zbMATH — the first resource for mathematics

Gluing methods for vortex dynamics in Euler flows. (English) Zbl 1439.35382
Summary: A classical problem for the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain is that of finding regular solutions with highly concentrated vorticities around \(N\) moving vortices. The formal dynamic law for such objects was first derived in the 19th century by G. Kirchhoff [Vorlesungen über mathematische Physik. I. Mechanik. Leipzig. Teubner (1876; JFM 08.0542.01)] and E. J. Routh [Proc. Lond. Math. Soc. 12, 73–89 (1881; JFM 13.0741.01)]. In this paper we devise a gluing approach for the construction of smooth \(N\)-vortex solutions. We capture in high precision the core of each vortex as a scaled finite mass solution of Liouville’s equation plus small, more regular terms. Gluing methods have been a powerful tool in geometric constructions by desingularization. We succeed in applying those ideas in this highly challenging setting.

35Q31 Euler equations
76B47 Vortex flows for incompressible inviscid fluids
35B65 Smoothness and regularity of solutions to PDEs
Full Text: DOI
[1] Baraket, S.; Pacard, F., Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var., 6, 1-38 (1998) · Zbl 0890.35047
[2] Bartsch, T.; Pistoia, A., Critical points of the N-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations, SIAM J. Appl. Math., 75, 2, 726-744 (2015) · Zbl 1317.35073
[3] Bartsch, T.; Dai, Q., Periodic solutions of the N-vortex Hamiltonian system in planar domains, J. Differ. Equ., 260, 3, 2275-2295 (2016) · Zbl 1337.37043
[4] Bartsch, T., Periodic solutions of singular first-order Hamiltonian systems of \(N\)-vortex type, Arch. Math., 107, 4, 413-422 (2016) · Zbl 1351.37232
[5] Bourgain, J.; Li, D., Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Invent. Math., 201, 1, 97-157 (2015) · Zbl 1320.35266
[6] Cao, D.; Liu, Z.; Wei, J., Regularization of point vortices pairs for the Euler equation in dimension two, Arch. Ration. Mech. Anal., 212, 1, 179-217 (2014) · Zbl 1293.35223
[7] Del Pino, M.; Kowalczyk, M.; Musso, M., Singular limits in Liouville-type equations, Calc. Var. Partial Differ. Equ., 24, 47-81 (2005) · Zbl 1088.35067
[8] De Lellis, C.; Székelyhidi, L., The Euler equations as a differential inclusion, Ann. Math., 170, 3, 1417-1436 (2009) · Zbl 1350.35146
[9] Di Perna, R.; Majda, A., Oscillations and concentrations in weak solutions of the incompressible fluid equations, Commun. Math. Phys., 108, 4, 667-689 (1987) · Zbl 0626.35059
[10] Esposito, P.; Grossi, M.; Pistoia, A., On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Lináire, 22, 2, 227-257 (2005) · Zbl 1129.35376
[11] Kiselev, A., Roquejoffre, J.-M., Ryzhik, L.: Appetizers in Nonlinear PDEs. 2017. http://math.stanford.edu/ ryzhik/STANFORD/STANF272-17/book-split-chapt1and12.pdf
[12] Kirchhoff, G., Vorlesungen über mathematische Physik (1876), Leipzig: Teubner, Leipzig · JFM 08.0542.01
[13] Lacave, C.; Miot, E., Uniqueness for the vortex-wave system when the vorticity is initially constant near the point vortex, SIAM J. Math. Anal., 41, 3, 1138-1163 (2009) · Zbl 1189.35259
[14] Lin, Cc, On the motion of vortices in 2D I. Existence of the Kirchhoff-Routh function, Proc. Natl. Acad. Sci., 27, 570-575 (1941) · Zbl 0063.03560
[15] Lopes Filho, Mc; Miot, E.; Nussenzveig Lopes, Hj, Existence of a weak solution in \(L^p\) to the vortex-wave system, J. Nonlinear Sci., 21, 5, 685-703 (2011) · Zbl 1226.76005
[16] Majda, Aj; Bertozzi, Al, Vorticity and Incompressible Flow, vol. 27 of Cambridge Texts in Applied Mathematics (2002), Cambridge: Cambridge University Press, Cambridge
[17] Marchioro, C.; Pulvirenti, M., Euler evolution for singular initial data and vortex theory, Commun. Math. Phys., 91, 4, 563-572 (1983) · Zbl 0529.76023
[18] Routh, Ej, Some applications of conjugate functions, Proc. Lond. Math. Soc., 12, 73-89 (1881) · JFM 13.0741.01
[19] Shnirelman, A., Weak solutions with decreasing energy of incompressible Euler equations, Commun. Math. Phys., 210, 3, 541-603 (2000) · Zbl 1011.35107
[20] Smets, D.; Van Schaftingen, J., Desingulariation of vortices for the Euler equation, Arch. Ration. Mech. Anal., 198, 869-925 (2010) · Zbl 1228.35171
[21] Yudovich, V., Nonstationary flow of an ideal incompressible liquid, Zh. Vych. Mat., 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.