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