Stability of noncharacteristic boundary layers in the standing-shock limit

We investigate one- and multi-dimensional stability of noncharacteristic boundary layers in the limit approaching a standing planar shock wave $\bar U(x_1)$, $x_1>0$, obtaining necessary conditions of (i) weak stability of the limiting shock, (ii) weak stability of the constant layer $u\equiv U_-:=\lim_{z\to -\infty} \bar U(z)$, 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 the stability of certain gas dynamical boundary layers, generalizing earlier work of Serre-Zumbrun and Costanzino-Humphreys-Nguyen-Zumbrun.