On the commutativity of flows of rough vector fields

In the class of Sobolev vector fields in R-n of bounded divergence, for which the theory of DiPerna and Lions provides a well defined notion of flow, we characterize the vector fields whose flow commutes in terms of the Lie bracket and of a regularity condition on the flows themselves. This extends a classical result of Frobenius in the smooth setting. (C) 2021 Elsevier Masson SAS. All rights reserved.