A unified treatment of ODEs under Osgood and Sobolev type conditions

TL;DR

This paper develops a unified approach to analyze the existence, uniqueness, and regularity of solutions to ODEs under Osgood and Sobolev conditions, extending the DiPerna-Lions theory using a direct method.

Contribution

It introduces a unified framework for ODEs under combined Osgood and Sobolev conditions, broadening the applicability of DiPerna-Lions flow analysis.

Findings

Proves existence and uniqueness of flows under almost everywhere Osgood continuity.

Establishes regularity results for the DiPerna-Lions flow.

Extends the direct method to a broader class of vector fields.

Abstract

In this paper we present a unified treatment for the ordinary differential equations under the Osgood and Sobolev type conditions, following Crippa and de Lellis's direct method. More precisely, we prove the existence, uniqueness and regularity of the DiPerna-Lions flow generated by a vector field which is "almost everywhere Osgood continuous".