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

François Bouchut (1) , Gianluca Crippa (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 Department of Mathematics [Basel]

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.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

François Bouchut, Gianluca Crippa. Lagrangian flows for vector fields with gradient given by a singular integral. Journal of Hyperbolic Differential Equations, 2013, 10 (2), pp.235-282. ⟨10.1142/S0219891613500100⟩. ⟨hal-00724586⟩