论文标题
确定性传输方程的良好特性，沿分数布朗路径扰动奇异速度场
Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths
论文作者
论文摘要
在本文中，我们从戴维（Davie \ Cite \ cite {davie07}和Shaposhnikov \ cite {shaposhnikov16}的davie \ cite {davie07}的意义上证明了途中的唯一性，因为SDE是由Brownian Motion驱动的。仅是界限和可衡量的。\ par 使用此结果，我们在$ w_ {loc}^{k，p} \ left（[0,1] \ times \ mathbb {r}^d \ right）$，$ p> d $，经典传输方程和连续性方程的单个速度型棕色路径上遇到单个速率。\ par。 后一个结果提供了一种系统的方式来产生奇异速度场的实例，无法通过diperna-Lyons \ cite {dipernalions89}，ambrosio \ ambrosio \ cite \ cite {ambrosio04}或crippa lellis lellis lellis \ cite \ cite {crippadeliss08}的规律性理论来处理。 我们的方法是基于通过一系列动荡向量场产生的流量水平的先验估计值，收敛到原始矢量场，并且相对于Mollification参数均匀。此外，我们使用基于\ cite {dmn92}的malliavin微积分的紧凑型标准以及上限浓度不平等。 \ emph {关键字}：传输方程，紧凑标准，单数向量场，正规化噪声。
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.
