# American Institute of Mathematical Sciences

December  2012, 5(6): 1091-1131. doi: 10.3934/dcdss.2012.5.1091

## Stability and interaction of vortices in two-dimensional viscous flows

 1 Université de Grenoble 1, Institut Fourier, UMR 5582, B.P. 74，38402 Saint-Martin-d'Hères, France

Received  December 2011 Revised  March 2012 Published  August 2012

The aim of these notes is to present in a comprehensive and relatively self-contained way some recent developments in the mathematical analysis of two-dimensional viscous flows. We consider the incompressible Navier-Stokes equations in the whole plane $\mathbb{R}$2, and assume that the initial vorticity is a finite measure. This general setting includes vortex patches, vortex sheets, and point vortices. We first prove the existence of a unique global solution, for any value of the viscosity parameter, and we investigate its long-time behavior. We next consider the particular case where the initial flow is a finite collection of point vortices. In that situation, we show that the solution behaves, in the vanishing viscosity limit, as a superposition of Oseen vortices whose centers evolve according to the Helmholtz-Kirchhoff point vortex system. The proof requires a careful stability analysis of the Oseen vortices in the large Reynolds number regime, as well as a precise computation of the deformations of the vortex cores due to mutual interactions.
Citation: Thierry Gallay. Stability and interaction of vortices in two-dimensional viscous flows. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1091-1131. doi: 10.3934/dcdss.2012.5.1091
##### References:
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Ben-Artzi:"Global solutions of two-dimensional Navier-Stokes and Eulerequations'',, Arch. Rational Mech. Anal., 128 (1994), 359. doi: 10.1007/BF00387713. Google Scholar [7] R. Caflisch and M. Sammartino, Vortex layers in the small viscosity limit,, “WASCOM 2005”-13th Conference on Waves and Stability in Continuous Media, (2005), 59. Google Scholar [8] E. Caglioti, M. Pulvirenti and F. Rousset, On a constrained 2-D Navier-Stokes equation,, Commun. Math. Phys., 290 (2009), 651. doi: 10.1007/s00220-008-0720-1. Google Scholar [9] M. Cannone and F. Planchon, Self-similar solutions for Navier-Stokes equations in$\mathbbR^3$,, Commun. Partial Differential Equations, 21 (1996), 179. doi: 10.1080/03605309608821179. Google Scholar [10] E. A. Carlen and M. Loss, Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the $2$-D Navier-Stokes equation,, Duke Math. J., 81 (1995), 135. doi: 10.1215/S0012-7094-95-08110-1. Google Scholar [11] J.-Y. Chemin, A remark on the inviscid limit for two-dimensional incompressible fluids,, Comm. Partial Diff. Equations, 21 (1996), 1771. doi: 10.1080/03605309608821245. Google Scholar [12] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics. An Introduction to Rotating Fluidsand the Navier-Stokes Equations,", Oxford Lecture Series in Mathematicsand its Applications, 32 (2006). Google Scholar [13] P. Constantin, On the Euler equations of incompressible fluids,, Bull. Amer. Math. Soc., 44 (2007), 603. doi: 10.1090/S0273-0979-07-01184-6. Google Scholar [14] P. Constantin and J. Wu, Inviscid limit for vortex patches,, Nonlinearity, 8 (1995), 735. Google Scholar [15] P. Constantin and J. Wu, The inviscid limit for non-smooth vorticity,, Indiana Univ. Math. J., 45 (1996), 67. doi: 10.1512/iumj.1996.45.1960. Google Scholar [16] G.-H. Cottet, Équations de Navier-Stokes dans le plan avec tourbillon initial mesure,, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 105. Google Scholar [17] Y. Couder, Observation expérimentale de la turbulencebidimensionnelle dans un film liquide mince,, C. R. Acad. Sci. Paris II, 297 (1983), 641. Google Scholar [18] R. Danchin, Poches de tourbillon visqueuses,, J. Math. Pures Appl., 76 (1997), 609. doi: 10.1016/S0021-7824(97)89964-3. Google Scholar [19] R. Danchin, Persistance de structures géométriques et limite non visqueuse pour les fluides incompressibles en dimension quelconque,, Bull. Soc. Math. France, 127 (1999), 179. Google Scholar [20] J.-M. Delort, Existence de nappes de tourbillon en dimension deux,, J. Amer. Math. Soc., 4 (1991), 553. doi: 10.1090/S0894-0347-1991-1102579-6. Google Scholar [21] W. Deng, Resolvent estimates for a two-dimensional non-selfadjoint operator,, to appear in Communications on Pure and Applied Analysis., (). Google Scholar [22] W. Deng, Pseudospectrum for Oseen vortices operators,, preprint, (). Google Scholar [23] P. G. Drazin and W. H. 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Anal., 200 (2011), 445. doi: 10.1007/s00205-010-0362-2. Google Scholar [29] Th. Gallay and L. M. Rodrigues, Sur le temps de vie de la turbulence bidimensionnelle,, Ann. Fac. Sci. Toulouse Math., 17 (2008), 719. doi: 10.5802/afst.1199. Google Scholar [30] Th. Gallay and C. E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on $\mathbbR^2$,, Arch. Ration. Mech. Anal., 163 (2002), 209. doi: 10.1007/s002050200200. Google Scholar [31] Th. Gallay and C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation,, Commun. Math. Phys., 255 (2005), 97. doi: 10.1007/s00220-004-1254-9. Google Scholar [32] Th. Gallay and C. E. Wayne, Existence and stability of asymmetric Burgers vortices,, J. Math. Fluid Mech., 9 (2007), 243. doi: 10.1007/s00021-005-0199-x. Google Scholar [33] Y. Giga and T. Kambe, Large time behavior of the vorticity of two dimensional viscous flow and its application to vortex formation,, Commun. Math. Phys., 117 (1988), 549. doi: 10.1007/BF01218384. Google Scholar [34] Y. Giga, T. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity,, Arch. Rational Mech. Anal., 104 (1988), 223. doi: 10.1007/BF00281355. Google Scholar [35] E. Grenier, On the nonlinear instability of Euler and Prandtl equations,, Comm. Pure Appl. Math., 53 (2000), 1067. doi: 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q. Google Scholar [36] H. von Helmholtz, Über Integrale des hydrodynamischen Gleichungen, welche die Wirbelbewegungen entsprechen,, J. reine angew. Math., 55 (1858), 25. doi: 10.1515/crll.1858.55.25. Google Scholar [37] T. Hmidi, Régularité höldérienne des poches de tourbillon visqueuses,, J. Math. Pures Appl., 84 (2005), 1455. doi: 10.1016/j.matpur.2005.01.004. Google Scholar [38] T. Hmidi, Poches de tourbillon singulières dans un fluide faiblement visqueux,, Rev. Mat. Iberoamericana, 22 (2006), 489. doi: 10.4171/RMI/464. Google Scholar [39] J. Jiménez, H. K. Moffatt and C. Vasco, The structure of the vortices in freely decaying two-dimensional turbulence,, J. Fluid Mech., 313 (1996), 209. doi: 10.1017/S0022112096002182. Google Scholar [40] F. Jing, E. Kanso and P. Newton, Viscous evolution of point vortex equilibria: The collinear state,, Phys. Fluids, 22 (2010). doi: 10.1063/1.3516637. Google Scholar [41] T. Kato, "Perturbation Theory for Linear Operators,'', Grundlehren der mathematischen Wissenschaften 132, 132 (1966). Google Scholar [42] T. Kato, Nonstationary flows of viscous and ideal fluids in $\mathbbR^3$,, J. Functional Analysis, 9 (1972), 296. doi: 10.1016/0022-1236(72)90003-1. Google Scholar [43] T. 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##### References:
 [1] H. Abidi and R. Danchin, Optimal bounds for the inviscid limit of Navier-Stokes equations,, Asymptot. Anal., 38 (2004), 35. Google Scholar [2] Th. Beale and A. Majda, Rates of convergence for viscous splitting of the Navier-Stokes equations,, Math. Comp., 37 (1981), 243. doi: 10.1090/S0025-5718-1981-0628693-0. Google Scholar [3] M. Ben-Artzi, Global solutions of two-dimensional Navier-Stokes andEuler equations,, Arch. Rational Mech. Anal., 128 (1994), 329. doi: 10.1007/BF00387712. Google Scholar [4] G. Benfatto, R. Esposito and M. Pulvirenti, Planar Navier-Stokes flow for singular initial data,, Nonlinear Anal., 9 (1985), 533. Google Scholar [5] A. Bracco, J. C. McWilliams, G. Murante, A. Provenzaleand J. B. Weiss, Revisiting freely-decaying two-dimensional turbulenceat millennial resolution,, Physics of Fluids, 12 (2000), 2931. doi: 10.1063/1.1290391. Google Scholar [6] H. Brezis, Remarks on the preceding paper by M. Ben-Artzi:"Global solutions of two-dimensional Navier-Stokes and Eulerequations'',, Arch. Rational Mech. Anal., 128 (1994), 359. doi: 10.1007/BF00387713. Google Scholar [7] R. Caflisch and M. Sammartino, Vortex layers in the small viscosity limit,, “WASCOM 2005”-13th Conference on Waves and Stability in Continuous Media, (2005), 59. Google Scholar [8] E. Caglioti, M. Pulvirenti and F. Rousset, On a constrained 2-D Navier-Stokes equation,, Commun. Math. Phys., 290 (2009), 651. doi: 10.1007/s00220-008-0720-1. Google Scholar [9] M. Cannone and F. Planchon, Self-similar solutions for Navier-Stokes equations in$\mathbbR^3$,, Commun. Partial Differential Equations, 21 (1996), 179. doi: 10.1080/03605309608821179. Google Scholar [10] E. A. Carlen and M. Loss, Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the $2$-D Navier-Stokes equation,, Duke Math. J., 81 (1995), 135. doi: 10.1215/S0012-7094-95-08110-1. Google Scholar [11] J.-Y. Chemin, A remark on the inviscid limit for two-dimensional incompressible fluids,, Comm. Partial Diff. Equations, 21 (1996), 1771. doi: 10.1080/03605309608821245. Google Scholar [12] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics. An Introduction to Rotating Fluidsand the Navier-Stokes Equations,", Oxford Lecture Series in Mathematicsand its Applications, 32 (2006). Google Scholar [13] P. Constantin, On the Euler equations of incompressible fluids,, Bull. Amer. Math. Soc., 44 (2007), 603. doi: 10.1090/S0273-0979-07-01184-6. Google Scholar [14] P. Constantin and J. Wu, Inviscid limit for vortex patches,, Nonlinearity, 8 (1995), 735. Google Scholar [15] P. Constantin and J. Wu, The inviscid limit for non-smooth vorticity,, Indiana Univ. Math. J., 45 (1996), 67. doi: 10.1512/iumj.1996.45.1960. Google Scholar [16] G.-H. Cottet, Équations de Navier-Stokes dans le plan avec tourbillon initial mesure,, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 105. 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Reid, "Hydrodynamic Stability,'', Second edition, (2004). doi: 10.1017/CBO9780511616938. Google Scholar [24] H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I.,, Arch. Rational Mech. Anal., 16 (1964), 269. doi: 10.1007/BF00276188. Google Scholar [25] I. Gallagher and Th. Gallay, Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity,, Math. Ann., 332 (2005), 287. doi: 10.1007/s00208-004-0627-x. Google Scholar [26] I. Gallagher, Th. Gallay and F. Nier, Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator,, Int. Math. Res. Notices, 2009 (2009), 2147. Google Scholar [27] I. Gallagher, Th. Gallay and P.-L. Lions, On the uniqueness of the solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity,, Math. Nachr., 278 (2005), 1665. doi: 10.1002/mana.200410331. Google Scholar [28] Th. Gallay, Interaction of vortices in weakly viscous planar flows,, Arch. Ration. Mech. Anal., 200 (2011), 445. doi: 10.1007/s00205-010-0362-2. Google Scholar [29] Th. Gallay and L. M. Rodrigues, Sur le temps de vie de la turbulence bidimensionnelle,, Ann. Fac. Sci. Toulouse Math., 17 (2008), 719. doi: 10.5802/afst.1199. Google Scholar [30] Th. Gallay and C. E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on $\mathbbR^2$,, Arch. Ration. Mech. Anal., 163 (2002), 209. doi: 10.1007/s002050200200. Google Scholar [31] Th. Gallay and C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation,, Commun. Math. Phys., 255 (2005), 97. doi: 10.1007/s00220-004-1254-9. Google Scholar [32] Th. Gallay and C. E. Wayne, Existence and stability of asymmetric Burgers vortices,, J. Math. Fluid Mech., 9 (2007), 243. doi: 10.1007/s00021-005-0199-x. Google Scholar [33] Y. Giga and T. Kambe, Large time behavior of the vorticity of two dimensional viscous flow and its application to vortex formation,, Commun. Math. Phys., 117 (1988), 549. doi: 10.1007/BF01218384. Google Scholar [34] Y. Giga, T. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity,, Arch. Rational Mech. Anal., 104 (1988), 223. doi: 10.1007/BF00281355. Google Scholar [35] E. Grenier, On the nonlinear instability of Euler and Prandtl equations,, Comm. Pure Appl. Math., 53 (2000), 1067. doi: 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q. Google Scholar [36] H. von Helmholtz, Über Integrale des hydrodynamischen Gleichungen, welche die Wirbelbewegungen entsprechen,, J. reine angew. Math., 55 (1858), 25. doi: 10.1515/crll.1858.55.25. Google Scholar [37] T. Hmidi, Régularité höldérienne des poches de tourbillon visqueuses,, J. Math. Pures Appl., 84 (2005), 1455. doi: 10.1016/j.matpur.2005.01.004. Google Scholar [38] T. Hmidi, Poches de tourbillon singulières dans un fluide faiblement visqueux,, Rev. Mat. Iberoamericana, 22 (2006), 489. doi: 10.4171/RMI/464. Google Scholar [39] J. Jiménez, H. K. Moffatt and C. Vasco, The structure of the vortices in freely decaying two-dimensional turbulence,, J. Fluid Mech., 313 (1996), 209. doi: 10.1017/S0022112096002182. Google Scholar [40] F. Jing, E. Kanso and P. Newton, Viscous evolution of point vortex equilibria: The collinear state,, Phys. Fluids, 22 (2010). doi: 10.1063/1.3516637. Google Scholar [41] T. Kato, "Perturbation Theory for Linear Operators,'', Grundlehren der mathematischen Wissenschaften 132, 132 (1966). Google Scholar [42] T. Kato, Nonstationary flows of viscous and ideal fluids in $\mathbbR^3$,, J. Functional Analysis, 9 (1972), 296. doi: 10.1016/0022-1236(72)90003-1. Google Scholar [43] T. 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