Desingularization of vortex rings in 3 dimensional Euler flows
Introduction
The motion of an incompressible steady Euler fluid in is governed by the following Euler equations where is the velocity field and P is the scalar pressure.
In this paper, we are concerned with desingularization of steady vortex rings of axisymmetric incompressible Euler system without swirl in several types of domains. Since the flow should be axisymmetric without swirl, the conservation of mass equation (1.2) is equivalent to the existence of the Stokes stream function ψ in cylindrical coordinates , satisfying and the associated vorticity is given by where is the usual cylindrical coordinate frame. Note that the conservation of momentum equation (1.1) can be rewritten as If for some vorticity function and , then Therefore the problem is thus reduced to the following semilinear elliptic problem Once we find the Stokes stream function ψ, the velocity of the flow is given by (1.3) and the pressure is given by .
The motion of vortex rings has been investigated since the work of Helmholtz [38] in 1858 and Kelvin [48] in 1867. In [39], Hill constructed an explicit particular translating flow of the Euler equation (called Hill's spherical vortex) whose support is a ball. Kelvin and Hicks showed that if the vortex ring with circulation κ has radius and its cross-section ε is small, then the vortex ring moves at the velocity (see [41], [48]) Fraenkel first proved that one can construct flows such that its vorticity is supported in an arbitrarily small toroidal region (see [32], [33], [34]). More precisely, he proved that for small , there exists a steady vortex ring whose vortex cross-section is of the order of ε and whose velocity satisfies asymptotically (1.5). Global existence of vortex rings with prescribed nonlinearity f was first established in [35]. For a detailed and historical description of this problem, we refer to [30], [35]. [43] is a good historical overview on the development of vortex dynamics.
Roughly speaking, there are two methods to investigate the problem of steady vortex rings. The first one is called the stream-function method, namely, finding a solution of (1.4) with the desired properties, see [3], [15], [30], [33], [34], [35], [44], [51] and references therein. By using the stream-function method, Fraenkel and Berger [35] constructed solutions of (1.4) in the whole space with prescribed constant velocity at far fields. Nonlinear desingularization for general free-boundary problems was studied in [15], but asymptotic behaviour of the solutions they constructed could not be studied precisely because of the presence of a Lagrange multiplier in the nonlinearity f. In [46], Tadie studied the asymptotic behaviour by letting the flux diverge. More steady vortex rings can also be obtained by using the mountain pass theorem proposed by Ambrosetti and Rabinowitz [2] (see [1], [3], [44] for example). Yang studied the asymptotic behaviour of a family of solutions of (1.4) with for some smooth function h [51]. However, their limiting objects are degenerate vortex rings with vanishing circulation. Recently, de Valeriola and Van Schaftingen obtained some desingularization results of steady vortex rings by using the stream-function method [30]. They proved that, for given , and , there exists a family of steady vortex rings of small cross-section with the circulation and the velocity satisfies at infinity as . Moreover, the steady vortex rings will concentrate at a circular vortex filament. Several types of domains were considered in [30].
Another method to study nonlinear desingularization of vortex rings is called the vorticity method, which solves variational problem for the potential vorticity (see [8], [13], [16], [17], [31], [37]). In contrast with the stream-function method, the vorticity method has strong physical motivation. In [13], Benjamin proposed a variational principle for the vorticity. The idea was to seek extremals of the energy relative to the set of rearrangements of a fixed function. In [37], Friedman and Turkington proved desingularization results of vortex rings in the whole space when the vorticity function f is a step function. They located the vortex rings by constraining the impulse of the flow to be a constant. Because of this, the velocities of the flows at far fields became Lagrange multipliers and hence were undetermined. Following Benjamin's idea, Burton et al. investigated the existence of vortex rings in various cases (see [8], [16], [17], [31]). We should mention that the approach they adopted is different in some important aspects from the one Benjamin envisaged. Very recently, Dekeyser used the vorticity method to study desingularization of a steady vortex pair in the lake equations of which the three-dimensional axisymmetric Euler equations are a particular case (see [28], [29]). Specifically, he constructed a family of steady solutions of the lake model which were proved to converge to a singular vortex pair. The precise localization of the asymptotic singular vortex pair depends on the depth function and the Coriolis parameter. Note that the lake domains therein were not necessarily regular.
In the present paper, we are interested in the case when the vorticity function is a step function, which has a simple discontinuity at . This simplest vorticity distribution has been a favourite for over a century (see [34]). It is also the vorticity of the well-known Prandtl-Batchelor theorem about the inviscid limit of flows with closed streamlines (refer to [11]). However, the discontinuity of f poses some challenging problems in analysis in the study of (1.4). For the case of the whole space, as mentioned before, Friedman and Turkington [37] obtained some results on the desingularization. However, the method we adopt here is quite different from theirs. We mention that their method seems to rely in essential way on the connectness of the vortex core, which was assumed to be true without proof in [37] (see also [36]). Our method gets rid of this limitation. In [29], Dekeyser studied the asymptotic behaviour of shrinking vortex pairs in the lake equations in bounded domains. In the case of vortex ring, our last result (see Theorem 2.11 in section 2) actually improves his result to some extent. To the best of our knowledge, there are no other results in this aspect. In this paper, we mainly use the vorticity method to study desingularization of steady vortex rings in several kinds of domains, namely, smooth bounded domains, infinite pipe, the whole space and exterior domain in . We adopt Burton's method to show the existence of vortex rings in brief. It is instructive to compare the previous solutions mentioned above with ours. For this aspect, one can refer to [8] for detailed description. Our focus is the asymptotic behaviour of those solutions. We note that in this case the method used in [30] seems cannot be applied, since one need assumption of continuity of the vorticity function f to ensure that the corresponding variational functional is Gateaux differentiable and the critical point theory can be used. Our strategy is to analyze the Green's function carefully and estimate the order of energy as optimally as possible. The key point is that in order to maximize the energy, those solutions must be concentrated. Our method is inspired by the works of [30], [33], [37], [49], [50]. It is worth noting that our method does not require the connectness of the vortex core. Note that in [15], [30], [37], [46], [51], the crucial estimates for the diameter of the cross-section via the vortex strength parameter depend on the connectness of the vortex core. We also remark that our method can also be applied to more general domains.
Although we only consider steady-state vortex rings here, the time evolution of vortex rings is also a matter of concern. One of the most important open questions is the vortex filament conjecture, which claims that one can find solutions of the Euler equations for which the vorticity remains close for a significant period of time to a given curve evolving by binormal curvature flow. We refer the reader to [12], [19], [27], [40] and references therein for some works on this problem. For the vortex desingularization problem of the (generalized) surface quasi-geostrophic equation, one can refer to [7].
We note that there is a similar situation with similar results in the study of vortex pairs for the two-dimensional Euler equation (see, for example, [4], [9], [10], [22], [23], [24], [26], [42], [45], [49]). Finally, what is worth mentioning is that the uniqueness of the vortex rings remains open. Amick and Fraenkel proved the uniqueness of Hill's spherical vortex in [5]. Without the uniqueness, one cannot verify whether the solutions constructed by the vorticity method or the stream-function method are the same. Several results can be found, see [5], [6], [24], [39].
The paper is organized as follows. In section 2, we state the main results and give some remarks. In section 3, we study vortex rings in an infinite pipe and the whole space since these two situations are similar. In section 4, we investigate vortex rings outside a ball which is a little different from other cases. In section 5, we consider the case of smooth bounded domains in brief.
Section snippets
Main results
Throughout the sequel we shall use the following notations: denotes the cylindrical coordinates of ; represents the associated standard orthonormal frame; denotes a meridional half-plane (θ=constant); Lebesgue measure on is denoted , and is to be understood as the measure defining any space and space, except when stated otherwise; ν denotes the measure on Π having density r with respect to , denotes ν measure; denotes the open
Vortex rings in an infinite cylinder and the whole space
In this section we consider the first two types of domains, that is, infinite cylinder and the whole space. To begin with, we need some estimates for Green's function of in D. Lemma 3.1 Let D be admissible, then we have where and is non-negative. Moreover, define then for all
Proof The calculation of Green's
Vortex rings outside a ball
In this section, we investigate vortex rings outside a ball. The approach here is a little different from the previous sections.
Let for some . Let . One can easily check that .
Let , be fixed and be a parameter. Consider the energy as follows We introduce the function as follows Let such that . It is easy to check that
Vortex rings in bounded domains
Now we turn to study vortex rings in bounded domains. Since the method is the same as before, we will only briefly describe some key steps here and omit other details.
Let or .
Due to the presence of the wall, it is natural to require that the fluid does not cross the boundary. Hence we consider the kinetic energy of the flow as follows We adopt the class of admissible functions as follows
Acknowledgements
The authors would like to thank the anonymous referee for the helpful comments and for bringing the three papers [7] [26] and [27] to their attention. This work was supported by NNSF of China (No. 11831009) and Chinese Academy of Sciences (No. QYZDJ-SSW-SYS021).
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