CONSTRUCTION OF AXISYMMETRIC STEADY STATES OF AN INVISCID INCOMPRESSIBLE FLUID BY SPATIALLY DISCRETIZED EQUATIONS FOR PSEUDO-ADVECTED VORTICITY

An infinite number of generalized solutions to the stationary Euler equations with axisymmetry and prescribed circulation are constructed by applying the finite difference method for spatial variables to an equation of pseudo-advected vorticity. They are proved to be different from exact solutions which are written with trigonometric functions and a Coulomb wave function.


Introduction
In a domain Ω (⊂ R 3 ), the velocity u (: Ω → R 3 ) of a steady-state inviscid incompressible fluid is described by the stationary Euler equations with a boundary condition: or equivalently, Here, p(: Ω → R) is the pressure, ∂Ω is the boundary, and n is the unit outward normal vector on ∂Ω.
In the axisymmetric case, the existence of solutions to (1.2) was discussed as the problem of vortex rings in, for example, [2,4,5,8].Their methods were based on a variational principle for kinetic energy.
By contrast, Vallis et al. [13] proposed a completely different approach to the solvability of (1.2).Assume that the pair (v, q) : Ω × {t > 0} → R 3 × R satisfies the nonstationary system globally in time t, where ω = ∇ × v, and α is a nonzero constant.They asserted the decay v t → 0 and the relaxation of (v, q) to the above (u, p) as t → ∞.For example, in the axisymmetric case (where v(•,•,t) is a function of the radial and the axial coordinates r, z and does not have the azimuthal component), the azimuthal component ω of ω(r,z,t) satisfies It means that the integral D f (ω/r)r drdz with any smooth function f is conserved in t, where D is the cross-section of Ω in the meridian plane.In addition, from (1.3), we can derive the decay D |v t | 2 r dr dz → 0 as t → ∞, whether α < 0 or α > 0, if D (ω/r) 2 r dr dz < ∞ at t = 0.This is worthy of remark, because we can obtain an axisymmetric solution to (1.2) which has iso-(ω/r)-lines "topologically accessible" from initially given lines, as was mentioned by Moffatt [7, Section 5].Some readers may criticize (1.3) saying that it is artificial and unphysical.They should note that in variational approaches to (1.2), all (physical or unphysical) divergence-free fields that deform streamlines or vortex lines are considered in order to obtain energy extrema (see [3, Chapter II, Section 2]).The method of Vallis et al. means that an energy extremum is automatically reached as t → ∞ if vortex lines are deformed by the divergence-free field v + αv t .
From a rigorous point of view, the theory of Vallis et al. has not been proved true in its entirety.Indeed, the nonlinearity of αω × v t seems too strong to obtain the temporally global solvability of (1.3) rigorously.
In order to make use of (1.3) and construct axisymmetric solutions to (1.2) in a rigorous manner, the author in [9] applied the Galerkin method.He approximated (1.3) with n basis functions in Ω and let n and t go to infinity simultaneously to evade the difficulty of the term αω × v t .(The equality (10) in [9] should be corrected as the inequality r −1 ∇ × u ≤ r −1 ∇ × v 0 .)Nevertheless, a question was left open in [9].As a set of basis functions, that is, an orthonormal system in a square-integrable space with the weight r, the author in [9] used {w (k) } k∈N such that each of its elements satisfies (1.2).He could not exclude the possibility of the trivial case in which every constructed solution to (1.2) is written in the form cw (k) with a constant c and some k.
In this paper, we note another system where P σ f = f + ∇Q with Q satisfying ∆Q = −∇ • f and (f + ∇Q) • n| ∂Ω = 0, and again α is a positive or negative constant.This system was introduced by the author in [11] in the two-dimensional context.It is based on the idea of Vallis et al.Indeed, in the axisymmetric case, it leads to our equation for pseudo-advected vorticity: Takahiro Nishiyama 3321 which has the same property as (1.4).Moreover, (1.5) yields the decay of dr dz) as t → ∞ if we assume its temporally global solvability (although it seems difficult to obtain as well as the solvability of (1.3)).Again, some readers may criticize (1.5) for its artificiality.As was mentioned above, it should be taken not as a physical model but as a substitute for variational methods for constructing stationary Euler flows.The aim of this paper is to approximate (1.6) by the finite difference method for r, z in a cylindrical domain Ω and prove that it generates an infinite number of generalized solutions to (1.2) which are axisymmetric, periodic in z, equipped with prescribed circulation, and different from the above cw (k) .The difficulty of proving the temporally global solvability of (1.6) is evaded by letting the lattice scale h → 0 and t → ∞ simultaneously.This is done and a generalized solution to (1.2) is constructed in Section 5 after some preparations in Sections 2, 3 and introducing the approximation of (1.6) in Section 4. For a fundamental theory of the finite difference method, we refer to [6,Chapter VI].By repeating the process used in Section 5, an infinite number of generalized solutions to (1.2) are generated in Section 6.
In our case, each element of {w (k) } k∈N = {w (m,n) } m∈N,n∈Z is concretely written with a trigonometric function and the regular Coulomb wave function of order zero, as is shown in Section 6.It satisfies (1.2).As far as the author knows, no paper introduced this set of exact solutions to (1.2).
An advantage of the finite difference method over the Galerkin method is that we have (5.7), which we mean by the above "prescribed circulation."By virtue of (5.7), we can show that our generalized solutions do not have the form cw (m,n) (see Theorem 6.1).
In [11], the author discussed the stationary Euler equations in a square domain in R 2 by using the finite difference method and proved a theorem analogous to Theorem 5.2.Although our axisymmetric case is more complex, it brings a better result, that is, the construction of an infinite number of generalized solutions in Theorem 6.1.A characteristic of our case is that a small r matches a small lattice scale h and we can prove (3.6), while such an estimate could not be obtained in the planar case in [11].

Preliminaries
Let us introduce our notation.We assume that Ω is a cylindrical domain with a constant radius a, that is, Ω = {(r, θ,z) | 0 ≤ r < a} and the flow is periodic in z.For simplicity, the period is set equal to a.The unit vectors in the r-, θ-, and z-directions in the cylindrical coordinate system are denoted by e r , e θ , and e z , respectively.
For h = a/N with a sufficiently large positive integer N, we define (2.1) The complements of Λ r h and Λr h in Z are denoted by (Λ r h ) c and ( Λr h ) c , respectively.
For { f j,k ∈ R} ( j,k)∈Z 2 , we define the difference quotients These operators D + h,r , D − h,r , D + h,z , and D − h,z are mutually commutative.For a set of vectors {f j,k = f r j,k e r + f z j,k e z } ( j,k)∈Z 2 , we define with s = r or z.The difference version of the gradient and the divergence operators are defined by (2.4) It is easy to verify ( Furthermore, we have We will often use in particular, = 0, and for s = z if f j,k = f j,k+N and g j,k = g j,k+N .It is also convenient to note the following inequalities.The last one (2.11) is known as the discrete Poincaré inequality.
Let {•, •} S and |{•}| S for sets of scalars { f j,k }, {g j,k } or of vectors {f j,k }, {g j,k } be defined by or by (2.14) In order to discuss the limit h → 0, it is convenient to use the interpolation operators A h and B h defined by (2.15) Here, [r/h] means the integer in (r/h − 1,r/h].These operators correspond to ũh and u h in [6,Chapter VI].They are also used for sets of vectors as (2.16) It is useful to note that B h { f j,k } is a continuous and piecewise bilinear function and that the inequalities (2.18) Moreover, we have (2.19) The scalar product •, • for scalar functions f (r,z), g(r,z) or vector functions f(r,z), g(r,z) is defined by The norm • is defined by Let X 0 and X 1 be spaces given by with the scalar product •, • and We consider the construction of solutions to (1.2) in X 1 , particularly in a subspace of itself: Here, X 0 is a subspace of X 0 : We also use the Sobolev space of the first order on (0,a) 2 , denoted by W 1 2 ((0,a) 2 ), and the space of vector functions whose components belong to W 1 2 ((0,a) 2 ), denoted by W 1  2 ((0,a) 2 ).We sometimes use

Difference operator Ξ h
In the axisymmetric case, the relation between the stream function φ and the vorticity ω is represented by where (see [2,4,5,8]).
Let the difference operator Ξ h be defined by It is a difference approximation of Ξ.
The following lemmas will be used in Section 5. From now on, we frequently denote positive constants independent of h by C or C without distinction.Lemma 3.1.Let { f j,k } be a set such that f j,k = 0 for all j ∈ (Λ r h ) c and f j,k = f j,k+N .Then, Proof.By (2.6) and (2.8), we have where (3.9) Therefore, using (2.11), we obtain 3 4 which leads to (3.4).

Spatially discretized equations for pseudo-advected vorticity
First, we introduce the linear system for {φ j,k } ( j,k)∈Z 2 with {ζ j,k } ( j,k)∈Λh given that ) It is uniquely solvable.Indeed, if ζ j,k = 0 is assumed for all (j,k), then φ j,k = 0 is derived from (3.4) with (2.11).Therefore, we can represent φ j,k in Λ h as ) is given for every ( j,k) ∈ Λ r h × Z so that f j,k = f j,k+N , and { f r 0,k | f r 0,k = f r 0,k+N } k∈Z is also given.We define the operator P σ,h by where Q j,k is determined by It is easily verified that grad + h Q j,k is uniquely determined for given {f j,k } and { f r 0,k }.Indeed, if f j,k | ( j,k)∈Λ r h ×Z = 0 with f r 0,k | k∈Z = 0 is assumed, then, by using (2.7), we have which means that As an approximation of (1.6) by the finite difference method for r and z, we present the system of ordinary differential equations for ζ j,k (t) ((j,k) ∈ Λ h ): Here α is a fixed positive or negative number, and with φ j,k given by (4.3) for (j,k) ∈ Λ h and by (4.2) for the others.
It should be noted that (4.1) is not always valid for j = 0 or N. The sets {ζ j,k } and {φ j,k } correspond to ω/r in (1.6) and φ in (3.1), respectively.As a result of the first condition of (4.2), the mean velocity in (0,a) 2 of our flows in Theorem 5.2 is equal to zero, while the case with φ j,k | j≤0 = 0 and φ j,k | j≥N = const.= 0 is open.Clearly, the system (4.8) is uniquely solvable at least locally in time if we give the initial data (4.10)

Construction of solutions to the stationary Euler equations
Let us define a generalized solution to (1.2).
for any f ∈ X 1 , then u is said to be an axisymmetric generalized solution to (1.2).
If this generalized solution belongs to the C 1 -class, then it is a classical solution to (1.2), according to the well-known orthogonality of the divergence-free and the gradient fields. Let Then we have the following theorem.
Then it is a smooth solution to (6.1) with µ = µ m,n , and {φ (m,n) } m∈N,n∈Z is a complete orthogonal system in L 2 ((0,a) 2 ) with the weight r.