Growth of Sobolev norms and loss of regularity in transport equations

We consider transport of a passive scalar advected by an irregular divergence-free vector field. Given any non-constant initial data [Formula: see text], [Formula: see text], we construct a divergence-free advecting velocity field [Formula: see text] (depending on [Formula: see text]) for which the unique weak solution to the transport equation does not belong to [Formula: see text] for any positive time. The velocity field [Formula: see text] is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space [Formula: see text] that does not embed into the Lipschitz class. The velocity field [Formula: see text] is constructed by pulling back and rescaling a sequence of sine/cosine shear flows on the torus that depends on the initial data. This loss of regularity result complements that in Ann. PDE, 5(1):Paper No. 9, 19, 2019. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.