Interaction des tourbillons dans les écoulements plans faiblement visqueux

Gallay, Thierry

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2007-2008), Exposé no. 13, 16 p.

[1] H. Abidi et R. Danchin. Optimal bounds for the inviscid limit of Navier-Stokes equations. Asymptot. Anal. 38 (2004), 35–46. | MR | Zbl

[2] Th. Beale et A. Majda. Rates of convergence for viscous splitting of the Navier-Stokes equations. Math. Comp. 37 (1981), 243–259. | MR | Zbl

[3] R. Caflisch et M. Sammartino. Vortex layers in the small viscosity limit. “WASCOM 2005”—13th Conference on Waves and Stability in Continuous Media, 59–70, World Sci. Publ., Hackensack, NJ, 2006.

[4] E. A. Carlen et 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, 135–157 (1996), 1995. | MR | Zbl

[5] J.-Y. Chemin. A remark on the inviscid limit for two-dimensional incompressible fluids. Comm. Partial Differential Equations 21 (1996), 1771–1779. | MR | Zbl

[6] P.-H. Chen et W.-L. Wang. Roll-up of a viscous vortex sheet. J. Chinese Inst. Engrs. 14 (1991), 507–517. | MR

[7] P. Constantin et J. Wu. Inviscid limit for vortex patches. Nonlinearity 8 (1995), 735–742. | MR | Zbl

[8] P. Constantin et J. Wu. The inviscid limit for non-smooth vorticity. Indiana Univ. Math. J. 45 (1996), 67–81. | MR | Zbl

[9] G.-H. Cottet. Équations de Navier-Stokes dans le plan avec tourbillon initial mesure. C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), 105–108. | MR | Zbl

[10] R. Danchin. Poches de tourbillon visqueuses. J. Math. Pures Appl. 76 (1997), 609–647. | MR | Zbl

[11] R. Danchin. Persistance de structures géométriques et limite non visqueuse pour les fluides incompressibles en dimension quelconque. Bull. Soc. Math. France 127 (1999), 179–227. | Numdam | MR | Zbl

[12] J.-M. Delort. Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc. 4 (1991), 553–586. | MR | Zbl

[13] D. Ebin et J. Marsden. Groups of diffeomorphisms and the notion of an incompressible fluid. Ann. of Math. 92 (1970), 102–163. | MR | Zbl

[14] I. Gallagher et Th. Gallay. Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity. Math. Ann. 332 (2005), 287–327. | MR | Zbl

[15] I. Gallagher, Th. Gallay, et 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), 1665–1672. | MR | Zbl

[16] Th. Gallay. Equations de Navier-Stokes dans le plan avec tourbillon initial mesure. Séminaire EDP de l’Ecole Polytechnique 2003-2004, exposé ${\mathrm{n}}^{\mathrm{o}}$XIV. | Numdam

[17] Th. Gallay et C. E. Wayne. Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on ${ℝ}^2$. Arch. Ration. Mech. Anal. 163 (2002), 209–258. | MR | Zbl

[18] Th. Gallay et C.E. Wayne. Global stability of vortex solutions of the two-dimensional Navier-Stokes equation. Comm. Math. Phys. 255 (2005), 97–129. | MR

[19] Th. Gallay et C.E. Wayne. Existence and stability of asymmetric Burgers vortices. J. Math. Fluid Mech. 9 (2007), 243–261. | MR | Zbl

[20] Y. Giga, T. Miyakawa, et H. Osada. Two-dimensional Navier-Stokes flow with measures as initial vorticity. Arch. Rational Mech. Anal. 104 (1988), 223–250. | MR | Zbl

[21] E. Grenier. On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. 53 (2000), 1067–1091. | MR | Zbl

[22] T. Hmidi. Régularité höldérienne des poches de tourbillon visqueuses. J. Math. Pures Appl. 84 (2005), 1455–1495. | MR | Zbl

[23] T. Hmidi. Poches de tourbillon singulières dans un fluide faiblement visqueux. Rev. Mat. Iberoamericana 22 (2006), 489–543. | MR | Zbl

[24] T. Kato : Nonstationary flows of viscous and ideal fluids in ${ℝ}^3$. J. Functional Analysis 9 (1972), 296–305. | MR | Zbl

[25] T. Kato. The Navier-Stokes equation for an incompressible fluid in ${ℝ}^2$ with a measure as the initial vorticity. Differential Integral Equations 7 (1994), 949–966. | MR | Zbl

[26] Y. Maekawa. Spectral properties of the linearization at the Burgers vortex in the high rotation limit. J. Math. Fluid Mech., to appear.

[27] Y. Maekawa. On the existence of Burgers vortices for high Reynolds numbers. J. Math. Analysis and Applications, to appear.

[28] A. Majda. Remarks on weak solutions for vortex sheets with a distinguished sign. Indiana Univ. Math. J. 42 (1993), 921–939. | MR | Zbl

[29] A. Majda et A. Bertozzi. Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics 27. Cambridge University Press, Cambridge, 2002. | MR | Zbl

[30] C. Marchioro. Euler evolution for singular initial data and vortex theory : a global solution. Comm. Math. Phys. 116 (1988), 45–55. | MR | Zbl

[31] C. Marchioro. On the vanishing viscosity limit for two-dimensional Navier-Stokes equations with singular initial data. Math. Methods Appl. Sci. 12 (1990), 463–470. | MR | Zbl

[32] C. Marchioro. On the inviscid limit for a fluid with a concentrated vorticity. Comm. Math. Phys. 196 (1998), 53–65. | MR | Zbl

[33] C. Marchioro et M. Pulvirenti. Vortices and localization in Euler flows. Comm. Math. Phys. 154 (1993), 49–61. | MR | Zbl

[34] C. Marchioro et M. Pulvirenti. Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences 96, Springer-Verlag, New York, 1994. | MR | Zbl

[35] N. Masmoudi. Remarks about the inviscid limit of the Navier-Stokes system. Comm. Math. Phys. 270 (2007), 777–788. | MR | Zbl

[36] H. K. Moffatt, S. Kida, et K. Ohkitani. Stretched vortices—the sinews of turbulence ; large-Reynolds-number asymptotics. J. Fluid Mech. 259 (1994), 241–264. | MR

[37] H. Osada. Diffusion processes with generators of generalized divergence form. J. Math. Kyoto Univ. 27 (1987), 597–619. | MR | Zbl

[38] M. Sammartino et R. Caflisch. 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 (1998), 433–461. II. Construction of the Navier-Stokes solution. Comm. Math. Phys. 192 (1998), 463–491. | MR | Zbl

[39] F. Sueur. Vorticity internal transition layers for the Navier-Stokes equations. Travail en préparation.

[40] H. Swann. The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in ${ℝ}^3$. Trans. Amer. Math. Soc. 157 (1971), 373–397. | MR | Zbl

[41] L. Ting et R. Klein. Viscous vortical flows. Lecture Notes in Physics 374. Springer-Verlag, Berlin, 1991. | MR | Zbl

[42] L. Ting et C. Tung. Motion and decay of a vortex in a nonuniform stream. Phys. Fluids 8 (1965), 1039–1051. | MR | Zbl

[43] M. Vishik. Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Ann. Sci. Ecole Norm. Sup. 32 (1999), 769–812. | Numdam | MR | Zbl

[44] V. Yudovich. Non-stationary flows of an ideal incompressible fluid. Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963), 1032–1066. | Zbl

[45] V. Yudovich. Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid. Math. Res. Lett. 2 (1995), 27–38. | MR | Zbl