On the global evolution of vortex filaments, blobs, and small loops in 3D ideal flows

TL;DR

This paper demonstrates that a broad class of approximate models for vortex evolution in 3D ideal fluids have globally smooth solutions, supporting their use in turbulence modeling and numerical simulations.

Contribution

It establishes the global regularity of various approximate vortex models using Hamiltonian functions, bridging theoretical analysis and computational turbulence.

Findings

Approximate vortex models have global smooth solutions.

Hamiltonian functions underpin the models' stability.

Results support the validity of turbulence simulation methods.

Abstract

We consider a wide class of approximate models of evolution of singular distributions of vorticity in three dimensional incompressible fluids and we show that they have global smooth solutions. The proof exploits the existence of suitable Hamiltonian functions. The approximate models we analyze (essentially discrete and continuous vortex filaments and vortex loops) are related to some problem of classical physics concerning turbulence and also to the numerical approximation of flows with very high Reynolds number. Finally, we interpret our results as a basis to theoretical validation of numerical methods used in state-of-the-art computations of turbulent flows.