×

Geometric hydrodynamics and infinite-dimensional Newton’s equations. (English) Zbl 1473.35444

Summary: We revisit the geodesic approach to ideal hydrodynamics and present a related geometric framework for Newton’s equations on groups of diffeomorphisms and spaces of probability densities. The latter setting is sufficiently general to include equations of compressible and incompressible fluid dynamics, magnetohydrodynamics, shallow water systems and equations of relativistic fluids. We illustrate this with a survey of selected examples, as well as with new results, using the tools of infinite-dimensional information geometry, optimal transport, the Madelung transform, and the formalism of symplectic and Poisson reduction.

MSC:

35Q35 PDEs in connection with fluid mechanics
76A02 Foundations of fluid mechanics
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Agrachev, A. A.; Caponigro, M., Controllability on the group of diffeomorphisms, Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire, 26, 6, 2503-2509 (2009) · Zbl 1188.93016 · doi:10.1016/j.anihpc.2009.07.003
[2] Anco, Stephen C.; Dar, Amanullah, Classification of conservation laws of compressible isentropic fluid flow in \(n>1\) spatial dimensions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465, 2108, 2461-2488 (2009) · Zbl 1186.35152 · doi:10.1098/rspa.2009.0072
[3] AnWe2018 S. C. Anco and G. M. Webb, Hierarchies of new invariants and conserved integrals in inviscid fluid flow, Phys. Fluids 32 (2020), 086104.
[4] Ar1965 V. I. Arnold, Variational principle for three-dimensional steady-state flows of an ideal fluid, Prikl. Mat. Mekh. 29 (1965). · Zbl 0163.19807
[5] Arnold, V., Sur la g\'{e}om\'{e}trie diff\'{e}rentielle des groupes de Lie de dimension infinie et ses applications \`a l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16, fasc. 1, 319-361 (1966) · Zbl 0148.45301
[6] Arnold, V. I., The asymptotic Hopf invariant and its applications, Selecta Math. Soviet., 5, 4, 327-345 (1986) · Zbl 0623.57016
[7] Arnold, V. I., Mathematical methods of classical mechanics, Graduate Texts in Mathematics 60, xvi+508 pp. (1989), Springer-Verlag, New York · Zbl 0692.70003 · doi:10.1007/978-1-4757-2063-1
[8] Arnold, Vladimir I.; Khesin, Boris A., Topological methods in hydrodynamics, Applied Mathematical Sciences 125, xvi+374 pp. (1998), Springer-Verlag, New York · Zbl 0902.76001
[9] Benamou, Jean-David; Brenier, Yann, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84, 3, 375-393 (2000) · Zbl 0968.76069 · doi:10.1007/s002110050002
[10] Besse, Nicolas; Frisch, Uriel, Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces, J. Fluid Mech., 825, 412-478 (2017) · Zbl 1374.76005 · doi:10.1017/jfm.2017.402
[11] Brenier, Yann, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Math. Soc., 2, 2, 225-255 (1989) · Zbl 0697.76030 · doi:10.2307/1990977
[12] Brenier, Yann, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44, 4, 375-417 (1991) · Zbl 0738.46011 · doi:10.1002/cpa.3160440402
[13] Brenier, Yann, Extended Monge-Kantorovich theory. Optimal transportation and applications, Martina Franca, 2001, Lecture Notes in Math. 1813, 91-121 (2003), Springer, Berlin · Zbl 1064.49036 · doi:10.1007/978-3-540-44857-0\_4
[14] Br\"{o}cker, Th., Differentiable germs and catastrophes, vi+179 pp. (1975), Cambridge University Press, Cambridge-New York-Melbourne · Zbl 0302.58006
[15] ChKnPiScWe2016 A. Chern, F. Kn\"oppel, U. Pinkall, P. Schr\"oder, and S. Weissmann, Schr\"odinger’s smoke, ACM Trans. Graph. 35 (2016), 77:1-77:13.
[16] De Lellis, Camillo; Sz\'{e}kelyhidi, L\'{a}szl\'{o}, Jr., High dimensionality and h-principle in PDE, Bull. Amer. Math. Soc. (N.S.), 54, 2, 247-282 (2017) · Zbl 1366.35120 · doi:10.1090/bull/1549
[17] Doebner, H.-D.; Goldin, Gerald A., On a general nonlinear Schr\"{o}dinger equation admitting diffusion currents, Phys. Lett. A, 162, 5, 397-401 (1992) · doi:10.1016/0375-9601(92)90061-P
[18] Dolzhansky, Felix V., Fundamentals of geophysical hydrodynamics, Encyclopaedia of Mathematical Sciences 103, xiv+272 pp. (2013), Springer, Heidelberg · Zbl 1286.86001 · doi:10.1007/978-3-642-31034-8
[19] Ebin, David G., Motion of a slightly compressible fluid, Proc. Nat. Acad. Sci. U.S.A., 72, 539-542 (1975) · doi:10.1073/pnas.72.2.539
[20] Ebin, David G.; Marsden, Jerrold, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2), 92, 102-163 (1970) · Zbl 0211.57401 · doi:10.2307/1970699
[21] Ebin, D. G.; Misio\l ek, G.; Preston, S. C., Singularities of the exponential map on the volume-preserving diffeomorphism group, Geom. Funct. Anal., 16, 4, 850-868 (2006) · Zbl 1105.35070 · doi:10.1007/s00039-006-0573-8
[22] Enciso, Alberto; Peralta-Salas, Daniel; Torres de Lizaur, Francisco, Helicity is the only integral invariant of volume-preserving transformations, Proc. Natl. Acad. Sci. USA, 113, 8, 2035-2040 (2016) · Zbl 1359.58006 · doi:10.1073/pnas.1516213113
[23] Friedlander, Susan; Shnirelman, Alexander, Instability of steady flows of an ideal incompressible fluid. Mathematical fluid mechanics, Adv. Math. Fluid Mech., 143-172 (2001), Birkh\"{a}user, Basel · Zbl 0984.35129
[24] Friedlander, Susan; Strauss, Walter; Vishik, Misha, Nonlinear instability in an ideal fluid, Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire, 14, 2, 187-209 (1997) · Zbl 0874.76026 · doi:10.1016/S0294-1449(97)80144-8
[25] Friedrich, Thomas, Die Fisher-Information und symplektische Strukturen, Math. Nachr., 153, 273-296 (1991) · Zbl 0792.62003 · doi:10.1002/mana.19911530125
[26] Gay-Balmaz, Fran\c{c}ois; Tronci, Cesare, Madelung transform and probability densities in hybrid quantum-classical dynamics, Nonlinearity, 33, 10, 5383-5424 (2020) · Zbl 1454.35310 · doi:10.1088/1361-6544/aba233
[27] Goldin, Gerald A.; Sharp, David H., Diffeomorphism groups and local symmetries: some applications in quantum physics. Symmetries in science, III, Vorarlberg, 1988, 181-205 (1989), Plenum, New York
[28] Gomes, Diogo Aguiar, A variational formulation for the Navier-Stokes equation, Comm. Math. Phys., 257, 1, 227-234 (2005) · Zbl 1080.37083 · doi:10.1007/s00220-004-1263-8
[29] Gui, Guilong; Liu, Yue; Zhu, Min, On the wave-breaking phenomena and global existence for the generalized periodic Camassa-Holm equation, Int. Math. Res. Not. IMRN, 21, 4858-4903 (2012) · Zbl 1252.35240 · doi:10.1093/imrn/rnr214
[30] Hamilton, Richard S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7, 1, 65-222 (1982) · Zbl 0499.58003 · doi:10.1090/S0273-0979-1982-15004-2
[31] Holm, Darryl D., Variational principles for stochastic fluid dynamics, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 471, 2176, 20140963, 19 pp. (2015) · Zbl 1371.35219 · doi:10.1098/rspa.2014.0963
[32] Holm, Darryl D.; Kupershmidt, Boris A., Relativistic fluid dynamics as a Hamiltonian system, Phys. Lett. A, 101, 1, 23-26 (1984) · doi:10.1016/0375-9601(84)90083-5
[33] Holm, Darryl D.; Marsden, Jerrold E.; Ratiu, Tudor S., The Euler-Poincar\'{e} equations and semidirect products with applications to continuum theories, Adv. Math., 137, 1, 1-81 (1998) · Zbl 0951.37020 · doi:10.1006/aima.1998.1721
[34] Holm, Darryl D.; Marsden, Jerrold E.; Ratiu, Tudor; Weinstein, Alan, Nonlinear stability of fluid and plasma equilibria, Phys. Rep., 123, 1-2, 116 pp. (1985) · Zbl 0717.76051 · doi:10.1016/0370-1573(85)90028-6
[35] Izosimov, Anton; Khesin, Boris; Mousavi, Mehdi, Coadjoint orbits of symplectic diffeomorphisms of surfaces and ideal hydrodynamics, Ann. Inst. Fourier (Grenoble), 66, 6, 2385-2433 (2016) · Zbl 1372.37106
[36] Khesin, Boris; Lee, Paul, A nonholonomic Moser theorem and optimal transport, J. Symplectic Geom., 7, 4, 381-414 (2009) · Zbl 1247.58005
[37] Khesin, Boris; Lenells, Jonatan; Misio\l ek, Gerard, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann., 342, 3, 617-656 (2008) · Zbl 1156.35082 · doi:10.1007/s00208-008-0250-3
[38] Khesin, B.; Lenells, J.; Misio\l ek, G.; Preston, S. C., Geometry of diffeomorphism groups, complete integrability and geometric statistics, Geom. Funct. Anal., 23, 1, 334-366 (2013) · Zbl 1275.58006 · doi:10.1007/s00039-013-0210-2
[39] Khesin, Boris; Misiolek, Gerard; Modin, Klas, Geometric hydrodynamics via Madelung transform, Proc. Natl. Acad. Sci. USA, 115, 24, 6165-6170 (2018) · Zbl 1416.58006 · doi:10.1073/pnas.1719346115
[40] Khesin, Boris; Misio\l ek, Gerard; Modin, Klas, Geometry of the Madelung transform, Arch. Ration. Mech. Anal., 234, 2, 549-573 (2019) · Zbl 1425.53107 · doi:10.1007/s00205-019-01397-2
[41] Kibble, T. W. B., Geometrization of quantum mechanics, Comm. Math. Phys., 65, 2, 189-201 (1979) · Zbl 0412.58006
[42] Landau, L. D.; Lifshitz, E. M., Fluid mechanics, Translated from the Russian by J. B. Sykes and W. H. Reid. Course of Theoretical Physics, Vol. 6, xii+536 pp. (1959), Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass.
[43] L\'{e}ger, Flavien; Li, Wuchen, Hopf-Cole transformation via generalized Schr\"{o}dinger bridge problem, J. Differential Equations, 274, 788-827 (2021) · Zbl 1455.35052 · doi:10.1016/j.jde.2020.10.029
[44] Lenells, Jonatan, Spheres, K\"{a}hler geometry and the Hunter-Saxton system, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469, 2154, 20120726, 19 pp. (2013) · Zbl 1404.53112 · doi:10.1098/rspa.2012.0726
[45] L\'{e}onard, Christian, A survey of the Schr\"{o}dinger problem and some of its connections with optimal transport, Discrete Contin. Dyn. Syst., 34, 4, 1533-1574 (2014) · Zbl 1277.49052 · doi:10.3934/dcds.2014.34.1533
[46] Li1890 S. Lie, Theorie der Transformationsgruppen: Abschnitt 2, Technische Informationsbibliothek (TIB), 1890.
[47] Lott, John, Some geometric calculations on Wasserstein space, Comm. Math. Phys., 277, 2, 423-437 (2008) · Zbl 1144.58007 · doi:10.1007/s00220-007-0367-3
[48] Ma1927 E. Madelung, Quantentheorie in hydrodynamischer form, Zeitschr. Phys. 40 (1927), 322-326. · JFM 52.0969.06
[49] Madelung, Erwin, Die mathematischen Hilfsmittel des Physikers, Siebente Auflage. Die Grundlehren der mathematischen Wissenschaften, Band 4, xx+536 pp. (1964), Springer-Verlag, Berlin · Zbl 0063.03698
[50] Marsden, Jerrold E.; Misio\l ek, Gerard; Ortega, Juan-Pablo; Perlmutter, Matthew; Ratiu, Tudor S., Hamiltonian reduction by stages, Lecture Notes in Mathematics 1913, xvi+519 pp. (2007), Springer, Berlin · Zbl 1129.37001
[51] Marsden, Jerrold E.; Ratiu, Tudor S., Introduction to mechanics and symmetry, Texts in Applied Mathematics 17, xviii+582 pp. (1999), Springer-Verlag, New York · Zbl 0933.70003 · doi:10.1007/978-0-387-21792-5
[52] Marsden, Jerrold E.; Ratiu, Tudor; Weinstein, Alan, Reduction and Hamiltonian structures on duals of semidirect product Lie algebras. Fluids and plasmas: geometry and dynamics, Boulder, Colo., 1983, Contemp. Math. 28, 55-100 (1984), Amer. Math. Soc., Providence, RI · Zbl 0546.58025 · doi:10.1090/conm/028/751975
[53] Marsden, Jerrold; Weinstein, Alan, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys., 5, 1, 121-130 (1974) · Zbl 0327.58005 · doi:10.1016/0034-4877(74)90021-4
[54] Marsden, Jerrold; Weinstein, Alan, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D, 7, 1-3, 305-323 (1983) · Zbl 0576.58008 · doi:10.1016/0167-2789(83)90134-3
[55] McCann, Robert J., Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal., 11, 3, 589-608 (2001) · Zbl 1011.58009 · doi:10.1007/PL00001679
[56] Mumford, David; Michor, Peter W., On Euler’s equation and “EPDiff”, J. Geom. Mech., 5, 3, 319-344 (2013) · Zbl 1274.35277 · doi:10.3934/jgm.2013.5.319
[57] Mi1996 G. Misioek, Conjugate points in \(D_\mu (T^2)\), Proc. Amer. Math. Soc. (1996), 977-982. · Zbl 0849.58004
[58] Modin, Klas, Generalized Hunter-Saxton equations, optimal information transport, and factorization of diffeomorphisms, J. Geom. Anal., 25, 2, 1306-1334 (2015) · Zbl 1330.58009 · doi:10.1007/s12220-014-9469-2
[59] Modin, Klas, Geometry of matrix decompositions seen through optimal transport and information geometry, J. Geom. Mech., 9, 3, 335-390 (2017) · Zbl 1368.15010 · doi:10.3934/jgm.2017014
[60] Molitor, Mathieu, On the relation between geometrical quantum mechanics and information geometry, J. Geom. Mech., 7, 2, 169-202 (2015) · Zbl 1328.53044 · doi:10.3934/jgm.2015.7.169
[61] Moser, J\"{u}rgen, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120, 286-294 (1965) · Zbl 0141.19407 · doi:10.2307/1994022
[62] Moser, J., Integrable Hamiltonian systems and spectral theory, Lezioni Fermiane. [Fermi Lectures], iv+85 pp. (1983), Scuola Normale Superiore, Pisa
[63] Ne1856 C. Neumann, De problemate quodam mechanico, quod ad primam integralium ultraellipticorum classem revocatur: Dissertatio inauguralis, Dalkowski, 1856.
[64] Otto, Felix, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26, 1-2, 101-174 (2001) · Zbl 0984.35089 · doi:10.1081/PDE-100002243
[65] OvKhCh1992 V. Y. Ovsienko, B. A. Khesin, and Y. V. Chekanov, Integrals of the Euler equations of multidimensional hydrodynamics and superconductivity, J.Sov.Math.59 (1992), 1096-1101. 1015703 · Zbl 0779.76103
[66] Po1901 H. Poincar\'e, Sur une forme nouvelle des \'equations de la m\'ecanique, C.R. Acad. Sci. 132 (1901), 369-371. · JFM 32.0715.01
[67] Preston, Stephen C., The geometry of barotropic flow, J. Math. Fluid Mech., 15, 4, 807-821 (2013) · Zbl 1286.35208 · doi:10.1007/s00021-013-0142-5
[68] Qu, Changzheng; Zhang, Ying; Liu, Xiaochuan; Liu, Yue, Orbital stability of periodic peakons to a generalized \(\mu \)-Camassa-Holm equation, Arch. Ration. Mech. Anal., 211, 2, 593-617 (2014) · Zbl 1287.35077 · doi:10.1007/s00205-013-0672-2
[69] Scheffer, Vladimir, An inviscid flow with compact support in space-time, J. Geom. Anal., 3, 4, 343-401 (1993) · Zbl 0836.76017 · doi:10.1007/BF02921318
[70] Shnirelman, A. I., The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Mat. Sb. (N.S.), 128(170), 1, 82-109, 144 (1985)
[71] Shnirelman, A., On the non-uniqueness of weak solution of the Euler equations. Journ\'{e}es “\'{E}quations aux D\'{e}riv\'{e}es Partielles”, Saint-Jean-de-Monts, 1996, Exp. No. XVIII, 10 pp. (1996), \'{E}cole Polytech., Palaiseau · Zbl 0881.35096
[72] Smolencev, N. K., The Maupertuis principle, Sibirsk. Mat. Zh., 20, 5, 1092-1098, 1167 (1979) · Zbl 0477.58007
[73] Smolentsev, N. K., Diffeomorphism groups of compact manifolds, Sovrem. Mat. Prilozh.. J. Math. Sci. (N.Y.), 146, 6, 6213-6312 (2007) · Zbl 1147.58012 · doi:10.1007/s10958-007-0471-0
[74] Tataru, Daniel, The wave maps equation, Bull. Amer. Math. Soc. (N.S.), 41, 2, 185-204 (2004) · Zbl 1065.35199 · doi:10.1090/S0273-0979-04-01005-5
[75] Ti\u{g}lay, Feride; Vizman, Cornelia, Generalized Euler-Poincar\'{e} equations on Lie groups and homogeneous spaces, orbit invariants and applications, Lett. Math. Phys., 97, 1, 45-60 (2011) · Zbl 1219.35198 · doi:10.1007/s11005-011-0464-2
[76] Tsirelson, Boris, Scaling limit, noise, stability. Lectures on probability theory and statistics, Lecture Notes in Math. 1840, 1-106 (2004), Springer, Berlin · Zbl 1056.60009 · doi:10.1007/978-3-540-39982-7\_1
[77] Villani, C\'{e}dric, Optimal transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338, xxii+973 pp. (2009), Springer-Verlag, Berlin · Zbl 1156.53003 · doi:10.1007/978-3-540-71050-9
[78] Vi\v{s}ik, S. M.; Dol\v{z}anski\u{\i }, F. V., Analogues, connected with Lie groups, of the Euler-Poisson equations and the equations of magnetohydrodynamics, Dokl. Akad. Nauk SSSR, 238, 5, 1032-1035 (1978)
[79] von Renesse, Max-K., An optimal transport view of Schr\"{o}dinger’s equation, Canad. Math. Bull., 55, 4, 858-869 (2012) · Zbl 1256.81072 · doi:10.4153/CMB-2011-121-9
[80] We2018 G. Webb, Magnetohydrodynamics and Fluid Dynamics: Action Principles and Conservation Laws, Springer, 2018. · Zbl 1397.76001
[81] Wu, Hao; Wunsch, Marcus, Global existence for the generalized two-component Hunter-Saxton system, J. Math. Fluid Mech., 14, 3, 455-469 (2012) · Zbl 1255.35188 · doi:10.1007/s00021-011-0075-9
[82] Younes, Laurent, Shapes and diffeomorphisms, Applied Mathematical Sciences 171, xviii+434 pp. (2010), Springer-Verlag, Berlin · Zbl 1205.68355 · doi:10.1007/978-3-642-12055-8
[83] Zambrini, J.-C., Variational processes and stochastic versions of mechanics, J. Math. Phys., 27, 9, 2307-2330 (1986) · Zbl 0623.60102 · doi:10.1063/1.527002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.