The point vortex model for the Euler equation

Carina Geldhauser; Marco Romito; Carina Geldhauser; Marco Romito

doi:10.3934/math.2019.3.534

AIMS Mathematics

2019, Volume 4, Issue 3: 534-575. doi: 10.3934/math.2019.3.534

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The point vortex model for the Euler equation

Carina Geldhauser ¹,
Marco Romito ^{2
,
,}

1.
Faculty of Mathematics, Technische Universit¨at Dresden, 01062 Dresden, Germany
2.
Dipartimento di Matematica, Universit`a di Pisa, Largo Bruno Pontecorvo 5, I–56127 Pisa, Italia

Received: 10 February 2019 Accepted: 15 May 2019 Published: 29 May 2019
MSC : 76B47, 60F05, 82C22, 35Q31, 35Q35

In this article we describe the system of point vortices, derived by Helmholtz from the Euler equation, and their associated Gibbs measures. We discuss solution concepts and available results for systems of point vortices with deterministic and random circulations, and further generalizations of the point vortex model.
- point vortex system,
- Euler equation,
- Gibbs measures,
- limit theorems and deviations,
- generalized SQG
Citation: Carina Geldhauser, Marco Romito. The point vortex model for the Euler equation[J]. AIMS Mathematics, 2019, 4(3): 534-575. doi: 10.3934/math.2019.3.534

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Abstract

In this article we describe the system of point vortices, derived by Helmholtz from the Euler equation, and their associated Gibbs measures. We discuss solution concepts and available results for systems of point vortices with deterministic and random circulations, and further generalizations of the point vortex model.

References

[1]	G. Kirchhoff, Vorlesungen über mathematische Physik, Monatsh. Math. Phys., 8 (1897), A29.
[2]	S. Albeverio and A. B. Cruzeiro, Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two-dimensional fluids, Commun. Math. Phys., 129 (1990), 431-444. doi: 10.1007/BF02097100
[3]	H. Aref, J. B. Kadtke, I. Zawadzki, et al. Point vortex dynamics: recent results and open problems, Fluid Dyn. Res., 3 (1988), 63-74. doi: 10.1016/0169-5983(88)90044-5
[4]	H. Aref and I. Zawadzki, Vortex interactions as a dynamical system, In: New approaches and concepts in turbulence (Monte Verità, 1991), Monte Verità, pp. 191-205. Birkhäuser, Basel, 1993.
[5]	G. Badin and A. M. Barry, Collapse of generalized Euler and surface quasi-geostrophic point-vortices, Phys. Rev. E, 98 (2018).
[6]	D. Bartolucci, Global bifurcation analysis of mean field equations and the Onsager microcanonical description of two-dimensional turbulence, Calc. Var. PDE, 58 (2019), 18.
[7]	D. Bartolucci, A. Jevnikar, Y. Lee and W. Yang Non-degeneracy, mean field equations and the Onsager theory of 2D turbulence, Arch. Ration. Mech. An., 230 (2018), 397-426. doi: 10.1007/s00205-018-1248-y
[8]	G. B. Arous and A. Guionnet, Large deviations for Wigner's law and Voiculescu's non-commutative entropy, Probab. Theory Rel. Fields, 108 (1997), 517-542. doi: 10.1007/s004400050119
[9]	G. Benfatto, P. Picco and M. Pulvirenti, On the invariant measures for the two-dimensional Euler flow, J. Stat. Phys., 46 (1987), 729-742. doi: 10.1007/BF01013382
[10]	T. Bodineau and A. Guionnet, About the stationary states of vortex systems, Ann. I. H. Poincare-PR, 35 (1999), 205-237. doi: 10.1016/S0246-0203(99)80011-9
[11]	F. Bouchet, C. Nardini and T. Tangarife, Non-equilibrium statistical mechanics of the stochastic Navier-Stokes equations and geostrophic turbulence, Warsaw University Press. 5th Warsaw School of Statistical Physics, 2014.
[12]	F. Bouchet and J. Sommeria, Emergence of intense jets and Jupiter's Great Red Spot as maximum-entropy structures, J. Fluid Mech., 464 (2002), 165-207.
[13]	E. Caglioti, P.-L. Lions, C. Marchioro, M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Commun. Math. Phys., 143 (1992), 501-525. doi: 10.1007/BF02099262
[14]	E. Caglioti, P.-L. Lions, C. Marchioro, M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Part II, Commun. Math. Phys., 174 (1995), 229-260. doi: 10.1007/BF02099602
[15]	G. Cavallaro, R. Garra and C. Marchioro, \newblock Localization and stability of active scalar flows, \newblock Riv. Math. Univ. Parma, 4 (2013), 175-196.
[16]	D. Chae, Weak solutions of 2-D incompressible Euler equations, Nonlinear Analysis, 23 (1994), 629-638. doi: 10.1016/0362-546X(94)90242-9
[17]	D. Chae, P. Constantin, D. Cόrdoba, F. Gancedo, J. Wu, Generalized surface quasi-geostrophic equations with singular velocities, Commun. Pure Appl. Math., 65 (2012), 1037-1066. doi: 10.1002/cpa.21390
[18]	D. Chae, P. Constantin and J. Wu, Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., 202 (2011), 35-62. doi: 10.1007/s00205-011-0411-5
[19]	P.-H. Chavanis, From Jupiter's great red spot to the structure of galaxies: Statistical mechanics of two-dimensional vortices and stellar systems, Annals of the New York Academy of Sciences, 867 (1998), 120-140. doi: 10.1111/j.1749-6632.1998.tb11254.x
[20]	P.-H. Chavanis, Statistical mechanics of two-dimensional vortices and stellar systems, In: Dynamics and thermodynamics of systems with long-range interactions, Vol. 602 of Lecture Notes in Phys., pp. 208-289. Springer, Berlin, 2002.
[21]	A. J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech., 57 (1973), 785-796. doi: 10.1017/S0022112073002016
[22]	A. J. Chorin, The evolution of a turbulent vortex, Commun. Math. Phys., 83 (1982), 517-535. doi: 10.1007/BF01208714
[23]	A. J. Chorin, Equilibrium statistics of a vortex filament with applications, Commun. Math. Phys., 141 (1991), 619-631. doi: 10.1007/BF02102820
[24]	A. J. Chorin, Vorticity and turbulence, Vol. 103 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994.
[25]	A. J. Chorin and J. H. Akao, Vortex equilibria in turbulence theory and quantum analogues, Physica D: Nonlinear Phenomena, 52 (1991), 403-414. doi: 10.1016/0167-2789(91)90136-W
[26]	X. Carton, Instability of surface quasigeostrophic vortices, J. Atmos. Sci., 66 (2009), 1051-1062. doi: 10.1175/2008JAS2872.1
[27]	P. Constantin, G. Iyer and J. Wu, Global regularity for a modified critical dissipative quasi-geostrophic equation, Indiana U. Math. J., 57 (2008), 2681-2692. doi: 10.1512/iumj.2008.57.3629
[28]	P. Constantin, A. J. Majda and E. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533. doi: 10.1088/0951-7715/7/6/001
[29]	G. Conti and G. Badin, Velocity statistics for point vortices in the local ɑ-models of turbulence, to appear on Geophysical & Astrophysical Fluid Dynamics, 2019
[30]	D. Cordoba, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. Math., 148 (1998), 1135-1152. doi: 10.2307/121037
[31]	D. Cordoba and C. Fefferman, Growth of solutions for QG and 2D Euler equations, J. Am. Math. Soc., 15 (2002), 665-670. doi: 10.1090/S0894-0347-02-00394-6
[32]	D. Cόrdoba, C. Fefferman and J. L. Rodrigo, Almost sharp fronts for the surface quasi-geostrophic equation, P. Natl. Acad. Sci. USA, 101 (2004), 2687-2691. doi: 10.1073/pnas.0308154101
[33]	D. Cόrdoba, M. A. Fontelos, A. M. Mancho and J. L. Rodrigo, Evidence of singularities for a family of contour dynamics equations, P. Natl. Acad. Sci. USA, 102 (2005), 5949-5952. doi: 10.1073/pnas.0501977102
[34]	D. Cόrdoba, J. Gόmez-Serrano and A. D. Ionescu, Global solutions for the generalized SQG patch equation, to appear on Arch. Rational. Mech. Anal., 2019.
[35]	C. De Lellis and L. Székelyhidi, Jr, The Euler equations as a differential inclusion, Ann. Math., 170 (2009), 1417-1436. doi: 10.4007/annals.2009.170.1417
[36]	J.-M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc., 4 (1991), 553-586. doi: 10.1090/S0894-0347-1991-1102579-6
[37]	J.-M. Delort, Existence des nappes de tourbillon de signe fixe en dimension deux. In: Nonlinear partial differential equations and their applications, Pitman Research Notes in Mathematics Series, 1994.
[38]	R. J. DiPerna and A. J. Majda, Concentrations in regularizations for 2-D incompressible flow, Commun. Pure Appl. Math., 40 (1987), 301-345. doi: 10.1002/cpa.3160400304
[39]	D. G. Dritschel, An exact steadily rotating surface quasi-geostrophic elliptical vortex, Geophys. Astro. Fluid, 105 (2011), 368-376. doi: 10.1080/03091929.2010.485997
[40]	G. L. Eyink and H. Spohn, Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence, J. Stat. Phys., 70 (1993), 833-886.
[41]	G. L. Eyink and K. R. Sreenivasan, Onsager and the theory of hydrodynamic turbulence, Rev. Mod. Phys., 78 (2006), 87-135. doi: 10.1103/RevModPhys.78.87
[42]	F. Flandoli, On a probabilistic description of small scale structures in 3D fluids, Ann. I. H. Poincare-Pr, 38 (2002), 207-228. doi: 10.1016/S0246-0203(01)01092-5
[43]	F. Flandoli, Weak vorticity formulation of 2D Euler equations with white noise initial condition, Commun. PDE, 43 (2018), 1102-1149. doi: 10.1080/03605302.2018.1467448
[44]	F. Flandoli and M. Gubinelli, The Gibbs ensemble of a vortex filament, Probab. Theory Rel. Fields, 122 (2002), 317-340. doi: 10.1007/s004400100163
[45]	F. Flandoli and M. Saal, mSQG equations in distributional spaces and point vortex approximation, to appear on J. Evol. Equat., 2019.
[46]	U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995.
[47]	J. Fröhlich and D. Ruelle, Statistical mechanics of vortices in an inviscid two-dimensional fluid, Commun. Math. Phys., 87 (1982), 1-36. doi: 10.1007/BF01211054
[48]	J. Fröhlich and E. Seiler, The massive Thirring-Schwinger model qed2: convergence of perturbation theory and particle structure, Helv. Phys. Acta, 49 (1976), 889-924.
[49]	R. Garra, Confinement of a hot temperature patch in the modified SQG model, Discrete Contin. Dynam. Syst. B, 2018.
[50]	C. Geldhauser and M. Romito, Limit theorems and fluctuations for point vortices of generalized Euler equations, arXiv:1810.12706, 2018.
[51]	C. Geldhauser and M. Romito, Point vortices for inviscid generalized surface quasi-geostrophic models, arXiv:1812.05166, 2018.
[52]	J. Goodman, T. Y. Hou and J. Lowengrub, Convergence of the point vortex method for the 2-D Euler equations, Comm. Pure Appl. Math., 43 (1990), 415-430. doi: 10.1002/cpa.3160430305
[53]	F. Grotto and M. Romito, A Central Limit Theorem for Gibbsian Invariant Measures of 2D Euler Equation, arXiv:1904.01871, 2019.
[54]	M. Hauray, Wasserstein distances for vortices approximation of Euler-type equations, Math. Models Methods Appl. Sci., 19 (2009), 1357-1384. doi: 10.1142/S0218202509003814
[55]	I. M. Held, R. T. Pierrehumbert, S. T. Garner and K. L. Swanson, Surface quasi-geostrophic dynamics, Journal of Fluid Mechanics, 282 (1995), 1-20. doi: 10.1017/S0022112095000012
[56]	I. M. Held, R. T. Pierrehumbert and K. L. Swanson, Spectra of local and nonlocal two-dimensional turbulence, Chaos, Solitons & Fractals, 4 (1994), 1111-1116.
[57]	H. Helmholtz, über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, J. Reine Angew. Math., 55 (1858), 25-55.
[58]	V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032-1066.
[59]	M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Commun. Pur. Appl. Math., 46 (1993), 27-56. doi: 10.1002/cpa.3160460103
[60]	M. K.-H. Kiessling and Y. Wang, Onsager's ensemble for point vortices with random circulations on the sphere, J. Stat. Phys., 148 (2012), 896-932. doi: 10.1007/s10955-012-0552-4
[61]	A. Kiselev and F. Nazarov, A variation on a theme of Caffarelli and Vasseur, Journal of Mathematical Sciences, 166 (2009), 31-39.
[62]	A. Kiselev, L. Ryzhik, Y. Yao and A. Zlatos, Finite time singularity for the modified SQG patch equation, Ann. Math., 184 (2016), 909-948. doi: 10.4007/annals.2016.184.3.7
[63]	R. Klein, A. J. Majda and K. Damodaran, Simplified equations for the interaction of nearly parallel vortex filaments, J. Fluid Mech., 288 (1995), 201-248. doi: 10.1017/S0022112095001121
[64]	R. H. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluids, 10 (1967), 1417-1423. doi: 10.1063/1.1762301
[65]	T. Leblé, S. Serfaty and O. Zeitouni, Large deviations for the two-dimensional two-component plasma, Commun. Math. Phys., 350 (2017), 301-360. doi: 10.1007/s00220-016-2735-3
[66]	C. C. Lim and A. J. Majda, Point vortex dynamics for coupled surface/interior QG and propagating heton clusters in models for ocean convection, Geophys. Astro. Fluid, 94 (2001), 177-220. doi: 10.1080/03091920108203407
[67]	C. C. Lin, On the motion of vortices in two dimensions: I. Existence of the Kirchhoff-Routh function, P. Natl. Acad. Sci. USA, 27 (1941), 570-575. doi: 10.1073/pnas.27.12.570
[68]	C. C. Lin, On the motion of vortices in two dimensions: II. Some further investigations on the Kirchhoff-Routh function, P. Natl. Acad. Sci. USA, 27 (1941), 575-577. doi: 10.1073/pnas.27.12.575
[69]	P.-L. Lions, On Euler equations and statistical physics, Cattedra Galileiana [Galileo Chair], Scuola Normale Superiore, Classe di Scienze, Pisa, 1998.
[70]	P.-L. Lions and A. Majda, Equilibrium statistical theory for nearly parallel vortex filaments, Commun. Pur. Appl. Math., 53 (2000), 76-142. doi: 10.1002/(SICI)1097-0312(200001)53:1<76::AID-CPA2>3.0.CO;2-L
[71]	A. J. Majda and E. G. Tabak, A two-dimensional model for quasi-geostrophic flow: comparison with the two-dimensional Euler flow, Physica D: Nonlinear Phenomena, 98 (1996), 515-522. doi: 10.1016/0167-2789(96)00114-5
[72]	F. Marchand, Existence and regularity of weak solutions to the quasi-geostrophic equations in the spaces $L^p$ or $\dot H^-1/2$, Commun. Math. Phys., 277 (2008), 45-67.
[73]	C. Marchioro and M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory, Commun. Math. Phys., 84 (1982), 483-503. doi: 10.1007/BF01209630
[74]	C. Marchioro and M. Pulvirenti, On the vortex-wave system. In: Mechanics, analysis and geometry: 200 years after Lagrange, North-Holland Delta Ser., pp. 79-95. North-Holland, Amsterdam, 1991.
[75]	C. Marchioro and M. Pulvirenti, Vortices and localization in Euler flows, Commun. Math. Phys., 154 (1993), 49-61. doi: 10.1007/BF02096831
[76]	C. Marchioro and M. Pulvirenti, Mathematical theory of incompressible nonviscous fluids, Springer-Verlag, New York, 1994.
[77]	J. C. McWilliams, The emergence of isolated coherent vortices in turbulent flow, J. Fluid Mech., 146 (1984), 21-43. doi: 10.1017/S0022112084001750
[78]	J. Miller, Statistical mechanics of Euler equations in two dimensions, Phys. Rev. Lett., 65 (1990), 2137-2140. doi: 10.1103/PhysRevLett.65.2137
[79]	D. Montgomery and G. Joyce, Statistical mechanics of ``negative temperature'' states, Phys. Fluids, 17 (1974), 1139-1145. doi: 10.1063/1.1694856
[80]	D. Montgomery, W. Matthaeus, W. Stribling, D. Martinez, and S. Oughton Relaxation in two dimensions and the sinh-Poisson equation, Phys. Fluids, 4 (1992), 3-6.
[81]	A. Nahmod, N. Pavlovic, G. Staffilani, et al. Global flows with invariant measures for the inviscid modified SQG equations, Stochastics and Partial Differential Equations: Analysis and Computations, 6 (2018), 184-210. doi: 10.1007/s40072-017-0106-5
[82]	C. Neri, Statistical mechanics of the $N$-point vortex system with random intensities on a bounded domain, Ann. I. H. Poincare-An, 21 (2004), 381-399. doi: 10.1016/j.anihpc.2003.05.002
[83]	C. Neri, Statistical mechanics of the $N$-point vortex system with random intensities on $\mathbbR^2$, Elec. J. Diff. Eq., 92 (2005), 1-26.
[84]	K. Ohkitani, Asymptotics and numerics of a family of two-dimensional generalized surface quasi-geostrophic equations, Phys. Fluids, 24 (2012), 095101.
[85]	L. Onsager, Statistical hydrodynamics, Nuovo Cim. (Suppl. 2), 6 (1949), 279-287. doi: 10.1007/BF02780991
[86]	E. A. Overman and N. J. Zabusky, Evolution and merger of isolated vortex structures, Phys. Fluids, 25 (1982), 1297-1305. doi: 10.1063/1.863907
[87]	F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods and Applications of Analysis, 9 (2002), 533-562. doi: 10.4310/MAA.2002.v9.n4.a4
[88]	M. Pulvirenti, On invariant measures for the 2-D Euler flow. In: Mathematical aspects of vortex dynamics, (Leesburg, VA, pp. 88-96, SIAM, 1989.
[89]	S. G. Resnick, Dynamical problems in non-linear advective partial differential equations, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.), The University of Chicago.
[90]	T. Ricciardi and R. Takahashi, Blow-up behavior for a degenerate elliptic sinh-Poisson equation with variable intensities, Calc. Var. PDE, 55 (2016), 152.
[91]	T. Ricciardi and R. Takahashi, On radial two-species onsager vortices near the critical temperature, arXiv:1706.06046, 2017.
[92]	T. Ricciardi, R. Takahashi, G. Zecca and X. Zhang On the existence and blow-up of solutions for a mean field equation with variable intensities, Rend. Lincei-Mat Appl., 27 (2016), 413-429.
[93]	R. Robert, états d'équilibre statistique pour l'écoulement bidimensionnel d'un fluide parfait, C. R. Acad. Sci. Paris Sér. I Math., 311 (1990), 575-578.
[94]	R. Robert, A maximum-entropy principle for two-dimensional perfect fluid dynamics, J. Stat. Phys., 65 (1991), 531-553. doi: 10.1007/BF01053743
[95]	R. Robert, On the statistical mechanics of 2D Euler equation, Commun. Math. Phys., 212 (2000), 245-256. doi: 10.1007/s002200000210
[96]	R. Robert and J. Sommeria, Statistical equilibrium states for two-dimensional flows, J. Fluid Mech., 229 (1991), 291-310. doi: 10.1017/S0022112091003038
[97]	J. L. Rodrigo, On the evolution of sharp fronts for the quasi-geostrophic equation, Commun. Pur. Appl. Math., 58 (2005), 821-866. doi: 10.1002/cpa.20059
[98]	K. Sawada and T. Suzuki, Rigorous Derivation of the Mean Field Equation for a Point Vortex System, Theoretical and Applied Mechanics Japan, 57 (2009), 233-239.
[99]	S. Schochet, The weak vorticity formulation of the 2-D Euler equations and concentration-cancellation, Commun. Part. Diff. Eq., 20 (1995), 1077-1104. doi: 10.1080/03605309508821124
[100]	N. Schorghofer, Energy spectra of steady two-dimensional turbulent flows, Phys. Rev. E, 61 (2000), 6572-6577. doi: 10.1103/PhysRevE.61.6572
[101]	R. K. Scott, A scenario for finite-time singularity in the quasigeostrophic model, J. Fluid Mech., 687 (2011), 492-502. doi: 10.1017/jfm.2011.377
[102]	R. K. Scott and D. G. Dritschel, Numerical simulation of a self-similar cascade of filament instabilities in the surface quasigeostrophic system, Phys. Rev. Lett., 112 (2014), 144505.
[103]	C. Taylor and S. G. Llewellyn Smith, Dynamics and transport properties of three surface quasigeostrophic point vortices, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 113117.
[104]	C. V. Tran, Nonlinear transfer and spectral distribution of energy in ɑ turbulence, Phys. D, 191 (2004), 137-155. doi: 10.1016/j.physd.2003.11.005
[105]	C. V. Tran, D. G. Dritschel and R. K. Scott, Effective degrees of nonlinearity in a family of generalized models of two-dimensional turbulence, Phys. Rev. E, 81 (2010), 016301.
[106]	A. Venaille, T. Dauxois and S. Ruffo, Violent relaxation in two-dimensional flows with varying interaction range, Phys. Rev. E, 92 (2015), 011001.
[107]	W. Wolibner, Un theorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long, Math. Z., 37 (1933), 698-726.

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