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doi: 10.3934/dcds.2022054
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The topology of Bott integrable fluids

Institut de Recherche Mathématique Avancée, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg, France

*Corresponding author: Robert Cardona

Received  November 2021 Early access April 2022

We construct non-vanishing steady solutions to the Euler equations (for some metric) with analytic Bernoulli function in each three-manifold where they can exist: graph manifolds. Using the theory of integrable systems, any admissible Morse-Bott function can be realized as the Bernoulli function of some non-vanishing steady Euler flow. This can be interpreted as an inverse problem to Arnold's structure theorem and yields as a corollary the topological classification of such solutions. Finally, we prove that the topological obstruction holds without the non-vanishing assumption: steady Euler flows with a Morse-Bott Bernoulli function only exist on graph three-manifolds.

Citation: Robert Cardona. The topology of Bott integrable fluids. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022054
References:
[1]

V. I. Arnold, Sur la topologie des écoulements stationnaires des fluides parfaits, C. R. Acad. Sci. Paris, 261 (1965), 17-20. 

[2]

V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag, New York, 1998.

[3]

A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian systems. Geometry, Topology and Classification, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203643426.

[4]

A. V. BolsinovA. T. Fomenko and S. V. Matveev, Topological classification of integrable Hamiltonian systems with two degrees of freedom. A list of systems of small complexity, Russian Math. Surveys, 45 (1990), 49-77.  doi: 10.1070/RM1990v045n02ABEH002344.

[5]

A. V. Brailov and A. T. Fomenko, The topology of integral submanifolds of completely integrable Hamiltonian systems, Math. USSR-Sb, 62 (1989), 373-383.  doi: 10.1070/SM1989v062n02ABEH003244.

[6]

R. Cardona, Steady Euler flows and Beltrami fields in high dimensions, Ergodic Theory Dynam. Systems, 41 (2021), 3610-3633.  doi: 10.1017/etds.2020.124.

[7]

R. Cardona, E. Miranda, D. Peralta-Salas and F. Presas, Universality of Euler flows and flexibility of Reeb embeddings, Preprint, 2019, arXiv: 1911.01963.

[8]

J. Casasayas, J. Martinez-Alfaro and A. Nunes, Knotted periodic orbits and integrability, Hamiltonian Systems and Celestial Mechanics(Guanajuato 1991), Adv. Ser. Nonlinear Dynam., World Sci. Pub., River Edge, NJ, 4 (1993), 35–44.

[9]

K. Cieliebak and E. Volkov, First steps in stable Hamiltonian topology, J. Eur. Math. Soc. (JEMS), 17 (2015), 321-404.  doi: 10.4171/JEMS/505.

[10]

K. Cieliebak and E. Volkov, A note on the stationary Euler equations of hydrodynamics, Ergodic Theory Dynam. Systems, 37 (2017), 454-480.  doi: 10.1017/etds.2015.50.

[11]

A. Enciso and D. Peralta-Salas, Knots and links in steady solutions of the Euler equation, Ann. of Math., 175 (2012), 345-367.  doi: 10.4007/annals.2012.175.1.9.

[12]

J. Etnyre and R. Ghrist, Stratified integrals and unknots in inviscid flows, Contemp. Math., 246 (1999), 99-111.  doi: 10.1090/conm/246/03777.

[13]

J. Etnyre and R. Ghrist, Contact topology and hydrodynamics I. Beltrami fields and the Seifert conjecture, Nonlinearity, 13 (2000), 441-458.  doi: 10.1088/0951-7715/13/2/306.

[14]

A. T. Fomenko, The topology of surfaces of constant energy in integrable Hamiltonian systems and obstructions to integrability, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 1276-1307. 

[15]

A. T. Fomenko and S. V. Matveev, Morse type theory for integrable Hamiltonian systems with tame integrals, Math. Notes, 43 (1988), 382-386.  doi: 10.1007/BF01158846.

[16]

A. T. Fomenko and H. Zieschang, On the topology of three-dimensional manifolds arising in Hamiltonian mechanics, Dokl. Akad. Nauk SSSR, 294 (1987), 283-287. 

[17]

A. T. Fomenko and H. Zieschang, On typical topological properties of integrable Hamiltonian systems, Izvest. Akad. Nauk SSSR, Ser. Matem., 52 (1988), 378-407. 

[18]

A. Izosimov and B. Khesin, Characterization of steady solutions to the 2D Euler equation, Int. Math. Res. Not. IMRN, 2017 (2017), 7459-7503.  doi: 10.1093/imrn/rnw230.

[19]

M. Jankins and W. D. Neumann, Lectures on Seifert Manifolds, Brandeis lecture notes 2, Brandeis University, Waltham, MA (1983).

[20]

B. KhesinS. Kuksin and D. Peralta-Salas, KAM theory and the 3D Euler equation, Adv. Math., 267 (2014), 498-522.  doi: 10.1016/j.aim.2014.09.009.

[21]

I. Kirillov, Classification of coadjoint orbits for symplectomorphism groups of surfaces, Int. Math. Res. Not. IMRN, (2022). doi: 10.1093/imrn/rnac041.

[22]

D. Peralta-Salas, Selected topics on the topology of ideal fluid flows, Int. J. Geom. Methods Mod. Phys., 13 (2016), 1630012, 23 pp. doi: 10.1142/S0219887816300129.

[23]

D. Peralta-SalasA. Rechtman and F. Torres de Lizaur, A characterization of 3D Euler flows using commuting zero-flux homologies, Ergodic Theory Dynam. Systems, 41 (2021), 2166-2181.  doi: 10.1017/etds.2020.25.

[24]

H. Seifert, Topologie dreidimensionaler gefaserter räume, Acta Math., 60 (1933), 147-238.  doi: 10.1007/BF02398271.

[25]

M. Shiota, Equivalence of differentiable mappings and analytic mappings, Inst. Hautes Études Sci. Publ. Math., (1981), 237–322.

[26]

F. Waldhausen, Eine Klasse von 3-dimensionalen Mannifaltigkeiten I, Invent. Math., 3 (1967), 308-333.  doi: 10.1007/BF01402956.

[27]

F. Waldhausen, Eine Klasse von 3-dimensionalen Mannifaltigkeiten II, Invent. Math., 4 (1967), 87-117. 

show all references

References:
[1]

V. I. Arnold, Sur la topologie des écoulements stationnaires des fluides parfaits, C. R. Acad. Sci. Paris, 261 (1965), 17-20. 

[2]

V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag, New York, 1998.

[3]

A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian systems. Geometry, Topology and Classification, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203643426.

[4]

A. V. BolsinovA. T. Fomenko and S. V. Matveev, Topological classification of integrable Hamiltonian systems with two degrees of freedom. A list of systems of small complexity, Russian Math. Surveys, 45 (1990), 49-77.  doi: 10.1070/RM1990v045n02ABEH002344.

[5]

A. V. Brailov and A. T. Fomenko, The topology of integral submanifolds of completely integrable Hamiltonian systems, Math. USSR-Sb, 62 (1989), 373-383.  doi: 10.1070/SM1989v062n02ABEH003244.

[6]

R. Cardona, Steady Euler flows and Beltrami fields in high dimensions, Ergodic Theory Dynam. Systems, 41 (2021), 3610-3633.  doi: 10.1017/etds.2020.124.

[7]

R. Cardona, E. Miranda, D. Peralta-Salas and F. Presas, Universality of Euler flows and flexibility of Reeb embeddings, Preprint, 2019, arXiv: 1911.01963.

[8]

J. Casasayas, J. Martinez-Alfaro and A. Nunes, Knotted periodic orbits and integrability, Hamiltonian Systems and Celestial Mechanics(Guanajuato 1991), Adv. Ser. Nonlinear Dynam., World Sci. Pub., River Edge, NJ, 4 (1993), 35–44.

[9]

K. Cieliebak and E. Volkov, First steps in stable Hamiltonian topology, J. Eur. Math. Soc. (JEMS), 17 (2015), 321-404.  doi: 10.4171/JEMS/505.

[10]

K. Cieliebak and E. Volkov, A note on the stationary Euler equations of hydrodynamics, Ergodic Theory Dynam. Systems, 37 (2017), 454-480.  doi: 10.1017/etds.2015.50.

[11]

A. Enciso and D. Peralta-Salas, Knots and links in steady solutions of the Euler equation, Ann. of Math., 175 (2012), 345-367.  doi: 10.4007/annals.2012.175.1.9.

[12]

J. Etnyre and R. Ghrist, Stratified integrals and unknots in inviscid flows, Contemp. Math., 246 (1999), 99-111.  doi: 10.1090/conm/246/03777.

[13]

J. Etnyre and R. Ghrist, Contact topology and hydrodynamics I. Beltrami fields and the Seifert conjecture, Nonlinearity, 13 (2000), 441-458.  doi: 10.1088/0951-7715/13/2/306.

[14]

A. T. Fomenko, The topology of surfaces of constant energy in integrable Hamiltonian systems and obstructions to integrability, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 1276-1307. 

[15]

A. T. Fomenko and S. V. Matveev, Morse type theory for integrable Hamiltonian systems with tame integrals, Math. Notes, 43 (1988), 382-386.  doi: 10.1007/BF01158846.

[16]

A. T. Fomenko and H. Zieschang, On the topology of three-dimensional manifolds arising in Hamiltonian mechanics, Dokl. Akad. Nauk SSSR, 294 (1987), 283-287. 

[17]

A. T. Fomenko and H. Zieschang, On typical topological properties of integrable Hamiltonian systems, Izvest. Akad. Nauk SSSR, Ser. Matem., 52 (1988), 378-407. 

[18]

A. Izosimov and B. Khesin, Characterization of steady solutions to the 2D Euler equation, Int. Math. Res. Not. IMRN, 2017 (2017), 7459-7503.  doi: 10.1093/imrn/rnw230.

[19]

M. Jankins and W. D. Neumann, Lectures on Seifert Manifolds, Brandeis lecture notes 2, Brandeis University, Waltham, MA (1983).

[20]

B. KhesinS. Kuksin and D. Peralta-Salas, KAM theory and the 3D Euler equation, Adv. Math., 267 (2014), 498-522.  doi: 10.1016/j.aim.2014.09.009.

[21]

I. Kirillov, Classification of coadjoint orbits for symplectomorphism groups of surfaces, Int. Math. Res. Not. IMRN, (2022). doi: 10.1093/imrn/rnac041.

[22]

D. Peralta-Salas, Selected topics on the topology of ideal fluid flows, Int. J. Geom. Methods Mod. Phys., 13 (2016), 1630012, 23 pp. doi: 10.1142/S0219887816300129.

[23]

D. Peralta-SalasA. Rechtman and F. Torres de Lizaur, A characterization of 3D Euler flows using commuting zero-flux homologies, Ergodic Theory Dynam. Systems, 41 (2021), 2166-2181.  doi: 10.1017/etds.2020.25.

[24]

H. Seifert, Topologie dreidimensionaler gefaserter räume, Acta Math., 60 (1933), 147-238.  doi: 10.1007/BF02398271.

[25]

M. Shiota, Equivalence of differentiable mappings and analytic mappings, Inst. Hautes Études Sci. Publ. Math., (1981), 237–322.

[26]

F. Waldhausen, Eine Klasse von 3-dimensionalen Mannifaltigkeiten I, Invent. Math., 3 (1967), 308-333.  doi: 10.1007/BF01402956.

[27]

F. Waldhausen, Eine Klasse von 3-dimensionalen Mannifaltigkeiten II, Invent. Math., 4 (1967), 87-117. 

Figure 1.  Height function in $ \Sigma_0 $
Figure 2.  Example of graph representation
Figure 3.  Level sets of $ h $
Figure 4.  Non simple $ 2 $-atom
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