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 exploits an anisotropic variant of the argument in [14,20] 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.
Keywords: Ordinary differential equations with non smooth vector fields, Continuity and transport equations, Regular Lagrangian flow, Maximal functions and singular integrals, Anisotropic regularity, Vlasov–Poisson equation
@article{AIHPC_2016__33_6_1409_0,
author = {Bohun, Anna and Bouchut, Fran\c{c}ois and Crippa, Gianluca},
title = {Lagrangian flows for vector fields with anisotropic regularity},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1409--1429},
year = {2016},
publisher = {Elsevier},
volume = {33},
number = {6},
doi = {10.1016/j.anihpc.2015.05.005},
zbl = {1353.35118},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.05.005/}
}
TY - JOUR AU - Bohun, Anna AU - Bouchut, François AU - Crippa, Gianluca TI - Lagrangian flows for vector fields with anisotropic regularity JO - Annales de l'I.H.P. Analyse non linéaire PY - 2016 SP - 1409 EP - 1429 VL - 33 IS - 6 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.05.005/ DO - 10.1016/j.anihpc.2015.05.005 LA - en ID - AIHPC_2016__33_6_1409_0 ER -
%0 Journal Article %A Bohun, Anna %A Bouchut, François %A Crippa, Gianluca %T Lagrangian flows for vector fields with anisotropic regularity %J Annales de l'I.H.P. Analyse non linéaire %D 2016 %P 1409-1429 %V 33 %N 6 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.05.005/ %R 10.1016/j.anihpc.2015.05.005 %G en %F AIHPC_2016__33_6_1409_0
Bohun, Anna; Bouchut, François; Crippa, Gianluca. Lagrangian flows for vector fields with anisotropic regularity. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1409-1429. doi: 10.1016/j.anihpc.2015.05.005
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