Fuzzy Fluid Mechanics in Three Dimensions

TL;DR

This paper introduces a rotation invariant non-commutative algebra for 3D ideal fluid mechanics, providing a new framework that preserves Hamiltonian structure and may aid in understanding singularities in Euler and Navier-Stokes equations.

Contribution

It proposes a novel non-commutative algebraic approach with a short-distance cutoff that maintains Hamiltonian structure in 3D fluid dynamics.

Findings

Established a non-commutative algebra of functions in position space.

Derived analogues of Euler equations within this new framework.

Potentially useful for studying singularities in fluid evolution.

Abstract

We introduce a rotation invariant short distance cut-off in the theory of an ideal fluid in three space dimensions, by requiring momenta to take values in a sphere. This leads to an algebra of functions in position space is non-commutative. Nevertheless it is possible to find appropriate analogues of the Euler equations of an ideal fluid. The system still has a hamiltonian structure. It is hoped that this will be useful in the study of possible singularities in the evolution of Euler (or Navier-Stokes) equations in three dimensions.