zbMATH — the first resource for mathematics

The sharp corner formation in 2D Euler dynamics of patches: infinite double exponential rate of merging. (English) Zbl 1308.35188
Summary: For the 2D Euler dynamics of patches, we investigate the convergence to the singular stationary solution in the presence of a regular strain. It is proved that the rate of merging can be double exponential infinitely in time and the estimates we obtain are sharp.

35Q31 Euler equations
Full Text: DOI arXiv
[1] Bertozzi, A.; Constantin, P., Global regularity for vortex patches, Commun. Math. Phys., 152, 19-28, (1993) · Zbl 0771.76014
[2] Bertozzi, A., Majda, A.: Vorticity and incompressible flow. In: Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002 · Zbl 0653.76020
[3] Caglioti, E.; Maffei, C., Scattering theory: a possible approach to the homogenization problem for the Euler equations, Rend. Mat. Appl. (7), 17, 445-475, (1997) · Zbl 0916.76012
[4] Chemin, J.-Y., Persistence of geometric structures in two-dimensional incompressible fluids, Ann. Sci. Ecole Norm. Sup., (4), 26, 517-542, (1993) · Zbl 0779.76011
[5] Chemin, J.-Y.: Perfect incompressible fluids. In: Oxford Lecture Series in Mathematics and its Applications, Vol. 14. The Clarendon Press, New York, 1998 · Zbl 0916.76012
[6] Cordoba, D.; Fontelos, M.; Mancho, A.; Rodrigo, J., Evidence of singularities for a family of contour dynamics equations, PNAS, 102, 5949-5952, (2005) · Zbl 1135.76315
[7] Cordoba, D., On the search for singularities in incompressible flows, Appl. Math., 51, 299-320, (2006) · Zbl 1164.76320
[8] Cordoba, D.; Fefferman, C., Behavior of several two-dimensional fluid equations in singular scenarios, Proc. Natl. Acad. Sci. USA, 98, 4311-4312, (2001) · Zbl 0965.35133
[9] Cordoba, D.; Fefferman, C., Scalars convected by a two-dimensional incompressible flow, Commun. Pure Appl. Math., 55, 255-260, (2002) · Zbl 1019.76013
[10] Cordoba, D.; Fefferman, C., Growth of solutions for QG and 2D Euler equations, J. Am. Math. Soc., 15, 665-670, (2002) · Zbl 1013.76011
[11] Cordoba, D., Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. Math. (2), 148, 1135-1152, (1998) · Zbl 0920.35109
[12] Denisov, S.: Double-exponential growth of the vorticity gradient for the two-dimensional Euler equation. In: Proceedings of the AMS (to appear, arXiv:1201.1771) · Zbl 1315.35150
[13] Mancho, A.M.: Numerical studies on the self-similar collapse of the α-patches problem, preprint, arXiv:0902.0706 (2009) · Zbl 0965.35133
[14] Marchioro, C., Pulvirenti, M.: Mathematical theory of incompressible nonviscous fluids. In: Applied Mathematical Sciences, Vol. 96. Springer, Berlin, 1994 · Zbl 0789.76002
[15] Melander, M.V.; Zabusky, N.J.; McWilliams, J.C., Symmetric vortex merger in two dimensions: causes and conditions, J. Fluid Mech., 195, 303-340, (1988) · Zbl 0653.76020
[16] Saffman, P.G.: Vortex dynamics. In: Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge, 1992 · JFM 59.1447.02
[17] Wolibner, W., Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long, (French) Mat. Z., 37, 698-726, (1933) · Zbl 0008.06901
[18] Yudovich, V.I., Non-stationary flow of an incompressible liquid, Zh. Vychils. Mat. Mat. Fiz., 3, 1032-1066, (1963)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.