Growth of Sobolev norms and loss of regularity in transport equations

TL;DR

This paper demonstrates that in transport equations with irregular divergence-free velocity fields, solutions can lose regularity instantly, even when the velocity is smooth except at a point, highlighting limitations of regularity preservation.

Contribution

It constructs explicit divergence-free velocity fields causing immediate loss of Sobolev regularity in solutions, extending previous results on regularity breakdown in transport equations.

Findings

Solutions do not belong to H^1_{loc} for any positive time.

Velocity fields are smooth except at one point, belonging to many Sobolev spaces.

Loss of regularity occurs even with controlled, almost Lipschitz velocity fields.

Abstract

We consider transport of a passive scalar advected by an irregular divergence free vector field. Given any non-constant initial data $\bar \rho \in H^1_\text{loc}({\mathbb R}^d)$, $d\geq 2$, we construct a divergence free advecting velocity field $v$ (depending on $\bar \rho$) for which the unique weak solution to the transport equation does not belong to $H^1_\text{loc}({\mathbb R}^d)$ for any positive positive time. The velocity field $v$ is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space $W^{s,p}$ that does not embed into the Lipschitz class. The velocity field $v$ is constructed by pulling back and rescaling an initial data dependent sequence of sine/cosine shear flows on the torus. This loss of regularity result complements that in [Ann. PDE, 5(1):Paper No. 9, 19, 2019].