Abstract
We consider the inviscid limit for the two-dimensional incompressible Navier–Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter ν, and we choose a time T > 0 such that the Helmholtz–Kirchhoff point vortex system is well-posed on the interval [0, T]. Under these assumptions, we prove that the solution of the Navier–Stokes equation converges, as ν → 0, to a superposition of Lamb–Oseen vortices whose centers evolve according to a viscous regularization of the point vortex system. Convergence holds uniformly in time, in a strong topology which allows us to give an accurate description of the asymptotic profile of each individual vortex. In particular, we compute to leading order the deformations of the vortices due to mutual interactions. This makes it possible to estimate the self-interactions, which play an important role in the convergence proof.
Similar content being viewed by others
References
Abidi H., Danchin R.: Optimal bounds for the inviscid limit of Navier–Stokes equations. Asymptot. Anal. 38, 35–46 (2004)
Beale T., Majda A.: Rates of convergence for viscous splitting of the Navier–Stokes equations. Math. Comput. 37, 243–259 (1981)
Benfatto G., Esposito R., Pulvirenti M.: Planar Navier–Stokes flow for singular initial data. Nonlinear Anal. 9, 533–545 (1985)
Caflisch, R., Sammartino, M.: Vortex layers in the small viscosity limit. In: WASCOM 2005—13th Conference on Waves and Stability in Continuous Media 59–70. World Scientific Publishing Company, Hackensack, 2006
Carlen E.A., Loss M.: Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the 2-D Navier–Stokes equation. Duke Math. J. 81, 135–157 (1996)
Chemin J.-Y.: A remark on the inviscid limit for two-dimensional incompressible fluids. Comm. Partial Differ. Equ. 21, 1771–1779 (1996)
Chen P.-H., Wang W.-L.: Roll-up of a viscous vortex sheet. J. Chin. Inst. Eng. 14, 507–517 (1991)
Constantin P., Wu J.: Inviscid limit for vortex patches. Nonlinearity 8, 735–742 (1995)
Constantin P., Wu J.: The inviscid limit for non-smooth vorticity. Indiana Univ. Math. J. 45, 67–81 (1996)
Cottet G.-H.: Équations de Navier–Stokes dans le plan avec tourbillon initial mesure. C. R. Acad. Sci. Paris Sér. I Math. 303, 105–108 (1986)
Couder Y.: Observation expérimentale de la turbulence bidimensionnelle dans un film liquide mince. C. R. Acad. Sci. Paris II 297, 641–645 (1983)
Danchin R.: Poches de tourbillon visqueuses. J. Math. Pures Appl. 76, 609–647 (1997)
Danchin R.: Persistance de structures géométriques et limite non visqueuse pour les fluides incompressibles en dimension quelconque. Bull. Soc. Math. France 127, 179–227 (1999)
Delort J.-M.: Existence de nappes de tourbillon en dimension deux. J. Am. Math. Soc. 4, 553–586 (1991)
Ebin D., Marsden J.: Groups of diffeomorphisms and the notion of an incompressible fluid. Ann. Math. 92, 102–163 (1970)
Gallagher I., Gallay T.: Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity. Math. Ann. 332, 287–327 (2005)
Gallagher I., Gallay T., Lions P.-L.: On the uniqueness of the solution of the two-dimensional Navier���Stokes equation with a Dirac mass as initial vorticity. Math. Nachr. 278, 1665–1672 (2005)
Gallay T., Wayne C.E.: Invariant manifolds and the long-time asymptotics of the Navier–Stokes and vorticity equations on \({{\mathbb R}^2}\). Arch. Rational Mech. Anal. 163, 209–258 (2002)
Gallay T., Wayne C.E.: Global stability of vortex solutions of the two-dimensional Navier–Stokes equation. Comm. Math. Phys. 255, 97–129 (2005)
Gallay T., Wayne C.E.: Existence and stability of asymmetric Burgers vortices. J. Math. Fluid Mech. 9, 243–261 (2007)
Giga Y., Miyakawa T., Osada H.: Two-dimensional Navier–Stokes flow with measures as initial vorticity. Arch. Rational Mech. Anal. 104, 223–250 (1988)
Grenier E.: On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. 53, 1067–1091 (2000)
von Helmholtz H.: Über Integrale des hydrodynamischen Gleichungen, welche die Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25–55 (1858)
Hmidi T.: Régularité höldérienne des poches de tourbillon visqueuses. J. Math. Pures Appl. 84, 1455–1495 (2005)
Hmidi T.: Poches de tourbillon singulières dans un fluide faiblement visqueux. Rev. Mat. Iberoamericana 22, 489–543 (2006)
Kato T.: Nonstationary flows of viscous and ideal fluids in \({{\mathbb R}^3}\) . J. Funct. Anal. 9, 296–305 (1972)
Kato T.: The Navier–Stokes equation for an incompressible fluid in \({{\mathbb R}^2}\) with a measure as the initial vorticity. Differ. Integral Equ. 7, 949–966 (1994)
Kirchhoff G.R.: Vorlesungen über Mathematische Physik. Mekanik. Teubner, Leipzig (1876)
Le Dizès S., Verga A.: Viscous interactions of two co-rotating vortices before merging. J. Fluid Mech. 467, 389–410 (2002)
Mc Williams J.C.: The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43 (1984)
Mc Williams J.C.: The vortices of two-dimensional turbulence. J. Fluid Mech. 219, 361–385 (1990)
Maekawa, Y.: Spectral properties of the linearization at the Burgers vortex in the high rotation limit. J. Math. Fluid Mech. (to appear)
Maekawa Y.: On the existence of Burgers vortices for high Reynolds numbers. J. Math. Anal. Appl. 349, 181–200 (2009)
Maekawa Y.: Existence of asymmetric Burgers vortices and their asymptotic behavior at large circulations. Math. Models Methods Appl. Sci. 19, 669–705 (2009)
Majda A.: Remarks on weak solutions for vortex sheets with a distinguished sign. Indiana Univ. Math. J. 42, 921–939 (1993)
Marchioro C.: Euler evolution for singular initial data and vortex theory: a global solution. Comm. Math. Phys. 116, 45–55 (1988)
Marchioro C.: On the vanishing viscosity limit for two-dimensional Navier–Stokes equations with singular initial data. Math. Methods Appl. Sci. 12, 463–470 (1990)
Marchioro C.: On the inviscid limit for a fluid with a concentrated vorticity. Comm. Math. Phys. 196, 53–65 (1998)
Marchioro C., Pulvirenti M.: Vortices and localization in Euler flows. Comm. Math. Phys. 154, 49–61 (1993)
Marchioro C., Pulvirenti M.: Mathematical theory of incompressible nonviscous fluids. In: Applied Mathematical Sciences, vol. 96. Springer-Verlag, New York, 1994
Masmoudi N.: Remarks about the inviscid limit of the Navier–Stokes system. Comm. Math. Phys. 270, 777–788 (2007)
Meunier P., Le Dizès S., Leweke T.: Physics of vortex merging. Comptes Rendus Physique 6, 431–450 (2005)
Moffatt H.K., Kida S., Ohkitani K.: Stretched vortices—the sinews of turbulence; large-Reynolds-number asymptotics. J. Fluid Mech. 259, 241–264 (1994)
Nagem R., Sandri G., Uminsky D., Wayne C.E.: Generalized Helmholtz– Kirchhoff model for two-dimensional distributed vortex motion. SIAM J. Appl. Dyn. Syst. 8, 160–179 (2009)
Newton P.: The N-vortex problem. Analytical techniques. In: Applied Mathematical Sciences, vol. 145. Springer-Verlag, New York, 2001
Osada H.: Diffusion processes with generators of generalized divergence form. J. Math. Kyoto Univ. 27, 597–619 (1987)
Sammartino, M., Caflisch, R.: Zero viscosity limit for analytic solutions of the Navier–Stokes equation on a half-space. I. Existence for Euler and Prandtl equations. Comm. Math. Phys. 192, 433–461 (1998). II. Construction of the Navier–Stokes solution. Comm. Math. Phys. 192, 463–491 (1998)
Sueur, F.: Vorticity internal transition layers for the Navier–Stokes equations. Preprint 2008. arXiv:0812.2145
Swann H.: The convergence with vanishing viscosity of nonstationary Navier–Stokes flow to ideal flow in \({{\mathbb R}^3}\) . Trans. Am. Math. Soc. 157, 373–397 (1971)
Ting, L., Klein, R.: Viscous vortical flows. In: Lecture Notes in Physics, vol. 374. Springer-Verlag, Berlin, 1991
Ting L., Tung C.: Motion and decay of a vortex in a nonuniform stream. Phys. Fluids 8, 1039–1051 (1965)
Vishik M.: Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Ann. Sci. Ecole Norm. Sup. 32, 769–812 (1999)
Yudovich V.: Non-stationary flows of an ideal incompressible fluid. Z̆. Vyčisl. Mat. i Mat. Fiz. 3, 1032–1066 (1963)
Yudovich V.: Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid. Math. Res. Lett. 2, 27–38 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Mielke
Rights and permissions
About this article
Cite this article
Gallay, T. Interaction of Vortices in Weakly Viscous Planar Flows. Arch Rational Mech Anal 200, 445–490 (2011). https://doi.org/10.1007/s00205-010-0362-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-010-0362-2