Interaction of Vortices in Weakly Viscous Planar Flows

Archive for Rational Mechanics and Analysis

Volumn 200, Issue 2, 2011, Pages 445-490

  Gallay, Thierry ^a  

^a INSTITUT FOURIER   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 79954608590     PISSN: 00039527     EISSN: 14320673     Source Type: Journal    
DOI: 10.1007/s00205-010-0362-2     Document Type: Article

References (54)

1. Abidi H., Danchin R.: Optimal bounds for the inviscid limit of Navier-Stokes equations. Asymptot. Anal. 38, 35-46 (2004).
2. Beale T., Majda A.: Rates of convergence for viscous splitting of the Navier-Stokes equations. Math. Comput. 37, 243-259 (1981).
3. Benfatto G., Esposito R., Pulvirenti M.: Planar Navier-Stokes flow for singular initial data. Nonlinear Anal. 9, 533-545 (1985).
4. 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.
5. 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).
6. Chemin J.-Y.: A remark on the inviscid limit for two-dimensional incompressible fluids. Comm. Partial Differ. Equ. 21, 1771-1779 (1996).
7. Chen P.-H., Wang W.-L.: Roll-up of a viscous vortex sheet. J. Chin. Inst. Eng. 14, 507-517 (1991).
8. Constantin P., Wu J.: Inviscid limit for vortex patches. Nonlinearity 8, 735-742 (1995).
9. Constantin P., Wu J.: The inviscid limit for non-smooth vorticity. Indiana Univ. Math. J. 45, 67-81 (1996).
10. 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).
11. Couder Y.: Observation expérimentale de la turbulence bidimensionnelle dans un film liquide mince. C. R. Acad. Sci. Paris II 297, 641-645 (1983).
12. Danchin R.: Poches de tourbillon visqueuses. J. Math. Pures Appl. 76, 609-647 (1997).
13. 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).
14. Delort J.-M.: Existence de nappes de tourbillon en dimension deux. J. Am. Math. Soc. 4, 553-586 (1991).
15. Ebin D., Marsden J.: Groups of diffeomorphisms and the notion of an incompressible fluid. Ann. Math. 92, 102-163 (1970).
16. Gallagher I., Gallay T.: Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity. Math. Ann. 332, 287-327 (2005).
17. 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).
18. 2. Arch. Rational Mech. Anal. 163, 209-258 (2002).
19. Gallay T., Wayne C. E.: Global stability of vortex solutions of the two-dimensional Navier-Stokes equation. Comm. Math. Phys. 255, 97-129 (2005).
20. Gallay T., Wayne C. E.: Existence and stability of asymmetric Burgers vortices. J. Math. Fluid Mech. 9, 243-261 (2007).
21. Giga Y., Miyakawa T., Osada H.: Two-dimensional Navier-Stokes flow with measures as initial vorticity. Arch. Rational Mech. Anal. 104, 223-250 (1988).
22. Grenier E.: On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. 53, 1067-1091 (2000).
23. von Helmholtz H.: Über Integrale des hydrodynamischen Gleichungen, welche die Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25-55 (1858).
24. Hmidi T.: Régularité höldérienne des poches de tourbillon visqueuses. J. Math. Pures Appl. 84, 1455-1495 (2005).
25. Hmidi T.: Poches de tourbillon singulières dans un fluide faiblement visqueux. Rev. Mat. Iberoamericana 22, 489-543 (2006).
26. 3. J. Funct. Anal. 9, 296-305 (1972).
27. 2 with a measure as the initial vorticity. Differ. Integral Equ. 7, 949-966 (1994).
28. Kirchhoff G. R.: Vorlesungen über Mathematische Physik. Mekanik. Teubner, Leipzig (1876).
29. Le Dizès S., Verga A.: Viscous interactions of two co-rotating vortices before merging. J. Fluid Mech. 467, 389-410 (2002).
30. Mc Williams J. C.: The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21-43 (1984).
31. Mc Williams J. C.: The vortices of two-dimensional turbulence. J. Fluid Mech. 219, 361-385 (1990).
32. Maekawa, Y.: Spectral properties of the linearization at the Burgers vortex in the high rotation limit. J. Math. Fluid Mech. (to appear).
33. Maekawa Y.: On the existence of Burgers vortices for high Reynolds numbers. J. Math. Anal. Appl. 349, 181-200 (2009).
34. Maekawa Y.: Existence of asymmetric Burgers vortices and their asymptotic behavior at large circulations. Math. Models Methods Appl. Sci. 19, 669-705 (2009).
35. Majda A.: Remarks on weak solutions for vortex sheets with a distinguished sign. Indiana Univ. Math. J. 42, 921-939 (1993).
36. Marchioro C.: Euler evolution for singular initial data and vortex theory: a global solution. Comm. Math. Phys. 116, 45-55 (1988).
37. 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).
38. Marchioro C.: On the inviscid limit for a fluid with a concentrated vorticity. Comm. Math. Phys. 196, 53-65 (1998).
39. Marchioro C., Pulvirenti M.: Vortices and localization in Euler flows. Comm. Math. Phys. 154, 49-61 (1993).
40. Marchioro C., Pulvirenti M.: Mathematical theory of incompressible nonviscous fluids. In: Applied Mathematical Sciences, vol. 96. Springer-Verlag, New York, 1994.
41. Masmoudi N.: Remarks about the inviscid limit of the Navier-Stokes system. Comm. Math. Phys. 270, 777-788 (2007).
42. Meunier P., Le Dizès S., Leweke T.: Physics of vortex merging. Comptes Rendus Physique 6, 431-450 (2005).
43. Moffatt H. K., Kida S., Ohkitani K.: Stretched vortices-the sinews of turbulence; large-Reynolds-number asymptotics. J. Fluid Mech. 259, 241-264 (1994).
44. 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).
45. Newton P.: The N-vortex problem. Analytical techniques. In: Applied Mathematical Sciences, vol. 145. Springer-Verlag, New York, 2001.
46. Osada H.: Diffusion processes with generators of generalized divergence form. J. Math. Kyoto Univ. 27, 597-619 (1987).
47. 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).
48. Sueur, F.: Vorticity internal transition layers for the Navier-Stokes equations. Preprint 2008. arXiv: 0812. 2145.
49. 3Trans. Am. Math. Soc. 157, 373-397 (1971).
50. Ting, L., Klein, R.: Viscous vortical flows. In: Lecture Notes in Physics, vol. 374. Springer-Verlag, Berlin, 1991.
51. Ting L., Tung C.: Motion and decay of a vortex in a nonuniform stream. Phys. Fluids 8, 1039-1051 (1965).
52. 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).
53. Yudovich V.: Non-stationary flows of an ideal incompressible fluid. Ž. Vyčisl. Mat. i Mat. Fiz. 3, 1032-1066 (1963).
54. Yudovich V.: Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid. Math. Res. Lett. 2, 27-38 (1995).