Energy-Minimising Parallel Flows with Prescribed Vorticity Distribution

This note concerns a nonlinear differential equation problem in which both the nonlinearity in the equation and its solution are determined by other constraints. The question under consideration arises from a study of two-dimensional steady parallel-flows of a perfect fluid governed by Euler's equations and a free-boundary condition, when the distribution of vorticity is arbitrary but prescribed.


Introduction
This is basically a study of the nonlinear ordinary differential equation −u = λ(u) when both the nonlinearity λ and the solution u are unknowns, to be determined by additional constraints and boundary conditions. To begin, we outline the motivation in hydrodynamic wave theory, for otherwise the problem treated in Section 2 might appear arbitrary, if not a little bizarre. Section 2 is self-contained and possibly of independent interest.

Preamble: Euler's equations
Euler's equations [6,13], which govern the velocity field v of an incompressible perfect fluid with pressure p and density in an external conservative force field −∇Φ are The first is Newton's law that the rate of change of momentum is the sum of external and internal forces and the second is the incompressibility condition. In two dimensions (when v is in the (x, y)-plane and k points in the z-direction) the vorticity, ω = ∇ × v := ω k satisfies ω × v = 0 and the transport equation ω t + v · ∇ω = 0. for any reasonable function f . Thus the vorticity distribution function is conserved by Euler's equations. In different words, for smooth solutions of Euler's equations the vorticity at time t is a rearrangement of the vorticity at time zero, in the following sense.
Equivalently the distribution functions Z 1 and Z 2 are equal. The set of rearrangements of a given function ω * is denoted by R(ω * ).
Let C(t) denote the position of a closed orientated curve evolving smoothly with the motion of its fluid particles governed by Euler's equations. Then the area within C(t) is conserved, by incompressibility. Moreover, This, conservation of circulation, is known as Kelvin's Circulation Theorem.
In a simply connected domain in R 2 , the incompressibility equation, ∇ · v = 0, implies that there is a stream function Ψ with ∇ ⊥ Ψ = − v where ∇ ⊥ = (−∂ y , ∂ x ). So Euler's equations can be re-written

Travelling waves with vorticity
A special case treated in [3,5,7,8,9,14] is that of periodic waves of permanent form travelling with speed c on the surface of water in a channel above a horizontal bottom. The domain Ω (t) occupied by the fluid at time t is the region between the bottom and the surface S (t) := {(x, y) : F (x − ct, y) = 0}, say, for some function F : R 2 → R which is periodic in x. Thus Ω (t) = Ω + t(c, 0), where Ω := Ω (0). If Ψ(·, ·, t) and ω(·, ·, t) : Ω (t) → R are written In particular, −∆ψ = ζ where ζ is constant on level sets of ψ; ψ is called the relative stream function.
Moreover, if a point (p(t), q(t)) of the fluid moves on the surface S (t), then F (p(t) − ct, q(t)) ≡ 0 and, since (ṗ(t),q(t)) = −∇ ⊥ ψ(p(t) − ct, q(t)), Therefore, formally speaking, ψ is constant on the fixed curve S (0) (henceforth denoted by S ) and the time-independent functions ψ, ζ : Ω → R, and the wave speed c, satisfy where C is an unspecified constant. Constraints on the pressure in the flow at the surface S and on the flow vorticity distribution must also be satisfied.
Remark 1. The theory in [7,8], and summarised in [14], focuses on cases in which the flow velocity relative to the moving frame is nowhere zero, equivalently ψ y − c does not change sign. Below this is called the monotone case because the relative stream function, ψ(x, y) − cy, is monotone in y.

A variational approach
We begin by summarising a variational approach [5] which led to the case c = 0 in (2). For the analogous variational treatment in which non-zero c arises as a Lagrange multiplier due to a constraint on the momentum, see [3]. Both lead to the problem is Section 2 which, in Sections 3.1 and 3.2, is interpreted in terms of the travelling-wave problem. Throughout, let µ ∈ R and ζ * ∈ L 2 (0, P ) × (0, Q) be given, where P, Q > 0 are fixed.
Furthermore consider only those S for which the area of Ω is P Q.
For any function ζ ∈ L 2 loc (Ω ) which is P -periodic in x with ζ| Ω ∈ R(ζ * ), let ψ = ψ(Ω, ζ) ∈ W 1,2 (Ω) be the weak solution of in which the constant C(ψ) is not prescribed. The classical water-wave problem is to find such a curve S and function ζ so that the vorticity ζ is constant on level sets of ψ(Ω, ζ), and The first of these comes from (2) with c = 0 and second is the classical Bernoulli condition which says that for a steady flow under gravity, the pressure at the free surface is constant atmospheric pressure. In this formulation ψ yields the stream function and S the free boundary, and µ ∈ R is the circulation on one period of the free boundary, as in the preceding section with c = 0.
More generally [1,15], if an elastic membrane that nonlinearly resists stretching and bending is in contact with the surface, in its simplest form the Bernoulli condition becomes where denotes differentiation with respect to arc length; (S) = length of S; E 0 is a coefficient of bending resistance; T 0 and β 1 measure nonlinear resistance to stretching and compression [5]. Note that (4c) includes the classical Bernoulli condition, E = T = 0, as a special case, and β = 1, T > 0 = E corresponds to simple surface tension.
An energy functional, the sum of the kinetic and potential energies of the fluid and the elastic energy of the surface, is defined by where, with σ denoting the curvature of S, Then it is shown in [5] that minimizing the energy F over admissible curves S and ζ ∈ R(ζ * ) yields a solution of (4). Moreover, a necessary condition for a minimiser is that (4b) holds in the strong sense that ζ = λ(ψ) for some non-increasing function λ. In other words, an energy minimiser satisfies −∆ψ = λ(ψ) = ζ ∈ R(ζ * ) for some non-increasing function λ.
The function λ is the infinite-dimensional Lagrange multiplier that arises from the constraint that the vorticity distribution coincides with that of ζ * . It is a priori unknown as is ψ: both are components of solutions. In the absence of information about its regularity, the key to the analysis (Lemma 2) is that λ is non-increasing, which follows from the fact that it arises from minimisation in [3,5], or from the analogous minimisation problem restricted to parallel flows. For suitable µ and positive values of the parameters, E, T it is shown in [3,5] that minimisers, with prescribed vorticity distribution function over arbitrary periodic domains, exist. However, in the absence of surface energy effects, when E = T = 0, these infima of the energies in [3,5] are not attained except in trivial circumstances [16]. (Minimizers in the narrow class of periodic flows, when surface energy plays no role, do exist.)

Parallel flows
A flow is called parallel if the stream function ψ and the vorticity ζ are functions of y only. In [5] it is remarked that 'for a given vorticity distribution it is easy to construct parallel-flow solutions by solving ordinary differential equations'. Consequently, the emphasis there was on finding non-parallel-flow solutions and on estimates to ensure that the minimisers found were not parallel flows. The purpose here is to examine the 'easy' task of finding which parallel flows, if any, could arise from minimizing energy, either over all domains, or over a strip domain on which admissible functions depend on y only.
For parallel flows S = {(x, Q) : x ∈ R}, Ω = (0, P ) × (0, Q), and ψ and ζ are functions of y only. Thus the P -periodicity in x of ψ and the generalised Bernoulli condition (4c) are satisfied automatically. Hence the problem reduces to −u = λ(u), where λ is non-increasing but otherwise unknown, except that λ(u) is a rearrangement of a known function on [0, Q]. We will return to this is Section 3.1 where the existence and behaviour of parallel-flow minimisers are shown to be determined by the parameters µ/P and Ω ζ * alone. The minimization problem [3], in which horizontal momentum is prescribed on a slightly different admissible set, is considered in Section 3.2.
In [12] there is an exhaustive account of the parallel solutions of (6) when a given function λ prescribes the functional dependence of vorticity on the stream function. Because prescription of λ has no role in the initial-value problem, we prefer the present approach where all the constraints (vorticity distribution function, surface circulation per period, cross-sectional area or horizontal momentum) are invariants for smooth solutions of Euler's equations with free-boundary conditions.

Stability
Conserved quantities often have significance for questions of stability [2,3,4]. However, linearised criteria, such as Rayleigh's instability criterion for parallel flows [13, p.122], presume that the total energy is that of a flowing liquid in a conservative force field. By contrast, when E, T > 0 the energy (5) is shared between the fluid and the deformed elastic membrane once the surface is not flat. So Rayleigh's criterion is not relevant to the possible parallel-flow solutions of the hydroelastic wave free-boundary problem under discussion here. If, however, these parallel flows are, as they may be, considered as solutions in a domain which is a fixed strip of constant width, Rayleigh's condition may be relevant and checked using the properties of the explicit solutions that have been found.

Conclusions
When the prescribed vorticity distribution ζ * is essentially one-signed, for all values of the circulation µ there is a unique parallel flow which satisfies the necessary conditions for an energy minimiser in [5]. If µ and µ + Ω ζ * have opposite signs (which is equivalent to the condition in Theorem 8), then that solution is non-monotone, otherwise it is monotone (Remark 1). However, when ζ * changes sign all solutions are monotone, but there are values for µ for which no parallel flow satisfies the necessary condition for a minimiser of the energy in [5].
For the energy in [3] there are parallel flows that satisfy the necessary condition for minimisers if and only if κ ∈ [k 0 , k 0 ], where κ in (23) depends only on the prescribed momentum and circulation, and k 0 , k 0 in (24) depend only on the given vorticity distribution ζ * . These parallel flows are monotone if and only if κ ∈ {k 0 , k 0 }, and there are infinitely many c with the same ψ corresponding to monotone solutions (Theorem 13).
If ζ * changes sign parallel flows that satisfy the necessary conditions for minimizers in [3], if any, must be monotone. If ζ * does not change sign, for every κ ∈ (k 0 , k 0 ) there is a unique solution and it is non-monotone (Theorem 15).
From given data, we calculate the stream function, the wave speed and the explicit dependence of vorticity on the stream function, for parallel-flow minimizers (indeed also for minimisers among the restricted class of parallel follows). For energy maximizers in the class of parallel flows, (6) holds with λ increasing. However, the analogue of Lemma 2 is not so straightforward because the regularity of the function λ is unknown. Note also, from Figure 1, that the stream function of non-monotone parallel-flow solutions may not be C 3 , even when the prescribed vorticity distribution is real-analytic.

An ODE with Rearrangement Constraints
All the observations in this section are elementary. Let ζ ∈ L 2 (0, Q) and suppose that and If α > 0, then either u > 0, u is concave and ζ > 0 is non-increasing on [0, α), or u < 0, u is convex and ζ < 0 is non-decreasing on [0, α).
Then u 1 , ζ 1 satisfies the same equation as u, ζ and the required conclusion about u on (β, Q) is equivalent to the preceding observation about u 1 on (0, α 1 ).
On intervals where u is monotone, ζ = λ(u) has the opposite monotonicity (not necessarily strict) to that of u, because λ is non-increasing.
We refer to (a)(ii) with 0 < α β < Q as the non-monotone case.

Prescribed Circulation µ
We begin with the energy in [5], discussed in Section 1.2, which leads to a solution of (2) with c = 0, and (6). If a parallel flow minimizer exists, then ψ(x, y) = u(y), x ∈ R, y ∈ [0, Q], where u is solution of (7) in which ζ is a rearrangement of ζ 0 , λ is non-increasing and P u (Q) = µ. Hence From Lemma 2 and Remark 4, there are only two possibilities: u is non-monotone, ζ 0 is essentially one-signed, u is monotone and ζ ∈ ζ 0 , ζ 0 .
It is obvious from (16) that if µ/P and µ/P + [ζ 0 ] are non-zero with opposite signs, no solution is monotone. If ζ 0 does not change sign and µ/P and µ/P + [ζ 0 ] have the same sign, then all solutions are monotone. When ζ 0 changes sign it is more complicated (Remark 12).
Theorem 8. A necessary and sufficient condition for the existence of a non-monotone solution is that ζ 0 is essentially one-signed with µ/P and [ζ 0 ] + µ/P non-zero of opposite signs. For such ζ 0 , P and µ, the solution of (7) with u (Q) = µ/P , and the function λ, are uniquely determined.
Remark 9. Note that for such values of P, µ and [ζ 0 ] another solution is uniquely determined, but with non-decreasing λ.
To relate the monotone and non-monotone cases, it is useful to recast the hypotheses of Theorem 8 using the parameter Turning now to monotone solutions, note that ζ 0 may change sign but that r(ζ 0 ) 0 defined below is zero when it does not (see Figures 2 and 3).
For such values of ρ, there is only one monotone solution.
Proof. Suppose a monotone solution u exists. Then ζ ∈ {ζ 0 , ζ 0 } and, since λ is nonincreasing, by (16), either u (y) = µ P + It follows from the identities after (13) that this is equivalent to either Hence there is a monotone solution in the form and in the form From the definition of r(ζ 0 ) it follows that (19) is a necessary condition for the existence of a monotone solution.

Prescribed Circulation µ and Momentum ν
Paper [3] considers the problem of minimizing an energy similar to (5) subject to an additional requirement that the horizontal momentum has the prescribed value ν: It is shown there that a minimizer satisfies the third condition in (2) in the strong sense that ζ = λ(ψ−cy) where λ is non-increasing and c is the Lagrange multiplier corresponding to the momentum constraint. Thus when a minimizer is a parallel flow, ψ(x, y) = u(y) and, for some constant c, −u (y) = ζ(y), ζ ∈ R(ζ 0 ), (22a) ζ(y) = λ(u(y) − cy) for some non-increasing function λ.
By a solution of (22) is meant a pair (u, c), and a solution is called monotone if u(y)−cy is monotone on [0, Q]. In Remark 16 we will see how there can exist non-monotone solutions for which ζ is monotone. Let Suppose (22a), (22b) have a solution u. Then Hence a necessary condition for the existence of a solution of (22a), (22b) is that If (22c) is also satisfied and g is defined in (15), then (b) If ζ 0 is constant, k 0 = k 0 = ζ 0 Q 2 /2, and the conclusion follows by direct calculation.
(d) The proof when ζ = ζ 0 is similar. This completes the proof.
However, when κ ∈ (k 0 , k 0 ) and the sign of [ζ 0 ] is given, the relevant equation from (28) has unique solutions in (0, Q) (in fact in (a 1 /2, Q − b 1 /2) where a 1 and b 1 are defined in Remark 7). The uniqueness is immediate from this construction.
An elementary adaptation of the proof of (a), in the light of Remark 7 yields (c) and (d), and completes the proof.
Remark 16. The solutions given by parts (c) and (d) involve {ζ (p) ,ζ (p) } which, by Remark 7, coincides with {ζ 0 , ζ 0 } in these cases. Note that althoughĝ orǧ are independent of p in these intervals,f andf , and consequently c, are not. Thus there are intervals of wave speeds for which the vorticity profiles are the same, as in part (a). Note also that parts (c) and (d) give non-monotone solutions for which the vorticity is monotone.