Non-uniqueness in plane fluid flows

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

Full-text PDF

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Additional Information

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.

References
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*Stabilization with relaxed controls*, Nonlinear Anal. **7** (1983), no. 11, 1163–1173. MR **721403**, DOI 10.1016/0362-546X(83)90049-4
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*Dissipative solutions and semiflow selection for the complete Euler system*, Comm. Math. Phys. **376** (2020), no. 2, 1471–1497. MR **4103973**, DOI 10.1007/s00220-019-03662-7
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*The initial value problem for the Euler equations of incompressible fluids viewed as a concave maximization problem*, Commun. Math. Phys. **365** (2018), 579–605.
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*Dissipative continuous Euler flows*, Invent. Math. **193** (2013), no. 2, 377–407. MR **3090182**, DOI 10.1007/s00222-012-0429-9
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*Maximal dissipation and well-posedness for the compressible Euler system*, J. Math. Fluid Mech. **16** (2014), no. 3, 447–461. MR **3247361**, DOI 10.1007/s00021-014-0163-8
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*Maximum entropy production as a necessary admissibility condition for the fluid Navier-Stokes and Euler equations*, SN Applied Sciences **2** (2020), 2160.
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*The common extremalities in biology and physics*, Elsevier Insights, Elsevier, 2011.
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*Some basic problems of the mathematical theory of elasticity. Fundamental equations, plane theory of elasticity, torsion and bending*, P. Noordhoff Ltd., Groningen, 1953. Translated by J. R. M. Radok. MR **58417**
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*Étude Thermodynamique des Phenomenes Irreversibiles*, Editions Desoer. Liege, 1947.
- James C. Robinson and Witold Sadowski,
*Almost-everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier-Stokes equations*, Nonlinearity **22** (2009), no. 9, 2093–2099. MR **2534294**, DOI 10.1088/0951-7715/22/9/002
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*A criterion for uniqueness of Lagrangian trajectories for weak solutions of the 3D Navier-Stokes equations*, Comm. Math. Phys. **290** (2009), no. 1, 15–22. MR **2520506**, DOI 10.1007/s00220-009-0819-z
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*An inviscid flow with compact support in space-time*, J. Geom. Anal. **3** (1993), no. 4, 343–401. MR **1231007**, DOI 10.1007/BF02921318
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*On the nonuniqueness of weak solution of the Euler equation*, Comm. Pure Appl. Math. **50** (1997), no. 12, 1261–1286. MR **1476315**, DOI 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.3.CO;2-4
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*Chemical reactions and the principle of maximal rate of entropy production*, Z. Angew. Math. Phys. **34** (1983), no. 6, 832–844 (English, with German summary). MR **732933**, DOI 10.1007/BF00949059

References
- Zvi Artstein,
*Stabilization with relaxed controls*, Nonlinear Anal. **7** (1983), no. 11, 1163–1173. MR **721403**, DOI 10.1016/0362-546X(83)90049-4
- Dominic Breit, Eduard Feireisl, and Martina Hofmanová,
*Dissipative solutions and semiflow selection for the complete Euler system*, Comm. Math. Phys. **376** (2020), no. 2, 1471–1497. MR **4103973**, DOI 10.1007/s00220-019-03662-7
- Y. Brenier,
*The initial value problem for the Euler equations of incompressible fluids viewed as a concave maximization problem*, Commun. Math. Phys. **365** (2018), 579–605.
- Tristan Buckmaster and Vlad Vicol,
*Nonuniqueness of weak solutions to the Navier-Stokes equation*, Ann. of Math. (2) **189** (2019), no. 1, 101–144. MR **3898708**, DOI 10.4007/annals.2019.189.1.3
- J. C. Burkill,
*The theory of ordinary differential equations*, 3rd ed., Longman Group, London, New York, 1975.
- Elisabetta Chiodaroli and Ondrej Kreml,
*On the energy dissipation rate of solutions to the compressible isentropic Euler system*, Arch. Ration. Mech. Anal. **214** (2014), no. 3, 1019–1049. MR **3269641**, DOI 10.1007/s00205-014-0771-8
- C. M. Dafermos,
*Maximal dissipation in equations of evolution*, J. Differential Equations **252** (2012), no. 1, 567–587. MR **2852218**, DOI 10.1016/j.jde.2011.08.006
- Camillo De Lellis and László Székelyhidi Jr.,
*The Euler equations as a differential inclusion*, Ann. of Math. (2) **170** (2009), no. 3, 1417–1436. MR **2600877**, DOI 10.4007/annals.2009.170.1417
- Camillo De Lellis and László Székelyhidi Jr.,
*On admissibility criteria for weak solutions of the Euler equations*, Arch. Ration. Mech. Anal. **195** (2010), no. 1, 225–260. MR **2564474**, DOI 10.1007/s00205-008-0201-x
- Camillo De Lellis and László Székelyhidi Jr.,
*Dissipative continuous Euler flows*, Invent. Math. **193** (2013), no. 2, 377–407. MR **3090182**, DOI 10.1007/s00222-012-0429-9
- Eduard Feireisl,
*Maximal dissipation and well-posedness for the compressible Euler system*, J. Math. Fluid Mech. **16** (2014), no. 3, 447–461. MR **3247361**, DOI 10.1007/s00021-014-0163-8
- J. Glimm, D. Lazarev, and G.-Q. G. Chen,
*Maximum entropy production as a necessary admissibility condition for the fluid Navier-Stokes and Euler equations*, SN Applied Sciences **2** (2020), 2160.
- Ernst Hairer, Christian Lubich, and Gerhard Wanner,
*Geometric numerical integration*, 2nd ed., Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2006. Structure-preserving algorithms for ordinary differential equations. MR **2221614**
- Robert Lasarzik,
*Maximally dissipative solutions for incompressible fluid dynamics*, Z. Angew. Math. Phys. **73** (2022), no. 1, Paper No. 1, 21. MR **4338593**, DOI 10.1007/s00033-021-01628-1
- P.-L. Lions,
*Mathematical topics in fluid mechanics*, Vol. I, The Clarendon Press, Oxford, New York, 1996.
- A. Moroz,
*The common extremalities in biology and physics*, Elsevier Insights, Elsevier, 2011.
- N. I. Muskhelishvili,
*Some basic problems of the mathematical theory of elasticity. Fundamental equations, plane theory of elasticity, torsion and bending*, P. Noordhoff Ltd., Groningen, 1953. Translated by J. R. M. Radok. MR **0058417**
- I. Prigogine,
*Étude Thermodynamique des Phenomenes Irreversibiles*, Editions Desoer. Liege, 1947.
- James C. Robinson and Witold Sadowski,
*Almost-everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier-Stokes equations*, Nonlinearity **22** (2009), no. 9, 2093–2099. MR **2534294**, DOI 10.1088/0951-7715/22/9/002
- James C. Robinson and Witold Sadowski,
*A criterion for uniqueness of Lagrangian trajectories for weak solutions of the 3D Navier-Stokes equations*, Comm. Math. Phys. **290** (2009), no. 1, 15–22. MR **2520506**, DOI 10.1007/s00220-009-0819-z
- Vladimir Scheffer,
*An inviscid flow with compact support in space-time*, J. Geom. Anal. **3** (1993), no. 4, 343–401. MR **1231007**, DOI 10.1007/BF02921318
- A. Shnirelman,
*On the nonuniqueness of weak solution of the Euler equation*, Comm. Pure Appl. Math. **50** (1997), no. 12, 1261–1286. MR **1476315**, DOI 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.3.CO;2-4
- Hans Ziegler,
*Chemical reactions and the principle of maximal rate of entropy production*, Z. Angew. Math. Phys. **34** (1983), no. 6, 832–844 (English, with German summary). MR **732933**, DOI 10.1007/BF00949059

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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