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An Eulerian–Lagrangian form for the Euler equations in Sobolev spaces
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Pooley, Benjamin C. and Robinson, James C. (2016) An Eulerian–Lagrangian form for the Euler equations in Sobolev spaces. Journal of Mathematical Fluid Mechanics, 18 (4). pp. 783-794. doi:10.1007/s00021-016-0271-8 ISSN 1422-6928.
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Official URL: http://dx.doi.org/10.1007/s00021-016-0271-8
Abstract
In 2000 Constantin showed that the incompressible Euler equations can be written in an “Eulerian–Lagrangian” form which involves the back-to-labels map (the inverse of the trajectory map for each fixed time). In the same paper a local existence result is proved in certain Hölder spaces C1,μ . We review the Eulerian–Lagrangian formulation of the equations and prove that given initial data in H s for n≥2 and s>n2+1 , a unique local-in-time solution exists on the n-torus that is continuous into H s and C 1 into H s-1. These solutions automatically have C 1 trajectories. The proof here is direct and does not appeal to results already known about the classical formulation. Moreover, these solutions are regular enough that the classical and Eulerian–Lagrangian formulations are equivalent, therefore what we present amounts to an alternative approach to some of the standard theory.
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Eulerian graph theory, Sobolev spaces, Lagrange equations | ||||||||
| Journal or Publication Title: | Journal of Mathematical Fluid Mechanics | ||||||||
| Publisher: | Birkhaeuser Verlag AG | ||||||||
| ISSN: | 1422-6928 | ||||||||
| Official Date: | December 2016 | ||||||||
| Dates: |
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| Volume: | 18 | ||||||||
| Number: | 4 | ||||||||
| Page Range: | pp. 783-794 | ||||||||
| DOI: | 10.1007/s00021-016-0271-8 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Date of first compliant deposit: | 20 July 2016 | ||||||||
| Date of first compliant Open Access: | 3 June 2017 |
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