Differentiability in Measure of the Flow Associated with a Nearly Incompressible BV Vector Field

We study the regularity of the flow X ( t , y ) , which represents (in the sense of Smirnov or as regular Lagrangian flow of Ambrosio) a solution ρ ∈ L ∞ ( R d + 1 ) of the continuity equation ∂ t ρ + div ( ρ b ) = 0 , with b ∈ L t 1 BV x . We prove that X is differentiable in measure in the sense of Ambrosio–Malý, that is X ( t , y + r z ) - X ( t , y ) r → r → 0 W ( t , y ) z in measure , where the derivative W ( t , y ) is a BV function satisfying the ODE d d t W ( t , y ) = ( D b ) y ( d t ) J ( t - , y ) W ( t - , y ) , where ( D b ) y ( d t ) is the disintegration of the measure ∫ D b ( t , · ) d t with respect to the partition given by the trajectories X ( t , y ) and the Jacobian J ( t , y ) solves d d t J ( t , y ) = ( div b ) y ( d t ) = Tr ( D b ) y ( d t ) . The proof of this regularity result is based on the theory of Lagrangian representations and proper sets introduced by Bianchini and Bonicatto in [ 16 ], on the construction of explicit approximate tubular neighborhoods of trajectories, and on estimates that take into account the local structure of the derivative of a BV vector field.