Propagation of singularities by Osgood vector fields and for 2D inviscid incompressible fluids

TL;DR

This paper investigates how specific singular structures in 2D inviscid incompressible fluids are transported by flows generated by Osgood velocity fields, establishing sharp conditions for their propagation and stability.

Contribution

It demonstrates the transport of certain singularities by Osgood velocity fields and proves their preservation and stability in 2D Euler flows, with sharp conditions on singularity types.

Findings

Certain singularities are transported by Osgood flows.

Log-log vortices are preserved and travel with the fluid.

Stability results for weak Euler solutions are established.

Abstract

We show that certain singular structures (H\"{o}lderian cusps and mild divergences) are transported by the flow of homeomorphisms generated by an Osgood velocity field. The structure of these singularities is related to the modulus of continuity of the velocity and the results are shown to be sharp in the sense that slightly more singular structures cannot generally be propagated. For the 2D Euler equation, we prove that certain singular structures are preserved by the motion, e.g. a system of $\log\log_+(1/|x|)$ vortices (and those that are slightly less singular) travel with the fluid in a nonlinear fashion, up to bounded perturbations. We also give stability results for weak Euler solutions away from their singular set.