%ACrippa, Gianluca%AElgindi, Tarek%AIyer, Gautam%AMazzucato, Anna%BJournal Name: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences; Journal Volume: 380; Journal Issue: 2225
%D2022%I
%JJournal Name: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences; Journal Volume: 380; Journal Issue: 2225
%K
%MOSTI ID: 10338847
%PMedium: X
%TGrowth of Sobolev norms and loss of regularity in transport equations
%XWe consider transport of a passive scalar advected by an irregular divergence-free vector field. Given any non-constant initial data ρ ¯ ∈ H loc 1 ( R d ) , d ≥ 2 , we construct a divergence-free advecting velocity field v (depending on ρ ¯ ) for which the unique weak solution to the transport equation does not belong to H loc 1 ( R d ) for any 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 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)’.
%0Journal Article