Stability of noncharacteristic boundary layers in the standing shock limit

TL;DR

This paper analyzes the stability of boundary layers near standing shock waves, establishing necessary and sufficient conditions for stability in various shock types, and applies these to gas dynamical boundary layers.

Contribution

It provides new stability criteria for noncharacteristic boundary layers approaching standing shocks, extending previous results to more complex shock types.

Findings

Necessary conditions for stability include weak shock and layer stability and a nonnegative Lopatinski determinant.

Sufficient conditions are derived for Lax 1-shocks.

Stability of certain gas dynamical boundary layers is established, generalizing prior work.

Abstract

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 stability of certain isentropic and full gas dynamical boundary-layers, generalizing earlier work of Serre--Zumbrun and Costanzino--Humphreys--Nguyen--Zumbrun.