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 \cite{Davie07} and Shaposhnikov \cite{Shaposhnikov16} for SDE's driven by a fractional Brownian motion with a Hurst parameter $H\in(0,\frac{1}{2})$, uniformly in the initial conditions, where the drift vector field is allowed to be merely bounded and measurable.\par Using this result, we construct weak unique regular solutions in $W_{loc}^{k,p}\left([0,1]\times\mathbb{R}^d\right)$, $p>d$ of the classical transport and continuity equations with singular velocity fields perturbed along fractional Brownian paths.\par The latter results provide a systematic way of producing examples of singular velocity fields, which cannot be treated by the regularity theory of DiPerna-Lyons \cite{DiPernaLions89}, Ambrosio \cite{Ambrosio04} or Crippa-De Lellis \cite{CrippaDeLellis08}.\par 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 \cite{DMN92} as well as supremum concentration inequalities. \emph{keywords}: Transport equation, Compactness criterion, Singular vector fields, Regularization by noise.