Nonuniqueness of Weak Solutions for the Transport Equation at Critical Space Regularity

Journal Article (Journal Article)

We consider the linear transport equations driven by an incompressible flow in dimensions d≥ 3. For divergence-free vector fields u∈Lt1W1,q, the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class Lt∞Lp when 1p+1q≤1. For such vector fields, we show that in the regime 1p+1q>1, weak solutions are not unique in the class Lt1Lp. One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integration scheme.

Full Text

Duke Authors

Cited Authors

  • Cheskidov, A; Luo, X

Published Date

  • June 1, 2021

Published In

Volume / Issue

  • 7 / 1

Electronic International Standard Serial Number (EISSN)

  • 2199-2576

International Standard Serial Number (ISSN)

  • 2524-5317

Digital Object Identifier (DOI)

  • 10.1007/s40818-020-00091-x

Citation Source

  • Scopus