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 sufficiently small Hurst parameter H?(0,12), uniformly initial conditions, where drift vector field is allowed to be merely bounded measurable. Using result, construct weak unique regular solutions Wlock,p([0,1]×Rd), p>d classical transport continuity equations singular velocity fields perturbed along paths. The latter results provide systematic way producing examples fields, which cannot treated regularity theory DiPerna-Lions [28], Ambrosio [2] or Crippa-De Lellis [23]. Our approach based on priori estimates at level flows generated sequence mollified converging original field, are uniform respect mollification parameter. addition, use compactness criterion Malliavin calculus from [24] as well supremum estimate time moments derivative flow SDE solutions.