On the role of continuous symmetries in the solution of the 3D Euler fluid equations and related models

TL;DR

This paper applies continuous symmetry methods to analyze the 3D Euler equations, identifying constants of motion, symmetries, and blowup behaviors, providing explicit solutions and insights into flow topology and singularity formation.

Contribution

It introduces a symmetry-based approach to solving and understanding the 3D Euler equations and related models, including explicit blowup characterizations.

Findings

Vorticity field is a flow symmetry, enabling new symmetry constructions.

Topology influences the existence of additional symmetries in steady flows.

Explicit blowup exponents and prefactors are derived for certain initial conditions.

Abstract

We review and apply the continuous symmetry approach to find the solution of the 3D Euler fluid equations in several instances of interest, via the construction of constants of motion and infinitesimal symmetries, without recourse to Noether's theorem. We show that the vorticity field is a symmetry of the flow, so if the flow admits another symmetry then a Lie algebra of new symmetries can be constructed. For steady Euler flows this leads directly to the distinction of (non-)Beltrami flows: an example is given where the topology of the spatial manifold determines whether extra symmetries can be constructed. Next, we study the stagnation-point-type exact solution of the 3D Euler fluid equations introduced by Gibbon et al. (Physica D, vol.132, 1999, pp.497-510) along with a one-parameter generalisation of it introduced by Mulungye et al. (J. Fluid Mech., vol.771, 2015, pp.468-502). Applying the symmetry approach to these models allows for the explicit integration of the fields along pathlines, revealing a fine structure of blowup for the vorticity, its stretching rate, and the back-to-labels map, depending on the value of the free parameter and on the initial conditions. Finally, we produce explicit blowup exponents and prefactors for a generic type of initial conditions.