Lagrangian flows for vector fields with gradient given by a singular integral

摘要

We prove quantitative estimates on flows of ordinary differential equations with vector field with gradient given by a singular integral of an L ~1 function. Such estimates allow to prove existence, uniqueness, quantitative stability and compactness for the flow, going beyond the BV theory. We illustrate the related well-posedness theory of Lagrangian solutions to the continuity and transport equations.