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.
@incollection{JEDP_2009____A4_0, author = {Christophe Lacave}, title = {Incompressible flow around thin obstacle, uniqueness for the wortex-wave system}, booktitle = {}, series = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, eid = {4}, pages = {1--17}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2009}, doi = {10.5802/jedp.57}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.57/} }
TY - JOUR AU - Christophe Lacave TI - Incompressible flow around thin obstacle, uniqueness for the wortex-wave system JO - Journées équations aux dérivées partielles PY - 2009 SP - 1 EP - 17 PB - Groupement de recherche 2434 du CNRS UR - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.57/ DO - 10.5802/jedp.57 LA - en ID - JEDP_2009____A4_0 ER -
%0 Journal Article %A Christophe Lacave %T Incompressible flow around thin obstacle, uniqueness for the wortex-wave system %J Journées équations aux dérivées partielles %D 2009 %P 1-17 %I Groupement de recherche 2434 du CNRS %U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.57/ %R 10.5802/jedp.57 %G en %F JEDP_2009____A4_0
Christophe Lacave. Incompressible flow around thin obstacle, uniqueness for the wortex-wave system. Journées équations aux dérivées partielles (2009), article no. 4, 17 p. doi : 10.5802/jedp.57. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.57/
[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. | MR
[2] Desjardins B., A few remarks on ordinary differential equations, Commun. in Part. Diff. Eq. ,21:11, 1667-1703, 1996. | MR | Zbl
[3] DiPerna R. J. and Lions P. L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98, 511-547, 1989. | MR | Zbl
[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. | MR | Zbl
[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. 12, 349-379. | MR | Zbl
[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. | MR | Zbl
[7] Kikuchi K., Exterior problem for the two-dimensional Euler equation, J Fac Sci Univ Tokyo Sect 1A Math 1983; 30(1):63-92. | MR | Zbl
[8] Kozono H. and Yamazaki M., Local and global unique solvability of the Navier-Stokes exterior problem with Cauchy data in the space , Houst. J. Math. 21(4), 755-799 (1995). | MR | Zbl
[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. | Numdam | MR | Zbl
[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. | MR
[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. | MR | Zbl
[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. | MR
[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. | MR | Zbl
[15] Marchioro C. and Pulvirenti M., Mathematical Theory of Incompressible Nonviscous Fluids, Springer-Verlag, 1991. | MR | Zbl
[16] Marchioro C. and Pulvirenti M., Vortices and Localization in Euler Flows, Commun. Math. Phy. 154,49-61, 1993. | MR | Zbl
[17] Monniaux S., Navier-Stokes Equations in Arbitrary Domains: the Fujita-Kato Scheme, Math. Res. Lett. 13 (2006), no. 3, 455-461. | MR | Zbl
[18] Pommerenke C., Univalent functions, Vandenhoeck Ruprecht, 1975. | MR | Zbl
[19] Pommerenke C., Boundary behaviour of conformal maps, BerlinNew York: Springer-Verlag, 1992. | MR | Zbl
[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. | MR | Zbl
[21] Yudovich V.I., Non-stationary flows of an ideal incompressible fluid, Zh Vych Mat, 3:1032-1066, 1963. | MR | Zbl
Cited by Sources: