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Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Non-uniqueness in plane fluid flows

Authors: Heiko Gimperlein, Michael Grinfeld, Robin J. Knops and Marshall Slemrod
Journal: Quart. Appl. Math.
MSC (2020): Primary 76N10; Secondary 34A12, 35Q35
Published electronically: June 15, 2023
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Abstract | References | Similar Articles | Additional Information


Examples of dynamical systems proposed by Z. Artstein and C. M. Dafermos admit non-unique solutions that track a one parameter family of closed circular orbits contiguous at a single point. Switching between orbits at this single point produces an infinite number of solutions with the same initial data. Dafermos appeals to a maximal entropy rate criterion to recover uniqueness.

These results are here interpreted as non-unique Lagrange trajectories on a particular spatial region. The corresponding special velocity is proved consistent with plane steady compressible fluid flows that for specified pressure and mass density satisfy not only the Euler equations but also the Navier-Stokes equations for specially chosen volume and (positive) shear viscosities. The maximal entropy rate criterion recovers uniqueness.

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Additional Information

Heiko Gimperlein
Affiliation: Engineering Mathematics, University of Innsbruck, Innsbruck, Austria; and Department of Mathematical, Physical and Computer Sciences, University of Parma, 43124 Parma, Italy
MR Author ID: 922641

Michael Grinfeld
Affiliation: Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, United Kingdom
MR Author ID: 233903
ORCID: 0000-0002-3897-8819

Robin J. Knops
Affiliation: The Maxwell Institute of Mathematical Sciences and School of Mathematical and Computing Sciences, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, United Kingdom
MR Author ID: 103430
ORCID: 0000-0001-9891-203X

Marshall Slemrod
Affiliation: Department of Mathematics, University of Wisconsin, Madison, WI 53706
MR Author ID: 163635
ORCID: 0000-0002-0514-9467

Received by editor(s): January 22, 2023
Received by editor(s) in revised form: March 24, 2023
Published electronically: June 15, 2023
Article copyright: © Copyright 2023 Brown University