the the nodes are connected by a family of overcornpressive profiles bounded by Lax profiles connecting saddles to nodes,with no undercornpressive shocks occurring.We carry out a systematic numerical Evans function analysis indicating that all of these two-dimensional shock profiles are linearly and nonlinearly stable,both with respect to two-and three-dimensional perturbations.For the same gas constants,but different viscosity ratios,we investigate also cases for which undercornpressive shocks appear; these are seen numerically to be stable as well,both with respect to two-dimensional and(in the neutral sense of convergence to nearby Riemann solutions) three-dimensional perturbations.