Existence of steady vortex rings in an ideal fluid.

*(English)*Zbl 0694.76012Consider 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.

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.

##### Keywords:

Stokes stream function; free boundary problem; existence of a solution; bounded, symmetric vortex core; semilinear Dirichlet boundary value problem; approximate problem; nontrivial, cylindrically symmetric solutions; “mountain pass” critical point; variational characterization of the “mountain pass” solution; a priori estimates
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\textit{A. Ambrosetti} and \textit{M. Struwe}, Arch. Ration. Mech. Anal. 108, No. 2, 97--109 (1989; Zbl 0694.76012)

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##### References:

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