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Existence of steady vortex rings in an ideal fluid. (English) Zbl 0694.76012
Consider an ideal fluid occupying all of \(R^ 3\) with axisymmetric velocity field q. A vortex ring \({\mathcal R}\) is a toroidal region in \(R^ 3\) such that curl q\(=0\) in \(R^ 3\setminus {\mathcal R}\) while curl \(q\neq 0\) in \({\mathcal R}.\)
In cylindrical coordinates, in terms of the Stokes stream function \(\Psi\) the problem can be reduced to a free boundary problem on the half plane \(\Pi =\{(r,z): r>0\}\) of the form (1) \(-L\Psi =0\) on \(\Pi\) \(\setminus A\), (2) \(-L\Psi =\lambda r^ 2f(\Psi)\) on A, (3) \(\Psi (0,z)=-k\leq 0\), (4) \(\Psi_{| \partial A}=0\), (5) \(\Psi_ r/r\to -W\), \(\Psi_ z/r\to 0\) as \(r^ 2+z^ 2\to \infty.\)
Above, L stands for a second order elliptic differential operator. A is the (a priori unknown) cross section of the vortex ring. f is called the “vorticity function” with coupling strength parameter \(\lambda >0\). k is the flux constant measuring the flow rate between the z-axis and \(\delta\) A. The constant \(W>0\) is the “propagation speed”, namely the limit of the velocity field q at infinity. Subscripts denote partial derivatives.
We do not know any existence results for vortex rings for given strength parameter \(\lambda\) and bounded, positive vorticity function f. The purpose of this paper is to study such a case. More precisely, in our main theorem we establish the existence of a solution \(\Psi\) of (1)-(5), corresponding to a bounded, symmetric vortex core A, under the assumptions that k, \(\lambda\), W are prescribed and the vorticity function f is bounded and positive, and so gives rise to a discontinuous nonlinearity.
Our approach would apply to superlinear f as well; also for this case in the present generality the existence of vortex rings would be new. However, to limit the paper to a reasonable length, we discuss in detail only the case of bounded vorticity, which seems to be the most interesting one.
Problem (1)-(5) is first approximated by a semilinear Dirichlet boundary value problem on a ball \(B_ R\) centered in 0, passing then to the limit as \(R\to \infty\). The approximate problem is accessible by variational methods and possesses, for R large, two nontrivial, cylindrically symmetric solutions: \(v_ R\), the absolute minimum of the associated energy; and \(u_ R\), corresponding to a “mountain pass” critical point.
It is worth noting that, strikingly, in the limit of energetically unstable solutions \(u_ R\) survive, while the stable ones, \(v_ R\), diverge. To perform the limit procedure we use the variational characterization of the “mountain pass” solution \(u_ R\) and derive, a uniform bound for \(| \nabla u_ R|\) in \(L^ 2\) for a sequence \(R_ m\to \infty\). When f is superlinear, this bound could be obtained by a more direct argument from the equation itself but the latter approach does not seem to work in the case of a bounded f. In contrast, the approach we use here could be employed to solve more general semilinear elliptic variational problems in \(R^ n\) under suitable symmetries.
The rest of the paper is divided into 4 sections. In § 1 the problem is described in more detail; in § 2 the existence of solutions of the approximating problems is derived; § 3 contains the a priori estimates which enable us to pass to the limit; in § 4 we state the main results.

MSC:
76B47 Vortex flows for incompressible inviscid fluids
35Q30 Navier-Stokes equations
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[1] Ambrosetti, A., & G. Mancini,On some free boundary problems. In ?Recent contributions to nonlinear partial Differential equations?, Ed.H. Berestycki & H. Brezis, Pitman 1981. · Zbl 0477.35084
[2] Ambrosetti, A., &P. H. Rabinowitz,Dual variational methods in critical point theory and applications, J. Funct. Anal.14 (1973), 349-381. · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[3] Ambrosetti, A., &R. E. L. Turner,Some discontinuous variational problems, Diff. and Integral Equat.1-3 (1988), 341-349. · Zbl 0728.35037
[4] Amick, C. J., &L. E. Fraenkel,The uniqueness of Hill’s spherical vortex, Archive Rational Mech. & Anal.92 (1986), 91-119. · Zbl 0609.76018
[5] Amick, C. J., & L. E. Fraenkel,The uniqueness of a family of steady vortex rings, Archive Rational Mech. & Anal. (1988), 207-241. · Zbl 0694.76011
[6] Amick, C. J., & R. E. L. Turner,A global branch of steady vortex rings, J. Reine Angew. Math. (to appear). · Zbl 0628.76032
[7] Bona, J. L., D. K. Bose &R. E. L. Turner,Finite amplitude steady waves in stratified fluids, J. de Math. Pures Appl.62 (1983), 389-439. · Zbl 0491.35049
[8] Cerami, G.,Soluzioni positive di problemi con parte nonlineare discontinua e applicazioni a un problema di frontiera libera, Boll. U. M. I.2 (1983), 321-338. · Zbl 0515.35025
[9] Chang, C. K.,Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl.80 (1981), 102-129. · Zbl 0487.49027 · doi:10.1016/0022-247X(81)90095-0
[10] Fraenkel, L. E., &M. S. Berger,A global theory of steady vortex rings in an ideal fluid, Acta Math.132 (1974), 13-51. · Zbl 0282.76014 · doi:10.1007/BF02392107
[11] Gidas, B., W. M. Ni &L. Nirenberg,Symmetry and related properties via the maximum principle, Comm. Math. Phys.68 (1979), 209-243. · Zbl 0425.35020 · doi:10.1007/BF01221125
[12] Hill, M. J. M.,On a spherical vortex, Phil. Trans. Roy. Soc. London185 (1894), 213-245. · JFM 25.1471.01 · doi:10.1098/rsta.1894.0006
[13] Ni, W. M.,On the existence of global vortex rings, J. d’Analyse Math.37 (1980), 208-247. · Zbl 0457.76020 · doi:10.1007/BF02797686
[14] Norbury, J.,A family of steady vortex rings, J. Fluid Mech.57 (1973), 417-431. · Zbl 0254.76018 · doi:10.1017/S0022112073001266
[15] Palais, R. S.,Lusternik-Schnirelman theory on Banach manifolds, Topology5 (1966), 115-132. · Zbl 0143.35203 · doi:10.1016/0040-9383(66)90013-9
[16] Struwe, M.,The existence of surfaces of constant mean curvature with free boundaries, Acta Math.160 (1988), 19-64. · Zbl 0646.53005 · doi:10.1007/BF02392272
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