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Critical points of the $$N$$-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations. (English) Zbl 1317.35073

##### MSC:
 35J60 Nonlinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 76B47 Vortex flows for incompressible inviscid fluids 35J08 Green’s functions for elliptic equations 35Q51 Soliton equations
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##### References:
 [1] H. Aref, Point vortex dynamics: A classical mathematical playground, J. Math. Phys., 48 (2007), pp. 1–22. · Zbl 1144.81308 [2] H. Aref, P. K. Newton, M. A. Strember, T. Tokieda, and D. Vainchtein, Vortex crystals, Adv. Appl. Mech., 39 (2002), pp. 1–79. [3] D. Bartolucci and A. Pistoia, Existence and qualitative properties of concentrating solutions for the sinh-Poisson equation, IMA J. Appl. Math., 72 (2007), pp. 706–729. · Zbl 1154.35072 [4] T. Bartsch and Q. Dai, Periodic Solutions of the $$N$$-Vortex Problem in Planar Domains, preprint, arXiv:1403.4533. · Zbl 1337.37043 [5] T. Bartsch, A. Pistoia, and T. Weth, N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-Poisson and the Lane-Emden-Fowler equations, Comm. Math. Phys., 297 (2010), pp. 653–687. · Zbl 1195.35250 [6] T. Bartsch, A. Pistoia, and T. Weth, Erratum to: N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-Poisson and the Lane-Emden-Fowler equations, Comm. Math. Phys., 333 (2015), p. 1107. · Zbl 1305.35111 [7] D. Cao, Z. Liu, and J. Wei, Regularization of point vortices pairs for the Euler equation in dimension two, Arch. Ration. Mech. Anal., 212 (2014), pp. 179–217. · Zbl 1293.35223 [8] D. Cao, Z. Liu, and J. Wei, Regularization of Point Vortices Pairs for the Euler Equation in Dimension Two, Part II, preprint, arXiv:1208.5540, 2012. [9] M. del Pino, M. Kowalczyk, and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), pp. 47–81. · Zbl 1088.35067 [10] P. Esposito, M. Musso, and A. Pistoia, Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Differential Equations, 227 (2006), pp. 29–68. · Zbl 1254.35083 [11] P. Esposito, M. Musso, and A. Pistoia, On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity, J. London Math. Soc., 94 (2007), pp. 497–519. · Zbl 1387.35219 [12] P. Esposito, M. Grossi, and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), pp. 227–257. · Zbl 1129.35376 [13] M. Flucher and B. Gustafsson, Vortex Motion in Two-Dimensional Hydromechanics, preprint, TRITA-MAT-1997-MA-02, Royal Institute of Technology, Stockholm, Sweden, 1997. [14] H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, J. Reine Angew. Math., 55 (1858), pp. 25–55. [15] G. Kirchhoff, Vorlesungen über mathematische Physik, Teubner, Leipzig 1876 · JFM 08.0542.01 [16] C. Kuhl, Equilibria for the N-vortex problem in a general bounded domain, preprint, arXiv:1502.06225, 2014. [17] C. Kuhl, Symmetric equilibria for the N-vortex problem, J. Fixed Point Theory Appl., to appear. · Zbl 1360.37179 [18] C. C. Lin, On the motion of vortices in 2D I. Existence of the Kirchhoff-Routh function, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), pp. 570–575. [19] C. C. Lin, On the motion of vortices in 2D II. Some further properties on the Kirchhoff-Routh function, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), pp. 575–577. [20] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, UK, 2001. [21] C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Appl. Math. Sci. 96, Springer-Verlag, New York, 1994. · Zbl 0789.76002 [22] P. K. Newton, The $$N$$-Vortex Problem, Springer-Verlag, Berlin, 2001. [23] E. J. Routh, Some applications of conjugate functions, Proc. London Mat. Soc., 12 (1881), pp. 73–89. · JFM 13.0720.01 [24] P. G. Saffman, Vortex Dynamics, Cambridge University Press, Cambridge, UK, 1992. · Zbl 0777.76004
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