Abstract

I will discuss recent work on quantitative estimates for the passive scalar transport equation when the advecting velocity field is below the Lipschitz regularity class. In this setting, the classical Cauchy-Lipschitz theory fails to give control on the rate at which the solution may grow, usually measured by Sobolev norms. Despite this, well-posedness of the transport equation has been known when the vector field lies in W ^{since 1989, and for vector fields in BV since 2004, under the additional assumption that the divergence remains bounded. It was conjectured by Bressan in 2003 that the same quantitative “mixing estimates” given by the Cauchy-Lipschitz theory should hold also in this setting. Since then, much progress has been made in recovering these estimates when the vector field lies in W}{1,p} with p>1, however the endpoint case p=1, and in particular BV, has remained a challenging open problem. In this talk I will discuss the first quantitative mixing and stability estimates for a passive scalar transported by W^{1,1} and BV autonomous vector fields with zero divergence. The proof involves quantitative weak harmonic estimates in BV, and a double pigeonhole principle to quantify the weak-* compactness of the solution space.