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Dissipation-Induced Instability Phenomena in Infinite-Dimensional Systems

Krechetnikov, Rouslan and Marsden, Jerrold E. (2009) Dissipation-Induced Instability Phenomena in Infinite-Dimensional Systems. Archive for Rational Mechanics and Analysis, 194 (2). pp. 611-668. ISSN 0003-9527. http://resolver.caltech.edu/CaltechAUTHORS:20091006-160910543

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Abstract

This paper develops a rigorous notion of dissipation-induced instability in infinite dimensions as an extension of the classical concept implicitly introduced by Thomson and Tait for finite degree of freedom mechanical systems over a century ago. Here we restrict ourselves to a particular form of infinite-dimensional systems—partial differential equations—whose inherent function-analytic differences from finite-dimensional systems make uncovering this notion more intricate. In building the concept of dissipation-induced instability in infinite dimensions we found Arnold’s and Yudovich’s nonlinear stability methods, for conservative and dissipative systems respectively, along with some new existence theory for solutions, to be the essential foundation. However, when proving the results for classical solutions, as motivated by their direct physical significance, we had to overcome a number of fundamental difficulties associated with existing stability analysis methods, which has led to new techniques. In particular, in this work we establish the connection of existence and general stability theories in strong and weak topologies and provide new insights into the physics and geometry of the dissipation-induced instability phenomena in infinite-dimensional systems. As a paradigm and the first infinite-dimensional example to be rigorously analyzed, we use a two-layer quasi-geostrophic beta-plane model, which describes the fundamental baroclinic instability in atmospheric and ocean dynamics; early formal linear approximate studies suggested that this system can be destabilized after the introduction of dissipation.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s00205-008-0193-6DOIUNSPECIFIED
http://www.springerlink.com/content/98v2188624605782/PublisherUNSPECIFIED
Additional Information:© 2009 Springer. Received: 7 July 2005; accepted: 30 June 2008; published online: 13 January 2009. The authors would like to thank Tapio Schneider, John Hart, Steve Shkoller, and Edriss Titi for helpful discussions. The authors were partially supported by NSF-ITR Grant ACI-0204932. R.K. also acknowledges partial support from NSERC 341849-2007.
Funders:
Funding AgencyGrant Number
NSFACI-0204932
Natural Sciences and Engineering Research Council of Canada (NSERC)341849-2007
Record Number:CaltechAUTHORS:20091006-160910543
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20091006-160910543
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ID Code:16195
Collection:CaltechAUTHORS
Deposited By: Jason Perez
Deposited On:07 Oct 2009 15:20
Last Modified:26 Dec 2012 11:26

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