A quantitative DiPerna-Lions theory and exponential mixing rate for passive scalar transport

Abstract

We consider the transport of a passive scalar $f\in\mathbb{R}$ along a divergence-free velocity vector field $u\in\mathbb{R}^d$ on the infinite space $\mathbb{R}^d$. We give a quantitative version of the DiPerna-Lions well-posedness theory for Sobolev vector fields $u \in L_t^1W_x^{1,p}$ when $1<p<\infty$ by giving a uniform decay rate of the DiPerna-Lions commutator. We recover a slightly more general form of the known exponential bound on the mixing rate by a Sobolev vector field $u \in L_t^1W_x^{1,p}$ when $1<p<\infty$.

Locations

• arXiv (Cornell University) - View - PDF