Stability of noncharacteristic boundary layers in the standing-shock limit

We investigate oneand multi-dimensional stability of noncharacteristic boundary layers in the limit approaching a standing planar shock wave Ū(x1), x1 > 0, obtaining necessary conditions of (i) weak stability of the limiting shock, (ii) weak stability of the constant layer u ≡ U− := limz→−∞ Ū(z), and (iii) nonnegativity of a modified Lopatinski determinant similar to that of the inviscid shock case. For Lax 1-shocks, we obtain equally simple sufficient conditions; for p-shocks, p > 1, the situation appears to be more complicated. Using these results, we determine stability of certain isentropic and full gas dynamical boundary-layers, generalizing earlier work of Serre–Zumbrun and Costanzino–Humphreys–Nguyen–Zumbrun.