Volume 29, Issue 4
On a Lagrangian Formulation of the Incompressible Euler Equation

J. Part. Diff. Eq., 29 (2016), pp. 320-359.

Published online: 2015-03

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• Abstract

In this paper we show that the incompressible Euler equation on the Sobolev space $H^s(\mathbb{R}^n), s › n ⁄ 2+1$, can be expressed in Lagrangian coordinates as a geodesic equation on an infinite dimensional manifold. Moreover the Christoffel map describing the geodesic equation is real analytic. The dynamics in Lagrangian coordinates is described on the group of volume preserving diffeomorphisms, which is an analytic submanifold of the whole diffeomorphism group. Furthermore it is shown that a Sobolev class vector field integrates to a curve on the diffeomorphism group.

• Keywords

Euler equation diffeomorphism group

35Q35

@Article{JPDE-29-320, author = {Inci , Hasan}, title = {On a Lagrangian Formulation of the Incompressible Euler Equation}, journal = {Journal of Partial Differential Equations}, year = {2015}, volume = {29}, number = {4}, pages = {320--359}, abstract = { In this paper we show that the incompressible Euler equation on the Sobolev space $H^s(\mathbb{R}^n), s › n ⁄ 2+1$, can be expressed in Lagrangian coordinates as a geodesic equation on an infinite dimensional manifold. Moreover the Christoffel map describing the geodesic equation is real analytic. The dynamics in Lagrangian coordinates is described on the group of volume preserving diffeomorphisms, which is an analytic submanifold of the whole diffeomorphism group. Furthermore it is shown that a Sobolev class vector field integrates to a curve on the diffeomorphism group.}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v29.n4.5}, url = {http://global-sci.org/intro/article_detail/jpde/5096.html} }
TY - JOUR T1 - On a Lagrangian Formulation of the Incompressible Euler Equation AU - Inci , Hasan JO - Journal of Partial Differential Equations VL - 4 SP - 320 EP - 359 PY - 2015 DA - 2015/03 SN - 29 DO - http://doi.org/10.4208/jpde.v29.n4.5 UR - https://global-sci.org/intro/article_detail/jpde/5096.html KW - Euler equation KW - diffeomorphism group AB - In this paper we show that the incompressible Euler equation on the Sobolev space $H^s(\mathbb{R}^n), s › n ⁄ 2+1$, can be expressed in Lagrangian coordinates as a geodesic equation on an infinite dimensional manifold. Moreover the Christoffel map describing the geodesic equation is real analytic. The dynamics in Lagrangian coordinates is described on the group of volume preserving diffeomorphisms, which is an analytic submanifold of the whole diffeomorphism group. Furthermore it is shown that a Sobolev class vector field integrates to a curve on the diffeomorphism group.