Differential Equations with singular fields

TL;DR

This paper studies the existence and uniqueness of solutions to differential equations with singular vector fields, focusing on flow properties under less restrictive conditions than divergence bounds, using compactness and BV/SBV regularity.

Contribution

It introduces a new approach to well-posedness of ODEs with singular fields by using compressibility and BV/SBV regularity instead of divergence bounds.

Findings

Existence of flows in dimension 2 with BV vector fields.

Existence of flows in higher dimensions with SBV vector fields.

Provides explicit compactness estimates for solutions.

Abstract

This paper investigates the well posedness of ordinary differential equations and more precisely the existence (or uniqueness) of a flow through explicit compactness estimates. Instead of assuming a bounded divergence condition on the vector field, a compressibility condition on the flow (bounded jacobian) is considered. The main result provides existence under the condition that the vector field belongs to $BV$ in dimension 2 and $SBV$ in higher dimensions.