Stability of noncharacteristic boundary layers in the standing-shock limit

Zumbrun, Kevin

doi:10.1090/S0002-9947-2010-05213-4

Stability of noncharacteristic boundary layers in the standing-shock limit
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Trans. Amer. Math. Soc. 362 (2010), 6397-6424 Request permission

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 the stability of certain gas dynamical boundary layers, generalizing earlier work of Serre–Zumbrun and Costanzino–Humphreys–Nguyen–Zumbrun.

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