Stability of viscous profiles: Proofs via dichotomies

Beyn W-J, Lorenz J (2006)
Journal of Dynamics and Differential Equations 18(1): 141-195.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
In this paper we give a self-contained approach to a nonlinear stability result, as t ->infinity, for a viscous profile corresponding to a strong shock of a system of conservation laws. The initial perturbation is assumed to be small and to have zero mass. As t ->infinity, the solution with perturbed initial data is shown to approach the viscous profile in maximum norm. A complete proof of the stability result is given under slightly weaker assumptions than those in [Comm. Pure Appl. Math. LI (1998) 1397]; our assumptions, techniques, and results also differ from those in [Indiana Univ. Math. J. 47 (1998) 741]. To derive resolvent estimates for a linearized problem, we use the theory of exponential dichotomies for ODES extensively. A main tool provided by this theory is a quantitative L-1 perturbation theorem for dichotomies, which yields the delicate resolvent estimates for s near zero. When showing that the resolvent estimates imply nonlinear stability, we essentially follow the arguments in [Comm. Pure App/. Math. L1 (1998) 1397; SIAM J. Math. Anal. 20 (1999) 401], but note some simplifications.
Stichworte
viscous conservation laws; stability; condition; viscous profiles; zero-mass
Erscheinungsjahr
2006
Zeitschriftentitel
Journal of Dynamics and Differential Equations
Band
18
Ausgabe
1
Seite(n)
141-195
ISSN
1040-7294
Page URI
https://pub.uni-bielefeld.de/record/1598550

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Beyn W-J, Lorenz J. Stability of viscous profiles: Proofs via dichotomies. Journal of Dynamics and Differential Equations. 2006;18(1):141-195.
Beyn, W. - J., & Lorenz, J. (2006). Stability of viscous profiles: Proofs via dichotomies. Journal of Dynamics and Differential Equations, 18(1), 141-195. doi:10.1007/s10884-005-9003-0
Beyn, W. - J., and Lorenz, J. (2006). Stability of viscous profiles: Proofs via dichotomies. Journal of Dynamics and Differential Equations 18, 141-195.
Beyn, W.-J., & Lorenz, J., 2006. Stability of viscous profiles: Proofs via dichotomies. Journal of Dynamics and Differential Equations, 18(1), p 141-195.
W.-J. Beyn and J. Lorenz, “Stability of viscous profiles: Proofs via dichotomies”, Journal of Dynamics and Differential Equations, vol. 18, 2006, pp. 141-195.
Beyn, W.-J., Lorenz, J.: Stability of viscous profiles: Proofs via dichotomies. Journal of Dynamics and Differential Equations. 18, 141-195 (2006).
Beyn, Wolf-Jürgen, and Lorenz, Jens. “Stability of viscous profiles: Proofs via dichotomies”. Journal of Dynamics and Differential Equations 18.1 (2006): 141-195.