Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths

In this article we prove path-by-path uniqueness in the sense of Davie [25] and Shaposhnikov [46] for SDE's driven by a fractional Brownian motion with a sufficiently small Hurst parameter H is an element of (0, 21), uniformly in the initial conditions, where the drift vector field is allowed to be merely bounded and measurable.Using this result, we construct weak unique regular solutions in Wk,p [0, 1] x Rd), p > d of the classical transport and continuity equations with singular velocity fields perturbed along fractional Brownian paths. The latter results provide a systematic way of producing examples of singular velocity fields, which cannot be treated by the regularity theory of DiPerna-Lions [28], Ambrosio [2] or Crippa-De Lellis [23].Our approach is based on a priori estimates at the level of flows generated by a sequence of mollified vector fields, converging to the original vector field, and which are uniform with respect to the mollification parameter. In addition, we use a compactness criterion based on Malliavin calculus from [24] as well as a supremum estimate in time of moments of the derivative of the flow of SDE solutions.(c) 2023 Elsevier Inc. All rights reserved.