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Weak dual pairs and jetlet methods for ideal incompressible fluid models in n >= 2 dimensions

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Title: Weak dual pairs and jetlet methods for ideal incompressible fluid models in n >= 2 dimensions
Authors: Cotter, CJ
Eldering, J
Holm, DD
Jacobs, HO
Meier, DM
Item Type: Journal Article
Abstract: We review the role of dual pairs in mechanics and use them to derive particle-like solutions to regularized incompressible fluid systems. In our case we have a dual pair resulting from the action of diffeomorphisms on point particles (essentially by moving the points). We then augment our dual pair by considering the action of diffeomorphisms on Taylor series, also known as jets. The augmented weak dual pairs induce a hierarchy of particle-like solutions and conservation laws with particles carrying a copy of a jet group. We call these augmented particles jetlets. The jet groups serve as finite-dimensional models of the diffeomorphism group itself, and so the jetlet particles serve as a finite-dimensional model of the self-similarity exhibited by ideal incompressible fluids. The conservation law associated to jetlet solutions is shown to be a shadow of Kelvin’s circulation theorem. Finally, we study the dynamics of infinite time particle mergers. We prove that two merging particles at the zeroth level in the hierarchy yield dynamics which asymptotically approach that of a single particle in the first level in the hierarchy. This merging behavior is then verified numerically as well as the exchange of angular momentum which must occur during a near collision of two particles. The resulting particle-like solutions suggest a new class of meshless methods which work in dimensions n≥2n≥2 and which exhibit a shadow of Kelvin’s circulation theorem. More broadly, this provides one of the first finite-dimensional models of self-similarity in ideal fluids.
Issue Date: 2-Jul-2016
Date of Acceptance: 14-Jun-2016
URI: http://hdl.handle.net/10044/1/43430
DOI: http://dx.doi.org/10.1007/s00332-016-9317-6
ISSN: 1432-1467
Publisher: Springer Verlag
Start Page: 1723
End Page: 1765
Journal / Book Title: Journal of Nonlinear Science
Volume: 26
Copyright Statement: © Springer-Verlag 2016. The final publication is available at Springer via http://dx.doi.org/10.1007/s00332-016-9317-6
Keywords: Science & Technology
Physical Sciences
Technology
Mathematics, Applied
Mechanics
Physics, Mathematical
Mathematics
Physics
Regularized fluids
Hamiltonian mechanics
Geometric mechanics
Dual pairs
COADJOINT ORBITS
EPDIFF EQUATION
DIFFEOMORPHISMS
HYDRODYNAMICS
physics.flu-dyn
math-ph
math.DS
math.MP
math.SG
37K65 (Primary), 37K05, 35Q35, 65P10 (Secondary)
Fluids & Plasmas
0102 Applied Mathematics
Publication Status: Published
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics