Growth of Sobolev norms and loss of regularity in transport equations

Gianluca Crippa, Tarek Elgindi, Gautam Iyer, Anna L. Mazzucato

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


We consider transport of a passive scalar advected by an irregular divergence-free vector field. Given any non-constant initial data ρ ∈ H1loc(Rd), 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 H1loc(Rd) 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 Ws,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)'.

Original languageEnglish (US)
Article number20210024
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2225
StatePublished - 2022

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • General Engineering
  • General Physics and Astronomy


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