Stability and Diffusive Dynamics on Extended Domains

Alexander Mielke, Guido Schneider and Hannes Uecker
Additional contact information
Alexander Mielke: Universität Stuttgart, Mathematisches Institut A
Guido Schneider: Universität Bayreuth, Mathematisches Institut
Hannes Uecker: Universität Bayreuth, Mathematisches Institut

A chapter in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, 2001, pp 563-583 from Springer

Abstract: Abstract We consider dissipative systems on the real axis in situations when the evolution is dominated by a dynamics similar to the one of a linear diffusion equation. It is surprising that such a diffusive behavior occurs in relatively complicated systems. After a discussion of the linear and nonlinear diffusion equation, we give a brief introduction into the methods which are available to describe diffusive behavior in nonlinear systems. These are L 1 -L∞.estimates, Lyapunov functions and discrete and continuous renormalization groups. In the second part of the paper we show examples, where such a diffusive dynamics can be seen. For the Ginzburg-Landau equation we consider the nonlinear stability of Eckhaus-stable equilibria and the diffusive mixing of two different Eckhaus-stable equilibria. Diffusive dynamics also occurs in pattern forming systems as the Swift-Hohenberg equation or hydrodynamical stability problems as Bénard’s problem. In such cases the method of reduced instability allows us to analyze the linearized problem. We close with an outlook on situations, where diffusive behavior is expected, but where a proof is still missing.

Keywords: Unbounded Domain; Nonlinear Stability; Diffusive Behavior; Diffusive Dynamics; Nonlinear Diffusion Equation (search for similar items in EconPapers)
Date: 2001
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-56589-2_24

Ordering information: This item can be ordered from
http://www.springer.com/9783642565892

DOI: 10.1007/978-3-642-56589-2_24

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().