Lagrangian flows for vector fields with anisotropic regularity

Anna Bohun (1) , François Bouchut (2) , Gianluca Crippa (1)

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

We prove quantitative estimates for flows of vector fields subject to anisotropic regu-larity 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 ex-ploits an anisotropic variant of the argument in [19, 13] 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.

Mots clés: transport equation; continuity equation; Ordinary differential equations with non smooth vector fields; regular Lagrangian flow; maximal functions; singular integrals; anisotropic regularity; Vlasov-Poisson equation

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Anna Bohun, François Bouchut, Gianluca Crippa. Lagrangian flows for vector fields with anisotropic regularity. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2016, 33 (6), pp.1409-1429. ⟨10.1016/j.anihpc.2015.05.005⟩. ⟨hal-01097370⟩