Details
| Original language | English |
|---|---|
| Pages (from-to) | 2010-2053 |
| Number of pages | 44 |
| Journal | Journal of Differential Equations |
| Volume | 258 |
| Issue number | 6 |
| Publication status | Published - 15 Mar 2015 |
Abstract
Of concern is the study of fractional order Sobolev-type metrics on the group of H∞-diffeomorphism of Rd and on its Sobolev completions Dq(Rd). It is shown that the Hs-Sobolev metric induces a strong and smooth Riemannian metric on the Banach manifolds Ds(Rd) for s>1+d2. As a consequence a global well-posedness result of the corresponding geodesic equations, both on the Banach manifold Ds(Rd) and on the smooth regular Fréchet-Lie group of all H∞-diffeomorphisms is obtained. In addition a local existence result for the geodesic equation for metrics of order 12≤s<1+d/2 is derived.
Keywords
- Diffeomorphism groups, EPDiff equation, Sobolev metrics of fractional order
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Applied Mathematics
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Journal of Differential Equations, Vol. 258, No. 6, 15.03.2015, p. 2010-2053.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Local and global well-posedness of the fractional order EPDiff equation on Rd
AU - Bauer, Martin
AU - Escher, Joachim
AU - Kolev, Boris
N1 - Funding information: The authors are grateful to Elmar Schrohe and Jörg Seiler for helpful discussions about various topics on translation invariant operators. It is also a pleasure to thank David Lannes for stimulating discussions concerning commutator estimates. Martin Bauer was supported by Austrian Science Fund ( FWF ) project P24625 .
PY - 2015/3/15
Y1 - 2015/3/15
N2 - Of concern is the study of fractional order Sobolev-type metrics on the group of H∞-diffeomorphism of Rd and on its Sobolev completions Dq(Rd). It is shown that the Hs-Sobolev metric induces a strong and smooth Riemannian metric on the Banach manifolds Ds(Rd) for s>1+d2. As a consequence a global well-posedness result of the corresponding geodesic equations, both on the Banach manifold Ds(Rd) and on the smooth regular Fréchet-Lie group of all H∞-diffeomorphisms is obtained. In addition a local existence result for the geodesic equation for metrics of order 12≤s<1+d/2 is derived.
AB - Of concern is the study of fractional order Sobolev-type metrics on the group of H∞-diffeomorphism of Rd and on its Sobolev completions Dq(Rd). It is shown that the Hs-Sobolev metric induces a strong and smooth Riemannian metric on the Banach manifolds Ds(Rd) for s>1+d2. As a consequence a global well-posedness result of the corresponding geodesic equations, both on the Banach manifold Ds(Rd) and on the smooth regular Fréchet-Lie group of all H∞-diffeomorphisms is obtained. In addition a local existence result for the geodesic equation for metrics of order 12≤s<1+d/2 is derived.
KW - Diffeomorphism groups
KW - EPDiff equation
KW - Sobolev metrics of fractional order
UR - http://www.scopus.com/inward/record.url?scp=84921604042&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2014.11.021
DO - 10.1016/j.jde.2014.11.021
M3 - Article
AN - SCOPUS:84921604042
VL - 258
SP - 2010
EP - 2053
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
IS - 6
ER -