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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Stability of noncharacteristic boundary layers in the standing-shock limit


Author: Kevin Zumbrun
Journal: Trans. Amer. Math. Soc. 362 (2010), 6397-6424
MSC (2010): Primary 35Q35; Secondary 35B35
Published electronically: July 14, 2010
MathSciNet review: 2678980
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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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Additional Information

Kevin Zumbrun
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: kzumbrun@indiana.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05213-4
PII: S 0002-9947(2010)05213-4
Received by editor(s): September 15, 2008
Published electronically: July 14, 2010
Additional Notes: The author’s research was partially supported under NSF grants number DMS-0070765 and DMS-0300487.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.