On the apparition of singularities of vector fields transported by volume preserving diffeomorphisms

TL;DR

This paper investigates the formation of singularities in vector fields transported by volume-preserving diffeomorphisms, with implications for understanding singularity development in fluid dynamics equations like Euler and MHD.

Contribution

It establishes relations between vector field directions and eigenvectors of the derivative of the back-to-label map, providing new invariants and insights into singularity mechanisms.

Findings

Relations between vector field directions and eigenvectors near singularities

Invariant properties of integral curves during motion

Implications for vortex and material stretching in fluid equations

Abstract

We consider possible generation of singularities of a vector field transported by diffeomorphisms with derivatives of uniformly bounded determinants. A particular case of volume preserving diffeomrphism is the most important, since it has direct applications to the incompressible, inviscid hydrodynamics. We find relations between the directions of the vector field and the eigenvectors of the derivative of the back-to-label map near the singularity. We also find an invariant when we follow the motion of the integral curves of the vector field. For the 3D Euler equations these results have immediate implications about the directions of the vortex stretching and the material stretching near the possible singularities. We also have similar applications to the other inviscid, incompressible fluid equations such as the 2D quasi-geostrophic equation and the 3D magnetohydrodynamics equations.