Curvatures of Sobolev metrics on diffeomorphism groups
2013 (English)In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 9, no 2, 291-332 p.Article in journal (Refereed) Published
Many conservative partial differential equations correspond to geodesic equations on groups of diffeomorphisms. Stability of their solutions can be studied by examining sectional curvature of these groups: negative curvature in all sections implies exponential growth of perturbations and hence suggests instability, while positive curvature suggests stability. In the first part of the paper we survey what we currently know about the curvature-stability relation in this context and provide detailed calculations for several equations of continuum mechanics associated to Sobolev H0 and H1 energies. In the second part we prove that in most cases (with some notable exceptions) the sectional curvature assumes both signs.
Place, publisher, year, edition, pages
2013. Vol. 9, no 2, 291-332 p.
Diffeomorphism groups, Euler-Arnold equations, Riemannian metrics, Sectional curvature, Stability
IdentifiersURN: urn:nbn:se:kth:diva-163794DOI: 10.4310/PAMQ.2013.v9.n2.a2ISI: 000327544000002ScopusID: 2-s2.0-84887385355OAI: oai:DiVA.org:kth-163794DiVA: diva2:802400
QC 201504172015-04-132015-04-122015-04-17Bibliographically approved