Dynamics of quasiconformal fields

TL;DR

This paper proves a uniqueness theorem for autonomous ODE systems with Sobolev vector fields exhibiting specific geometric properties, ensuring a unique integral curve through each non-critical point.

Contribution

It introduces a new uniqueness result for ODEs with Sobolev vector fields possessing delta-monoticity or reduced quasiconformality.

Findings

Unique integral curves through non-critical points

Extension of classical ODE uniqueness to Sobolev vector fields

Application of geometric structures to ensure solution uniqueness

Abstract

A uniqueness theorem is established for autonomous systems of ODEs, $\dot{x}=f(x)$, where $f$ is a Sobolev vector field with additional geometric structure, such as delta-monoticity or reduced quasiconformality. Specifically, through every non-critical point of $f$ there passes a unique integral curve.