Exponential self-similar mixing and loss of regularity for continuity equations

TL;DR

This paper demonstrates exponential mixing decay and loss of regularity in solutions to the continuity equation with Sobolev velocity fields, highlighting optimal bounds and the effects of non-Lipschitz velocities.

Contribution

It provides explicit examples of exponential mixing and constructs solutions with Sobolev but non-Lipschitz velocities showing instantaneous regularity loss.

Findings

Exponential decay of mixing scale shown for Sobolev velocity fields.

Constructed solutions exhibit instantaneous loss of fractional Sobolev regularity.

Examples confirm the optimality of known lower bounds on mixing decay.

Abstract

We consider the mixing behaviour of the solutions of the continuity equation associated with a divergence-free velocity field. In this announcement we sketch two explicit examples of exponential decay of the mixing scale of the solution, in case of Sobolev velocity fields, thus showing the optimality of known lower bounds. We also describe how to use such examples to construct solutions to the continuity equation with Sobolev but non-Lipschitz velocity field exhibiting instantaneous loss of any fractional Sobolev regularity.