Weak Dual Pairs and Jetlet Methods for Ideal Incompressible Fluid Models in $$n \ge 2$$ n ≥ 2 Dimensions

Abstract

We review the role of dual pairs in mechanics and use them to derive particlelike 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 ≥ 2 and which exhibit a shadow of Kelvin's circulation theorem. More broadly, this provides one of the first finite-dimensional models of selfsimilarity in ideal fluids.

Correct Definition of Sound Speed and Its Consequences in the Tasks of Hydrodynamics

“…This is evidenced by the fact that they are scarcely cited. For example, we give several studies [7][8][9][10][11][12][13][14][15][16][17][18], published after 2012, and, as before, consider liquids as incompressible and their motion as potential. Work [12] is worthy of particular regret, as it represents lecture notes and is designed for students.…”

Nonlinear dispersion in wave-current interactions

“…The Euler-Poincaré equation (12) for the variational derivatives in (17) yields the CH2 equation in (16).…”

“…In contrast, point vortices in Euler flow define a symplectic momentum map[36] which also generalises to higher-order derivatives,[24,13,12].…”

Nonlinear dispersion in wave-current interactions

“…In contrast, point vortices in Euler flow define a symplectic momentum map[34] which also generalises to higher-order derivatives,[22,13,12]. 3 See[38] for the definitive discussion of dual pairs.…”