Details
| Original language | English |
|---|---|
| Title of host publication | Progress in Nonlinear Differential Equations and Their Application |
| Publisher | Springer US |
| Pages | 193-209 |
| Number of pages | 17 |
| Publication status | Published - 10 Jun 2011 |
Publication series
| Name | Progress in Nonlinear Differential Equations and Their Application |
|---|---|
| Volume | 80 |
| ISSN (Print) | 1421-1750 |
| ISSN (electronic) | 2374-0280 |
Abstract
We consider the periodic μDP equation (a modified version of the Degasperis-Procesi equation) as the geodesic flow of a right-invariant affine connection ∇ on the Fréchet Lie group Diff∞(S1) of all smooth and orientation-preserving diffeomorphisms of the circle S1 = ℝ/ℤ. On the Lie algebra C∞(S1) of Diff∞(S1), this connection is canonically given by the sum of the Lie bracket and a bilinear operator. For smooth initial data, we show the short time existence of a smooth solution of μDP which depends smoothly on time and on the initial data. Furthermore, we prove that the exponential map defined by ∇ is a smooth local diffeomorphism of a neighbourhood of zero in C∞(S1) onto a neighbourhood of the unit element in Diff∞(S1). Our results follow from a general approach on non-metric Euler equations on Lie groups, a Banach space approximation of the Fréchet space C∞(S1), and a sharp spatial regularity result for the geodesic flow.
Keywords
- Degasperis–Procesi equation, Euler equation, Geodesic flow
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Engineering(all)
- Computational Mechanics
- Mathematics(all)
- Mathematical Physics
- Mathematics(all)
- Control and Optimization
- Mathematics(all)
- Applied Mathematics
Cite this
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Progress in Nonlinear Differential Equations and Their Application. Springer US, 2011. p. 193-209 (Progress in Nonlinear Differential Equations and Their Application; Vol. 80).
Research output: Chapter in book/report/conference proceeding › Contribution to book/anthology › Research › peer review
}
TY - CHAP
T1 - Geometric Aspects of the Periodic μ-Degasperis-Procesi Equation
AU - Escher, Joachim
AU - Kohlmann, Martin
AU - Kolev, Boris
PY - 2011/6/10
Y1 - 2011/6/10
N2 - We consider the periodic μDP equation (a modified version of the Degasperis-Procesi equation) as the geodesic flow of a right-invariant affine connection ∇ on the Fréchet Lie group Diff∞(S1) of all smooth and orientation-preserving diffeomorphisms of the circle S1 = ℝ/ℤ. On the Lie algebra C∞(S1) of Diff∞(S1), this connection is canonically given by the sum of the Lie bracket and a bilinear operator. For smooth initial data, we show the short time existence of a smooth solution of μDP which depends smoothly on time and on the initial data. Furthermore, we prove that the exponential map defined by ∇ is a smooth local diffeomorphism of a neighbourhood of zero in C∞(S1) onto a neighbourhood of the unit element in Diff∞(S1). Our results follow from a general approach on non-metric Euler equations on Lie groups, a Banach space approximation of the Fréchet space C∞(S1), and a sharp spatial regularity result for the geodesic flow.
AB - We consider the periodic μDP equation (a modified version of the Degasperis-Procesi equation) as the geodesic flow of a right-invariant affine connection ∇ on the Fréchet Lie group Diff∞(S1) of all smooth and orientation-preserving diffeomorphisms of the circle S1 = ℝ/ℤ. On the Lie algebra C∞(S1) of Diff∞(S1), this connection is canonically given by the sum of the Lie bracket and a bilinear operator. For smooth initial data, we show the short time existence of a smooth solution of μDP which depends smoothly on time and on the initial data. Furthermore, we prove that the exponential map defined by ∇ is a smooth local diffeomorphism of a neighbourhood of zero in C∞(S1) onto a neighbourhood of the unit element in Diff∞(S1). Our results follow from a general approach on non-metric Euler equations on Lie groups, a Banach space approximation of the Fréchet space C∞(S1), and a sharp spatial regularity result for the geodesic flow.
KW - Degasperis–Procesi equation
KW - Euler equation
KW - Geodesic flow
UR - http://www.scopus.com/inward/record.url?scp=84989230983&partnerID=8YFLogxK
U2 - 10.1007/978-3-0348-0075-4_10
DO - 10.1007/978-3-0348-0075-4_10
M3 - Contribution to book/anthology
AN - SCOPUS:84989230983
T3 - Progress in Nonlinear Differential Equations and Their Application
SP - 193
EP - 209
BT - Progress in Nonlinear Differential Equations and Their Application
PB - Springer US
ER -