# Preprint 2015-022

# Lagrangian flows for vector fields with anisotropic regularity

## Anna Bohun, François Bouchut and Gianluca Crippa

**Abstract:**
We prove quantitative estimates for flows of vector fields
subject to anisotropic regularity 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 [20, 14]
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.

**References**

[14] F. Bouchut & G. Crippa: Lagrangian flows for vector fields with gradient given by a singular integral.

*J. Hyperbolic Differ. Equ.*,

**10**(2013), 235–282 [MR3078074].

[20] G. Crippa & C. De Lellis: Estimates and regularity results for the DiPerna-Lions flow.

*J. Reine Angew. Math.*,

**616**(2008), 15–46 [MR2369485].