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For the two-dimensional Navier--Stokes equations of isentropicmagnetohydrodynamics (MHD) with $\gamma$-law gas equation of state, $\gamma\ge1$, and infinite electrical resistivity, we carry out a global analysiscategorizing all possible viscous shock profiles. Precisely, we show that thephase portrait of the traveling-wave ODE generically consists of either tworest points connected by a viscous Lax profile, or else four rest points, twosaddles and two nodes. In the latter configuration, which rest points areconnected by profiles depends on the ratio of viscosities, and can involve Lax,overcompressive, or undercompressive shock profiles. For the monatomic anddiatomic cases $\gamma=5/3$ and $\gamma= 7/5$, with standard viscosity ratiofor a nonmagnetic gas, we find numerically that the the nodes are connected bya family of overcompressive profiles bounded by Lax profiles connecting saddlesto nodes, with no undercompressive shocks occurring. We carry out a systematicnumerical Evans function analysis indicating that all of these two-dimensionalshock profiles are linearly and nonlinearly stable, both with respect to two-and three-dimensional perturbations. For the same gas constants, but differentviscosity ratios, we investigate also cases for which undercompressive shocksappear; these are seen numerically to be stable as well

Existence and stability of viscous shock profiles for 2-D isentropic MHD with infinite electrical resistivity