×

Singular vortex patches in a fluid of low viscosity. (Poches de tourbillon singulières dans un fluide faiblement visqueux.) (French) Zbl 1127.35037

The author studies the motion of a bidimensional, incompressible, small viscous fluid as the Navier-Stokes equations
\[ \partial_t v_\nu+ v_\nu\nabla v_\nu- \nu\Delta v_\nu= -\nabla p_\nu,\quad \operatorname{div} v_\nu= 0,\quad v_\nu|_{t= 0}= v^0,\tag{1} \]
near the motion of a fluid of null viscosity in the system of Euler equations
\[ \partial_t v+v\nabla v\neq -\nabla p,\quad\operatorname{div} v= 0,\quad v|_{t= 0}= v^0, \tag{2} \]
The used notations are evident. The motions are in the presence of singular vortex patches. The author considers only small viscosity, because he is interested in the comparison of the results for \(\nu\neq 0\) and \(\nu= 0\). He considers \(\lim_{\nu\to 0} v_\nu\) obtained from (1) and compares it with \(v\) obtained from (2). The singularities of the vortex patches concern the singularities which can appear on the boundary \(\partial\Omega\) of the domain \(\Omega\) occupied by the vortex patch: coin, cusp or other singularities.
The history of the presented problem is sketched in the introduction, followed by five chapters and an appendix, where there are analysed the following subjects: the velocity field, the singularities on \(\partial\Omega\), the dynamics of the singular patches and the comparison between the motion in (1) and (2). The paper is very rich in content and requires a high level of mathematical knowledge.

MSC:

35Q30 Navier-Stokes equations
76D17 Viscous vortex flows
35A20 Analyticity in context of PDEs
42B25 Maximal functions, Littlewood-Paley theory
76B47 Vortex flows for incompressible inviscid fluids
PDFBibTeX XMLCite
Full Text: DOI Euclid EuDML

References:

[1] Bony, J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. (4) 14 (1981), 209-246. · Zbl 0495.35024
[2] Bahouri, H. and Chemin, J.-Y.: Equations de transport relatives à des champs de vecteurs non-lipschitziens et mécanique des fluides. Arch. Rational Mech. Anal. 127 (1994), 159-181. · Zbl 0821.76012 · doi:10.1007/BF00377659
[3] Chemin, J.-Y.: Perfect incompressible fluids . Translated from the 1995 french original by Isabelle Gallagher and Dragos Iftimie. Oxford Lecture Series in Mathematics and its Applications, 14 . The Clarendon Press, Oxford University Press, New York, 1998. · Zbl 0927.76002
[4] Chemin, J.-Y.: A remark on the inviscid limit for two-dimensional incompressible fluids. Comm. Partial Differential Equations 21 (1996), 1771-1779. · Zbl 0876.35087 · doi:10.1080/03605309608821245
[5] Chemin, J.-Y.: Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel. J. Anal. Math. 77 (1999), 27-50. · Zbl 0938.35125 · doi:10.1007/BF02791256
[6] Constantin, P. and Wu, J.: The inviscid limit for non-smooth vorticity. Indiana Univ. Math. J. 45 (1996), 67-81. · Zbl 0859.76015 · doi:10.1512/iumj.1996.45.1960
[7] Hmidi, T.: Transport-diffusion et viscosité évanescente. C. R. Math. Acad. Sci. Paris 337 (2003), 309-312. · Zbl 1034.35007 · doi:10.1016/S1631-073X(03)00351-0
[8] Hmidi, T.: Régularité höldérienne des poches de tourbillon visqueuses. J. Math. Pures Appl. (9) 84 (2005), 1455-1495. · Zbl 1095.35024 · doi:10.1016/j.matpur.2005.01.004
[9] Danchin, R.: Poches de tourbillon visqueuses. J. Math. Pures Appl. (9) 76 (1997), 609-647. · Zbl 0903.76020 · doi:10.1016/S0021-7824(97)89964-3
[10] Danchin, R.: Évolution temporelle d’une poche de tourbillon singulière. Comm. Partial Differential Equations 22 (1997), 685-721. · Zbl 0882.35093 · doi:10.1080/03605309708821280
[11] Danchin, R.: Évolution d’une singularité de type cusp dans une poche de tourbillon. Rev. Mat. Iberoamericana 16 (2000), 281-329. · Zbl 1158.35404 · doi:10.4171/RMI/276
[12] Planchon, F.: Sur une inégalité de type Poincaré. C. R. Acad. Sci. Paris Sér. I Math 330 (2000), 21-23. · Zbl 0953.46020 · doi:10.1016/S0764-4442(00)88138-0
[13] Majda, A.: Vorticity and the mathematical theory of incompressible fluid flow. Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S187-S220. · Zbl 0595.76021 · doi:10.1002/cpa.3160390711
[14] Vishik, M.: Hydrodynamics in Besov Spaces. Arch. Ration. Mech. Anal 145 (1998), 197-214. · Zbl 0926.35123 · doi:10.1007/s002050050128
[15] Yudovich, V.I.: Non-stationnary flows of an ideal incompressible fluid. Zh. Vychisl. Mat. Fiz. (1963), no. 3, 1032-1066.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.