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.