Quantitative Estimates in Passive Scalar Transport: A Unified Approach in $W^{1,p}$ via Christ-Journ\'e Commutator Estimates

Abstract

This paper establishes new applications of the Christ-Journ\'e singular integral estimate to the transport equation for divergence-free vector fields in the Sobolev class $W^{1,p}$ with $p>1$. Our main result is a stability estimate for the Cauchy problem, which quantifies the continuous dependence of solutions on initial data in the weak topology by giving bounds on the transfer of mass from high frequencies to low frequencies. Among other applications to stability and vanishing diffusion, this implies a mixing bound valid for all initial data in the DiPerna-Lions well-posedness class, with the mixing rate expressed in terms of a dimensionless ratio of the passive scalar and vector field. We also demonstrate that the passive scalar propagates a full "logarithm of a derivative" and finally discuss connections between the commutator estimates developed in this paper and uniform decay rates for the standard DiPerna-Lions commutator.