Metastability for nonlinear convection–diffusion equations

Metastability for nonlinear convection–diffusion equations We study the metastable dynamics of solutions to nonlinear evolutive equations of parabolic type, with a particular attention to the case of the viscous scalar Burgers equation with small viscosity $$\varepsilon $$ ε . In order to describe rigorously such slow motion, we adapt the strategy firstly proposed in Mascia and Strani (SIAM J Math Anal 45:3084–3113, 2013) by linearizing the original equation around a metastable state and by studying the system obtained for the couple $$(\xi ,v)$$ ( ξ , v ) , where $$\xi $$ ξ is the position of the internal shock layer and v is a perturbative term. The main result of this paper provides estimates for the speed of the shock layer and for the error v; in particular, in the case of the viscous Burgers equation, we prove they are exponentially small in $$\varepsilon $$ ε . As a consequence, the time taken for the solution to reach the unique stable steady state is exponentially large, and we have exponentially slow motion. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nonlinear Differential Equations and Applications NoDEA Springer Journals

Metastability for nonlinear convection–diffusion equations

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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by Springer International Publishing AG
Subject
Mathematics; Analysis
ISSN
1021-9722
eISSN
1420-9004
D.O.I.
10.1007/s00030-017-0459-5
Publisher site
See Article on Publisher Site

Abstract

We study the metastable dynamics of solutions to nonlinear evolutive equations of parabolic type, with a particular attention to the case of the viscous scalar Burgers equation with small viscosity $$\varepsilon $$ ε . In order to describe rigorously such slow motion, we adapt the strategy firstly proposed in Mascia and Strani (SIAM J Math Anal 45:3084–3113, 2013) by linearizing the original equation around a metastable state and by studying the system obtained for the couple $$(\xi ,v)$$ ( ξ , v ) , where $$\xi $$ ξ is the position of the internal shock layer and v is a perturbative term. The main result of this paper provides estimates for the speed of the shock layer and for the error v; in particular, in the case of the viscous Burgers equation, we prove they are exponentially small in $$\varepsilon $$ ε . As a consequence, the time taken for the solution to reach the unique stable steady state is exponentially large, and we have exponentially slow motion.

Journal

Nonlinear Differential Equations and Applications NoDEASpringer Journals

Published: Jun 13, 2017

References

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