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Extreme temporal
intermittency in the linear Sobolev transport: Almost smooth
nonunique solutions
Alexey Cheskidov and Xiaoyutao Luo |
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Vol. 17 (2024), No. 6, 2161–2177
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Abstract |
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We revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity for all in space dimensions whose transport equations admit nonunique weak solutions belonging to for all and . In particular, our result shows that the time-integrability assumption in the uniqueness of the DiPerna–Lions theory is essential. The same result also holds for transport-diffusion equations with diffusion operators of arbitrarily large order in any dimensions . |
Keywords
transport equation, nonuniqueness, convex integration
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Mathematical Subject Classification
Primary: 35A02, 35D30, 35Q35
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Milestones
Received: 26 May 2022
Revised: 11 November 2022
Accepted: 10 February 2023
Published: 19 July 2024
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Authors |
| Alexey Cheskidov | |
| Department of Mathematics,
Statistics and Computer Science University of Illinois at Chicago Chicago, IL United States |
School of Mathematics Institute for Advanced Study Princeton, NJ United States |
| Xiaoyutao Luo | |
| Department of Mathematics Duke University Durham, NC United States |
School of Mathematics Institute for Advanced Study Princeton, NJ United States |
| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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