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Regularization of point vortices pairs for the Euler equation in dimension two. (English) Zbl 1293.35223
Summary: In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem \[ \begin{cases} -\varepsilon^2\Delta u=\sum\limits_{i=1}^m\chi_{\Omega_i^+}\biggl(u-q-\frac{\kappa_i^+}{2\pi}\ln\frac{1}{\varepsilon}\biggl)_+^p\\ -\sum_{j=1}^n\chi_{\Omega_j^-}\biggl(q-\frac{\kappa_j^-}{2\pi}\ln\frac{1}{\varepsilon}-u\biggl)_+^p,\qquad & x\in\Omega,\\ u=0,\quad & x\in\partial\Omega,\end{cases} \] where \(p>1\), \(\Omega\subset\mathbb R^2\) is a bounded domain, \(\Omega_i^+\) and \(\Omega_j^-\) are mutually disjoint subdomains of \(\Omega\) and \(\chi_{\Omega_i^+}\) (resp. \(\chi_{\Omega_j^-}\)) are characteristic functions of \(\Omega_i^+\) (resp. \(\Omega_j^-\)), \(q\) is a harmonic function. We show that if \(\Omega\) is a simply-connected smooth domain, then for any given \(C^1\)-stable critical point of Kirchhoff-Routh function \(\mathcal W(x_1^+,\dots, x_m^+,x_1^-,\dots,x_n^-)\) with \(\kappa^+_i> 0\) \((i=1,\dots,m)\) and \(\kappa^-_j>0\) \((j=1,\dots,n)\), there is a stationary classical solution approximating stationary \(m+n\) points vortex solution of incompressible Euler equations with total vorticity \(\sum_{i=1}^m\kappa^+_i-\sum_{j=1}^n\kappa_j^-\). The case that \(n=0\) can be dealt with in the same way as well by taking each \(\Omega_j^-\) as an empty set and set \(\chi_{\Omega_j^-}\equiv 0\), \(\kappa^-_j=0\).

MSC:
35Q31 Euler equations
76B47 Vortex flows for incompressible inviscid fluids
35B65 Smoothness and regularity of solutions to PDEs
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References:
[1] Ambrosetti, A.; Struwe, M., Existence of steady vortex rings in an ideal fluid, Arch. Rational Mech. Anal., 108, 97-109, (1989) · Zbl 0694.76012
[2] Ambrosetti, A., Yang, J.: Asymptotic behaviour in planar vortex theory. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 1, 285-291 (1990) · Zbl 0709.76027
[3] Arnold, V.I., Khesin, B.A.: Topological methods in hydrodynamics. In: Applied Mathematical Sciences, vol. 125. Springer, New York, 1998 · Zbl 0902.76001
[4] Badiani, T.V., Existence of steady symmetric vortex pairs on a planar domain with an obstacle, Math. Proc. Cambridge Philos. Soc., 123, 365-384, (1998) · Zbl 0903.76016
[5] Bartsch, T.; Pistoia, A.; Weth, T., N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-Poisson and the Lane-Emden-Fowler equations, Commun. Math. Phys., 297, 653-686, (2010) · Zbl 1195.35250
[6] Benjamin, T.B.: The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics. In: Applications of Methods of Functional Analysis to Problems in Mechanics. Lecture Notes in Mathematics, vol. 503, pp. 8-29, 1976 · Zbl 0658.76016
[7] Berger, M.S.; Fraenkel, L.E., Nonlinear desingularization in certain free-boundary problems, Commun. Math. Phys., 77, 149-172, (1980) · Zbl 0454.35087
[8] Burton, G.R., Vortex rings in a cylinder and rearrangements, J. Differ. Equ., 70, 333-348, (1987) · Zbl 0648.35029
[9] Burton, G.R., Steady symmetric vortex pairs and rearrangements, Proc. R. Soc. Edinburgh., 108A, 269-290, (1988) · Zbl 0658.76016
[10] Burton, G.R., Variational problems on classes of rearrangements and multiple configurations for steady vortices., Ann. Inst. Henri Poincare. Analyse Nonlineare,, 6, 295-319, (1989) · Zbl 0677.49005
[11] Burton, G.R., Rearrangements of functions, saddle points and uncountable families of steady configurations for a vortex., Acta Math., 163, 291-309, (1989) · Zbl 0695.76016
[12] Caffarelli, L.; Friedman, A., Asymptotic estimates for the plasma problem, Duke Math. J., 47, 705-742, (1980) · Zbl 0466.35033
[13] Cao, D.; Küpper, T., On the existence of multi-peaked solutions to a semilinear Neumann problem, Duke Math. J., 97, 261-300, (1999) · Zbl 0942.35085
[14] Cao, D.; Peng, S.; Yan, S., Multiplicity of solutions for the plasma problem in two dimensions, Adv. Math., 225, 2741-2785, (2010) · Zbl 1200.35130
[15] Dancer, E.N.; Yan, S., The lazer-mckenna conjecture and a free boundary problem in two dimensions, J. Lond. Math. Soc., 78, 639-662, (2008) · Zbl 1202.35088
[16] Elcrat, A.R.; Miller, K.G., Steady vortex flows with circulation past asymmetric obstacles, Commun. Partial Differ. Equ.,, 12, 1095-1115, (1987) · Zbl 0628.76031
[17] Elcrat, A.R.; Miller, K.G., Rearrangements in steady vortex jows with circulation, Proc. Am. Math. Soc.,, 111, 1051-1055, (1991) · Zbl 0731.35083
[18] Elcrat, A.R.; Miller, K.G., Rearrangements in steady multiple vortex flows, Commun. Partial Differ. Equ., 20, 1481-1490, (1995) · Zbl 0841.35086
[19] Fraenkel, L.E.; Berger, M.S., Global theory of steady vortex rings in an ideal fluid, Acta Math., 132, 13-51, (1974) · Zbl 0282.76014
[20] Kulpa, W., The Poincaré-miranda theorem, Am. Math. Mon., 104, 545-550, (1997) · Zbl 0891.47040
[21] Li, G.; Yan, S.; Yang, J., An elliptic problem related to planar vortex pairs, SIAM J. Math. Anal., 36, 1444-1460, (2005) · Zbl 1076.35039
[22] Li, Y.; Peng, S., Multiple solutions for an elliptic problem related to vortex pairs, J. Differ. Equ., 250, 3448-3472, (2011) · Zbl 1214.31005
[23] Lin, C.C., On the motion of vortices in two dimension - I, Existence of the Kirchhoff-Routh function. Proc. Natl. Acad. Sci. USA, 27, 570-575, (1941) · Zbl 0063.03560
[24] Miranda, C., Un’osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3, 5-7, (1940) · JFM 66.0217.01
[25] Marchioro, C.; Pulvirenti, M., Euler evolution for singular initial data and vortex theory, Commun. Math. Phys., 91, 563-572, (1983) · Zbl 0529.76023
[26] Miller, K.G., Stationary corner vortex configurations, Z. Angew. Math. Phys., 47, 39-56, (1996) · Zbl 0841.76015
[27] Ni, W.-M., On the existence of global vortex rings, J. Anal. Math., 37, 208-247, (1980) · Zbl 0457.76020
[28] Norbury, J., Steady planar vortex pairs in an ideal fluid, Commun. Pure Appl. Math., 28, 679-700, (1975) · Zbl 0338.76015
[29] Saffman, P.G.; Sheffield, J., Flow over a wing with an attached free vortex, Stud. Appl. Math., 57, 107-117, (1977) · Zbl 0385.76033
[30] Smets, D.; Van Schaftingen, J., Desingulariation of vortices for the Euler equation, Arch. Rational Mech. Anal., 198, 869-925, (2010) · Zbl 1228.35171
[31] Turkington, B.: On steady vortex flow in two dimensions. I, II. Comm. Partial Differ. Equ. 8, 999-1030, 1031-1071 (1983) · Zbl 0523.76014
[32] De Valeriola S., Van Schaftingen, J.: Desingularization of vortex rings and shallow water vortices by semilinear elliptic problem, arXiv:12093988v1 · Zbl 1294.35083
[33] Yang, J., Existence and asymptotic behavior in planar vortex theory, Math. Models Methods Appl. Sci., 1, 461-475, (1991) · Zbl 0738.76009
[34] Yang, J., Global vortex rings and asymptotic behaviour., Nonlinear Anal., 25, 531-546, (1995) · Zbl 0845.76017
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