Incompressible flow around thin obstacle, uniqueness for the wortex-wave system

Christophe Lacave[1]

  • [1] Université Paris-Diderot (Paris 7), Institut de Mathématiques de Jussieu, UMR 7586 - CNRS, 175 rue du Chevaleret, 75013 Paris, France

Journées Équations aux dérivées partielles (2009)

  • page 1-17
  • ISSN: 0752-0360

Abstract

top
We present here the results concerning the influence of a thin obstacle on the behavior of incompressible flow. We extend the works made by Itimie, Lopes Filho, Nussenzveig Lopes and Kelliher where they consider that the obstacle shrinks to a point. We begin by working in two-dimension, and thanks to complex analysis we treat the case of ideal and viscous flows around a curve. Next, we consider three-dimensional viscous flow in the exterior of a surface/curve. We finish by giving uniqueness of the vortex-wave system with a single point vortex introduced by Marchioro and Pulvirenti, in the case where the initial vorticity is constant near the point vortex. This last result gives, in particular, the uniqueness of the limit system obtained in the case of a perfect fluid around a point. We choose here to give the main steps of this uniqueness result, obtained in collaboration with E. Miot.

How to cite

top

Lacave, Christophe. "Incompressible flow around thin obstacle, uniqueness for the wortex-wave system." Journées Équations aux dérivées partielles (2009): 1-17. <http://eudml.org/doc/116375>.

@article{Lacave2009,
abstract = {We present here the results concerning the influence of a thin obstacle on the behavior of incompressible flow. We extend the works made by Itimie, Lopes Filho, Nussenzveig Lopes and Kelliher where they consider that the obstacle shrinks to a point. We begin by working in two-dimension, and thanks to complex analysis we treat the case of ideal and viscous flows around a curve. Next, we consider three-dimensional viscous flow in the exterior of a surface/curve. We finish by giving uniqueness of the vortex-wave system with a single point vortex introduced by Marchioro and Pulvirenti, in the case where the initial vorticity is constant near the point vortex. This last result gives, in particular, the uniqueness of the limit system obtained in the case of a perfect fluid around a point. We choose here to give the main steps of this uniqueness result, obtained in collaboration with E. Miot.},
affiliation = {Université Paris-Diderot (Paris 7), Institut de Mathématiques de Jussieu, UMR 7586 - CNRS, 175 rue du Chevaleret, 75013 Paris, France},
author = {Lacave, Christophe},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
month = {6},
pages = {1-17},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Incompressible flow around thin obstacle, uniqueness for the wortex-wave system},
url = {http://eudml.org/doc/116375},
year = {2009},
}

TY - JOUR
AU - Lacave, Christophe
TI - Incompressible flow around thin obstacle, uniqueness for the wortex-wave system
JO - Journées Équations aux dérivées partielles
DA - 2009/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 17
AB - We present here the results concerning the influence of a thin obstacle on the behavior of incompressible flow. We extend the works made by Itimie, Lopes Filho, Nussenzveig Lopes and Kelliher where they consider that the obstacle shrinks to a point. We begin by working in two-dimension, and thanks to complex analysis we treat the case of ideal and viscous flows around a curve. Next, we consider three-dimensional viscous flow in the exterior of a surface/curve. We finish by giving uniqueness of the vortex-wave system with a single point vortex introduced by Marchioro and Pulvirenti, in the case where the initial vorticity is constant near the point vortex. This last result gives, in particular, the uniqueness of the limit system obtained in the case of a perfect fluid around a point. We choose here to give the main steps of this uniqueness result, obtained in collaboration with E. Miot.
LA - eng
UR - http://eudml.org/doc/116375
ER -

References

top
  1. Chemin J-Y., Desjardins B., Gallagher I. and Grenier E., Mathematical Geophysics: An introduction to rotating fluids and to the Navier-Stokes equations, Oxford University Press, 2006. Zbl1205.86001MR2228849
  2. Desjardins B., A few remarks on ordinary differential equations, Commun. in Part. Diff. Eq. ,21:11, 1667-1703, 1996. Zbl0899.35022MR1421208
  3. DiPerna R. J. and Lions P. L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98, 511-547, 1989. Zbl0696.34049MR1022305
  4. Iftimie D. and Kelliher J., Remarks on the vanishing obstacle limit for a 3D viscous incompressible fluid, to appear in Proceedings of the AMS, 2008. Zbl1156.76018MR2448591
  5. Iftimie D., Lopes Filho M.C. and Nussenzveig Lopes H.J., Two Dimensional Incompressible Ideal Flow Around a Small Obstacle, Comm. Partial Diff. Eqns. 28 (2003), no. 1 & 2, 349-379. Zbl1094.76007MR1974460
  6. Iftimie D., Lopes Filho M.C. and Nussenzveig Lopes H.J., Two Dimensional Incompressible Viscous Flow Around a Small Obstacle, Math. Annalen. 336 (2006), 449-489. Zbl1169.76016MR2244381
  7. Kikuchi K., Exterior problem for the two-dimensional Euler equation, J Fac Sci Univ Tokyo Sect 1A Math 1983; 30(1):63-92. Zbl0517.76024MR700596
  8. Kozono H. and Yamazaki M., Local and global unique solvability of the Navier-Stokes exterior problem with Cauchy data in the space L n , ∞ , Houst. J. Math. 21(4), 755-799 (1995). Zbl0848.35099MR1368344
  9. Lacave C., Two Dimensional Incompressible Ideal Flow Around a Thin Obstacle Tending to a Curve, Annales de l’IHP, Analyse non linéaire 26 (2009), 1121-1148. Zbl1166.76300MR2542717
  10. Lacave C., Two Dimensional Incompressible Viscous Flow Around a Thin Obstacle Tending to a Curve, Proc. Roy. Soc. Edinburgh Sect. A, Vol. 139 (2009), No. 6, pp. 1237-1254. Zbl1259.76007MR2557320
  11. Lacave C. and Miot E., Uniqueness for the vortex-wave system when the vorticity is constant near the point vortex, SIAM Journal on Mathematical Analysis, Vol. 41 (2009), No. 3, pp. 1138-1163. Zbl1189.35259MR2529959
  12. Lopes Filho M.C., Vortex dynamics in a two dimensional domain with holes and the small obstacle limit, SIAM Journal on Mathematical Analysis, 39(2)(2007) : 422-436. Zbl1286.76018MR2338413
  13. Lopes Filho M.C., Nussenzveig Lopes H.J. and Zheng Y., Weak solutions to the equations of incompressible, ideal flow, Text of minicourse for the 22nd Brazilian Colloquium of Mathematics, (1999), http://www.ime.unicamp.br/ mlopes/publications.html 
  14. Marchioro C. and Pulvirenti M., On the vortex-wave system, Mechanics, analysis, and geometry: 200 years after Lagrange, M. Francaviglia (ed), Elsevier Science, Amsterdam, 1991. Zbl0733.76015MR1098512
  15. Marchioro C. and Pulvirenti M., Mathematical Theory of Incompressible Nonviscous Fluids, Springer-Verlag, 1991. Zbl0789.76002MR1245492
  16. Marchioro C. and Pulvirenti M., Vortices and Localization in Euler Flows, Commun. Math. Phy. 154,49-61, 1993. Zbl0774.35058MR1220946
  17. Monniaux S., Navier-Stokes Equations in Arbitrary Domains: the Fujita-Kato Scheme, Math. Res. Lett. 13 (2006), no. 3, 455-461. Zbl1126.35044MR2231130
  18. Pommerenke C., Univalent functions, Vandenhoeck & Ruprecht, 1975. Zbl0298.30014MR507768
  19. Pommerenke C., Boundary behaviour of conformal maps, BerlinNew York: Springer-Verlag, 1992. Zbl0762.30001MR1217706
  20. Starovoitov V.N., Uniqueness of a solution to the problem of evolution of a point vortex, Siberian Mathematical Journal, Vol. 35, no. 3, 1994. Zbl0854.76021MR1292230
  21. Yudovich V.I., Non-stationary flows of an ideal incompressible fluid, Zh Vych Mat, 3:1032-1066, 1963. Zbl0129.19402MR158189

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.