TY - JOUR

T1 - Growth of Sobolev norms and loss of regularity in transport equations

AU - Crippa, Gianluca

AU - Elgindi, Tarek

AU - Iyer, Gautam

AU - Mazzucato, Anna L.

N1 - Funding Information:
G.C. was partially supported by the ERC Starting grant no. 676675 FLIRT. The remaining authors were partially supported by the US National Science Foundation through grant nos DMS 2043024 and DMS 2124748 to T.E., DMS 1814147 and DMS 2108080 to G. Iyer, and DMS 1909103 to A.L.M. Acknowledgements
Publisher Copyright:
© 2022 The Author(s).

PY - 2022

Y1 - 2022

N2 - 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)'.

AB - 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)'.

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U2 - 10.1098/rsta.2021.0024

DO - 10.1098/rsta.2021.0024

M3 - Article

C2 - 35465718

AN - SCOPUS:85128844579

SN - 0962-8428

VL - 380

JO - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

IS - 2225

M1 - 20210024

ER -