Loss of regularity for the continuity equation with non-Lipschitz velocity field

TL;DR

This paper demonstrates that even with divergence-free velocity fields that are bounded in Sobolev spaces, solutions to the continuity equation can lose regularity instantly, revealing limitations of regularity preservation.

Contribution

It constructs explicit examples of velocity fields in Sobolev spaces causing immediate loss of regularity in solutions to the continuity equation, advancing understanding of regularity behavior.

Findings

Solutions can lose regularity instantly despite smooth initial data.

Constructed velocity fields in Sobolev spaces cause non-preservation of regularity.

Examples based on optimal mixers show sharpness of regularity loss.

Abstract

We consider the Cauchy problem for the continuity equation in space dimension ${d \geq 2}$. We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces $W^{1,p}$, for $1 \leq p<\infty$, and a smooth compactly supported initial datum such that the unique solution to the continuity equation with this initial datum and advecting field does not belong to any Sobolev space of positive fractional order at any positive time. We also construct velocity fields in $W^{r,p}$, with $r>1$, and solutions of the continuity equation with these velocities that exhibit some loss of regularity, as long as the Sobolev space $W^{r,p}$ does not embed in the space of Lipschitz functions. Our constructions are based on examples of optimal mixers from the companion paper "Exponential self-similar mixing by incompressible flows" (J. Amer. Math. Soc. 32 (2019), no. 2), and have been announced in "Exponential self-similar mixing and loss of regularity for continuity equations" (C. R. Math. Acad. Sci. Paris, 352 (2014), no. 11).