Lagrangian flows for vector fields with gradient given by a singular   integral
  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. 
Fran\c{c}ois Bouchut (LAMA), Gianluca Crippa
math.AP math.FA