On the flow map for 2D Euler equations with unbounded vorticity

TL;DR

This paper investigates the behavior of flow maps in 2D Euler equations with unbounded vorticity, showing that solutions can have flow maps with extremely poor regularity, and explores inverse problems related to this phenomenon.

Contribution

It constructs examples of initial velocities leading to flow maps with no positive Holder continuity and examines inverse problems for such irregular flows.

Findings

Flow maps can lack positive Holder regularity for solutions with unbounded vorticity.

Constructed initial velocities produce flow maps with arbitrarily poor modulus of continuity.

Explored inverse problems related to irregular flow map construction.

Abstract

In Part I, we construct a class of examples of initial velocities for which the unique solution to the Euler equations in the plane has an associated flow map that lies in no Holder space of positive exponent for any positive time. In Part II, we explore inverse problems that arise in attempting to construct an example of an initial velocity producing an arbitrarily poor modulus of continuity of the flow map.