Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Instantaneous shock location and one-dimensional nonlinear stability of viscous shock waves

Author: Kevin Zumbrun
Journal: Quart. Appl. Math. 69 (2011), 177-202
MSC (2010): Primary 35B35
DOI: https://doi.org/10.1090/S0033-569X-2011-01221-6
Published electronically: January 19, 2011
MathSciNet review: 2807984
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Abstract: We illustrate in a simple setting the instantaneous shock tracking approach to stability of viscous conservation laws introduced by Howard, Mascia, and Zumbrun. This involves a choice of the definition of instantaneous location of a viscous shock. We show that this choice is time-asymptotically equivalent both to the natural choice of least-squares fit pointed out by Goodman and to a simple phase condition used by Guès, Métivier, Williams, and Zumbrun in other contexts. More generally, we show that it is asymptotically equivalent to any location defined by a localized projection.

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Additional Information

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

DOI: https://doi.org/10.1090/S0033-569X-2011-01221-6
Keywords: Viscous shock waves; nonlinear stability; pointwise Green function bounds
Received by editor(s): September 28, 2009
Published electronically: January 19, 2011
Additional Notes: The author’s research was partially supported under NSF grants no. DMS-0300487 and DMS-0801745
Article copyright: © Copyright 2011 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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