Lagrangian flows for vector fields with anisotropic regularity

TL;DR

This paper establishes quantitative estimates for flows of vector fields with anisotropic regularity, crucial for understanding solutions to equations like Vlasov-Poisson with measure densities, using an anisotropic approach that avoids regularization.

Contribution

It introduces an anisotropic variant of the Crippa-De Lellis argument, providing explicit estimates for vector fields with mixed regularity, advancing the analysis of Lagrangian flows.

Findings

Quantitative estimates for anisotropic vector fields

Recovery of well-posedness for ODEs and PDEs with measure-based regularity

Application to Vlasov-Poisson equation with measure density

Abstract

We prove quantitative estimates for flows of vector fields subject to anisotropic regularity conditions: some derivatives of some components are (singular integrals of) measures, while the remaining derivatives are (singular integrals of) integrable functions. This is motivated by the regularity of the vector field in the Vlasov-Poisson equation with measure density. The proof exploits an anisotropic variant of the argument of Crippa and De Lellis and suitable estimates for the difference quotients in such anisotropic context. In contrast to regularization methods, this approach gives quantitative estimates in terms of the given regularity bounds. From such estimates it is possible to recover the well posedness for the ordinary differential equation and for Lagrangian solutions to the continuity and transport equations.