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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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