Non-uniqueness in plane fluid flows

Gimperlein, Heiko; Grinfeld, Michael; Knops, Robin; Slemrod, Marshall

doi:10.1090/qam/1670

Authors: Heiko Gimperlein, Michael Grinfeld, Robin J. Knops and Marshall Slemrod
Journal: Quart. Appl. Math. 82 (2024), 535-561
MSC (2020): Primary 76N10; Secondary 34A12, 35Q35
DOI: https://doi.org/10.1090/qam/1670
Published electronically: June 15, 2023
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Abstract:

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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