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The uniqueness of Hill’s spherical vortex. (English) Zbl 0609.76018
The authors study the free boundary problem

$r(\frac{1}{r}\psi_ r)_ r+\psi_{zz}= \begin{cases} -\lambda r^ 2f_ 0(\psi) &\text{ in $$A;$$} \\ 0 &\text{ in $$\Pi \setminus A,$$}\end{cases}$ $$\psi |_{r-0}=-k,\quad |_{\partial A}=0$$ together with certain asymptotics at infinity.
Here $$\Pi =\{(r,z)|$$ $$r>0$$, $$z\in {\mathbb{R}}\}$$, $$f_ 0\geq 0$$, and $$\psi$$ is a Stokes stream function in cylindrical co-ordinates (no dependence on $$\theta)$$. The set $$A\subset \Pi$$ is bounded and open, but a priori unknown. A special case of the problem is Hill’s problem, in which an explicit solution is known. It is proven that any weak solution to the problem is the explicit solution modulo a translation in z. Such solutions may be obtained as local maximizers of functional.
Reviewer: G.Warnecke

##### MSC:
 76B47 Vortex flows for incompressible inviscid fluids 35J25 Boundary value problems for second-order elliptic equations 35R35 Free boundary problems for PDEs
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##### References:
 [2] Agmon, S., Douglis, A., & Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I · Zbl 0093.10401 · doi:10.1002/cpa.3160120405 [3] Ambrosetti, A., & Mancini, G., On some free boundary problems. In Recent contributions to nonlinear partial differential equations (edited by H. Berestycki & H. Brézis). Pitman, 1981. · Zbl 0477.35084 [4] Amick, C. J., & Fraenkel, L. E., The uniqueness of Norbury’s perturbation of Hill’s spherical vortex. To appear. · Zbl 0694.76011 [5] Amick, C. J., & Fraenkel, L. E., Note on the equivalence of two variational principles for certain steady vortex rings. To appear. · Zbl 0694.76011 [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 of mechanics, Lecture notes in math. 503. Springer, 1976. [7] Berestycki, H., Some free boundary problems in plasma physics and fluid mechanics. In Applications of nonlinear analysis in the physical sciences (edited by H. Amann, N. Bazley & K. Kirchgässner). Pitman, 1981. · Zbl 0503.76127 [8] Caffarelli, L. A., & Friedman, A., Asymptotic estimates for the plasm · Zbl 0466.35033 · doi:10.1215/S0012-7094-80-04743-2 [9] Chandrasekhar, S., Hydrodynamic and hydromagnetic stability. Oxford, 1961. · Zbl 0142.44103 [10] Ekeland, I., & Temam, R., Convex analysis and variational problems. North-Holland, 1976. · Zbl 0322.90046 [11] Esteban, M. J., Nonlinear elliptic problems in strip-like domains: symmetry of positive vortex rings. Nonlinear Analysis, Theory, · Zbl 0513.35035 · doi:10.1016/0362-546X(83)90090-1 [12] Fraenkel, L. E., & Berger, M. S., A global theory of steady vortex rings in an · Zbl 0282.76014 · doi:10.1007/BF02392107 [13] Friedman, A., & Turkington, B., Vortex rings: existence and asymptotic estimates. · Zbl 0497.76031 · doi:10.1090/S0002-9947-1981-0628444-6 [14] Gidas, B., Ni, W.-M., & Nirenberg, L., Symmetry and related properties via the maximum prin · Zbl 0425.35020 · doi:10.1007/BF01221125 [15] Gilbarg, D., & Trudinger, N. S., Elliptical partial differential equations of second order. Springer, 1977. · Zbl 0361.35003 [16] Giles, J. R., Convex analysis with application in differentiation of convex functions. Pitman, 1982. · Zbl 0486.46001 [17] Hill, M. J. M., On a spherical vortex. Philos. Trans. Roy. Soc. London A 185 (1894), 213–245. · JFM 25.1471.01 · doi:10.1098/rsta.1894.0006 [18] Keady, G., & Kloeden, P. E., Maximum principles and an application to an elliptic boundary-value problem with a discontinuous nonlinearity. Research report, Dept. of Math., University of Western Australia, 1984. · Zbl 0647.35029 [19] Kinderlehrer, D., & Stampacchia, G., An introduction to variational inequalities and their applications. Academic Press, 1980. · Zbl 0457.35001 [20] Ni, W.-M., On the existence of global vortex r · Zbl 0457.76020 · doi:10.1007/BF02797686 [21] Nirenberg, L., On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115–162. · Zbl 0088.07601 [22] Norbury, J., A steady vortex ring close to Hill’s spherical vortex. Proc · Zbl 0256.76016 · doi:10.1017/S0305004100047083 [23] Norbury, J., A family of steady vort · Zbl 0254.76018 · doi:10.1017/S0022112073001266 [24] Serrin, J., A symmetry problem in potential theory. Ar · Zbl 0222.31007 · doi:10.1007/BF00250468
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