Weak dual pairs and jetlet methods for ideal incompressible fluid models in $n\geq 2$ dimensions

TL;DR

This paper introduces jetlet methods, a hierarchy of finite-dimensional particle-like solutions derived from dual pairs and jets, modeling self-similarity and conservation laws in ideal incompressible fluids across multiple dimensions.

Contribution

It develops a novel framework of jetlet particles using augmented weak dual pairs, providing finite-dimensional models of fluid self-similarity and Kelvin's circulation theorem.

Findings

Jetlet solutions exhibit conservation laws related to Kelvin's theorem.

Particles merging asymptotically behave like a single particle at a higher hierarchy level.

Numerical simulations confirm the merging dynamics and angular momentum exchange.

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 $n \geq 2$ 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.