Arnold’s variational principle and its application to the stability of planar vortices

We consider variational principles related to V. I. Arnold’s stability criteria for steady-state solutions of the two-dimensional incompressible Euler equation. Our goal is to investigate under which conditions the quadratic forms deﬁned by the second variation of the associated functionals can be used in the stability analysis, both for the Euler evolution and for the the Navier-Stokes equation at low viscosity. In particular, we revisit the classical example of Oseen’s vortex, providing a new stability proof with stronger geometric ﬂavor. Our analysis involves a fairly detailed functional-analytic study of the inviscid case, which may be of independent interest, and a careful investigation of the inﬂuence of the viscous term in the particular example of the Gaussian vortex.


Introduction
We investigate the applicability of V. I. Arnold's geometric methods to certain stability problems related to Navier-Stokes vortices at high Reynolds number.Our main goal is a "proof of concept" that such applications are possible, at least in simple cases, even though much of the geometric structure behind the inviscid stability analysis does not survive the addition of the viscosity term.In particular, we give a new proof of a known result concerning the stability of Oseen's vortex as a steady state of the Navier-Stokes equation in self-similar variables.We expect that the approach we advertise here will be useful to tackle stability problems involving solutions that are less symmetric and less explicit than the classical Oseen vortex.In such cases one may not have good alternative methods for proving stability in the presence of viscosity.Our investigation leads to a detailed study of the quadratic forms naturally arising in Arnold's approach.Some of their functional-analytic properties, which are established in the course of our analysis, may be of independent interest.

1A.
A finite-dimensional model.Following the seminal paper [Arnold 1965], we first illustrate the issues we want to address in a model situation where the "phase space" is finite-dimensional.We consider the ordinary differential equation where b is a smooth vector field in ‫ޒ‬ n .Let us assume that f, g 1 , . . ., g m : ‫ޒ‬ n → ‫ޒ‬ are (sufficiently smooth) conserved quantities for the evolution (1-1), with m < n.This means f ′ (x)b(x) = 0 and g ′ j (x)b(x) = 0, x ∈ ‫ޒ‬ n , j = 1, . . ., m, (1-2) 682 THIERRY GALLAY AND VLADIMÍR ŠVERÁK where we adopt the standard notation f ′ (x) for the linear form given by the first derivative of f at x.The situation we have ultimately in mind is somewhat more specific: it corresponds to the case where the phase space ‫ޒ‬ n is equipped with a Poisson bracket { • , • }, where system (1-1) is of the form and where g 1 , . . ., g m are Casimir functions.The Poisson structure is of course important in many respects, but for our arguments here it does not play a big role.We can therefore proceed in the general context of (1-1) and (1-2).
For any c = (c 1 , . . ., c m ) ∈ ‫ޒ‬ m , let us define X c = {x ∈ ‫ޒ‬ n : g 1 (x) = c 1 , . . ., g m (x) = c m }.We assume that, for some c ∈ ‫ޒ‬ m , the function f attains a nondegenerate local maximum on X c at some point x ∈ X c and that the derivatives g ′ 1 ( x), . . ., g ′ m ( x) are linearly independent.The stationarity condition at x gives the linear relation for some Lagrange multipliers λ 1 , . . ., λ m ‫.ޒ∈‬Moreover, the second-order differential1 of the function f | X c (the restriction of f to X c ) at x is given by the restriction to the tangent space T x X c of the quadratic form where we denote by f ′′ ( x) the quadratic form given by the Hessian of f at x, and similarly for g ′′ 1 ( x), . . ., g ′′ m ( x).Our nondegeneracy assumption means that the restriction of the form Q to T x X c is strictly negative definite.Now, let B = b ′ ( x) be the n × n matrix corresponding to the linearization of (1-1) at the point x, which is a steady state by construction [Arnold 1965].If we differentiate twice the relations (1-2) and use (1-4) together with b( x) = 0, we see that the evolution defined by the linearized equation ξ = Bξ leaves the form Q invariant.In other words, d dt Q(ξ, ξ ) = Q(Bξ, ξ ) + Q(ξ, Bξ ) = 0 for all ξ ∈ ‫ޒ‬ n . (1-6) The above structure2 gives various options for the stability analysis of the equilibrium x of (1-1), depending on the index of the quadratic form Q in (1-5).Our assumptions readily imply that x is stable in the sense of Lyapunov with respect to perturbations on the invariant submanifold X c .Moreover, since a neighborhood of x in ‫ޒ‬ n is foliated by submanifolds of this form for nearby values of the parameter c = (c 1 , . . ., c m ), one can show that x is in fact Lyapunov stable with respect to small unconstrained perturbations [Arnold 1965].The perspective changes qualitatively if we add to the vector field b in (1-1) a small "dissipative" term, with the effect that the quantities f and g 1 , . . ., g m are no longer exactly conserved under the modified evolution.This is in the spirit of what we intend to do in the infinitedimensional case, when we consider the Navier-Stokes equation as a perturbation of the Euler equation.Since the evolution no longer takes place on the manifolds X c , the argument above leading to unconstrained Lyapunov stability is not applicable anymore.However, in good situations, stability can still be obtained if the quadratic form Q in (1-5) happens to be negative definite not just on T x X c , but on larger subspaces as well, for instance on the whole space ‫ޒ‬ n .This is, roughly speaking, the idea we shall pursue in the infinite-dimensional case, to study the stability of vortex-like solutions of the Navier-Stokes equation.
To conclude with the (unmodified) evolution (1-1), we emphasize that the problem of determining the index of the form (1-5) is also very natural from the viewpoint of the usual constrained optimization theory.Clearly, the "Lagrange function" (1-7) when considered on the whole space ‫ޒ‬ n , has a critical point at x (and a local maximum at x when restricted to X c ).The form Q will be strictly negative definite3 in the whole space ‫ޒ‬ n if and only if L has a nondegenerate unconstrained maximum at x.As is explained in Section 2D, this is related to the concavity of the function (c 1 , . . ., c m ) −→ M(c 1 , . . ., c m ) := sup x∈X c f (x). (1-8) 1B. Arnold's geometric view of the two-dimensional incompressible Euler equation.V. I. Arnold [1966b;1966a] (see also [Arnold and Khesin 1998]) carried out the analogue of the above calculations in an infinite-dimensional setting to handle in particular the two-dimensional incompressible Euler equation ∂ t ω + u • ∇ω = 0, where u denotes the velocity of the fluid and ω = curl u is the associated vorticity.In this case the evolution is generated by the Hamiltonian function, which represents the kinetic energy of the fluid, and the constraints are given by the Casimir functionals where ⊂ ‫ޒ‬ 2 is the fluid domain and is an "arbitrary" function on ‫.ޒ‬The idea of maximizing or minimizing the energy on the set of vorticities satisfying suitable constraints has been widely used since then to study the stability of steady-state solutions of the two-dimensional Euler equations and related fluid models; see [Arnold and Khesin 1998;Burton 2005;Cao et al. 2019].
Let us briefly recall the setup relevant for our goals here, making the similarities with the finitedimensional case as transparent as possible.Our main objects will be the following: (1) The phase space P = {ω : ‫ޒ‬ 2 → (0, ∞) : ω is smooth and decays "sufficiently fast" at ∞}.This is our infinite-dimensional replacement for the manifold ‫ޒ‬ n in the finite-dimensional model.We restrict ourselves to positive vorticity distributions defined on = ‫ޒ‬ 2 , because this is the appropriate framework to study the stability of radially symmetric vortices in the whole plane.Admittedly, the definition above is somewhat vague, but it serves only as a motivation and our results will be independent of the vague parts of the definitions.There is a natural Poisson structure on P that is relevant for the Euler equation, see Section A5, but here we only need some of its Casimir functionals (to be specified now).
(2) The Casimir functionals, which play the role of the constraints g j in the finite-dimensional example.These are linear combinations of elementary functionals of the form (1-10) where χ = 1 (0,∞) is the indicator function of (0, ∞).
Here and in what follows, we denote by |S| the Lebesgue measure of any (Borel) set S ⊂ ‫ޒ‬ 2 .Due to our assumptions on the vorticities in P, the functions a → h(a, ω) are finite and nonincreasing on (0, ∞).In general, they do not have to be continuous in a but they will have this property in the examples considered later.Similarly, the functionals ω → h(a, ω) may in general not be differentiable in every direction, but they will be in our examples.It is useful to single out the quantity (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11) which will be referred to as the "mass" of the vorticity distribution ω ∈ P.
(3) The orbits defined for any ω ∈ P by O ω = {ω ∈ P : h(a, ω) = h(a, ω) for all a ∈ (0, ∞)}. (1-12) These subsets of the phase space are the analogues of the manifolds X c defined by the constraints and can be considered as a measure-theoretical replacement for the symplectic leaves where SDiff denotes the group of area-preserving diffeomorphisms in ‫ޒ‬ 2 .In contrast to O SDiff ω , the orbit O ω does not carry any topological information about ω, since ω ∈ O ω as soon as ω is a measure-preserving rearrangement of ω.
(4) The Hamiltonian (or energy functional) E : P → ‫,ޒ‬ given by where ψ = −1 ω is the stream function defined by This is an analogue of the function f in the finite-dimensional example.Note that the usual kinetic energy defined by 1 2 ‫ޒ‬ 2 |u| 2 dx, where u = ∇ ⊥ ψ, is infinite for ω ∈ P.However, both definitions of the energy coincide when ‫ޒ‬ 2 ω dx = 0, which is the case for instance if ω is the difference of two vorticities in P with the same mass.It is also worth observing that the functional E is not invariant under the scaling transformation ω(x) → ω (λ) (x) := λ 2 ω(λx) when M 0 = ‫ޒ‬ 2 ω dx ̸ = 0.In fact, one can easily check that log λ for all λ > 0.
(5) The conserved quantities induced by Euclidean symmetries.These are the first-order moments M 1 , M 2 and the symmetric second-order moment I defined by (1-15) Note that M 1 , M 2 are associated to the translational symmetry, via Noether's theorem, and I to the rotational symmetry.
With these definitions, the Euler equation can be written in the form ∂ t ω = {E(ω), ω}, where { • , • } denotes the Poisson bracket on P; see Section A5.Any steady state ω ∈ P is a critical point of the Hamiltonian E on the orbit O ω. Stability can be inferred when the restriction of the energy E to O ω has a strict local extremum at ω.In what follows, we focus on the maximizers of the energy, which correspond to radially symmetric vortices.
1C.The constrained maximization of the energy in P.Under our assumptions, it is easy to determine the maximizers of the Hamiltonian E under the constraints given by the functions h(a, ω) for a ∈ (0, ∞).Indeed, for any ω ∈ P, the orbit O ω contains a unique element ω * that is radially symmetric and nonincreasing in the radial direction; this is the symmetric decreasing rearrangement of ω [Lieb and Loss 1997].The Riesz's rearrangement inequality then shows that E(ω) ≤ E(ω * ) for all ω ∈ O ω * , with equality if and only if ω is a translate of ω * ; see [Carlen and Loss 1992, Lemma 2].Of course ω * is a stationary solution of the Euler equation, which represents a radially symmetric vortex with nonincreasing vorticity profile.Our main focus here will be on the analogue of the quadratic form (1-5) for the steady state ω = ω * .
First, the analogue of the Lagrange function (1-7) is where the quantities (a) for a ∈ (0, ∞) can be thought of as the Lagrange multipliers.The role of the discrete index j in (1-7) is now played by the continuous parameter a > 0. Defining4 we see that the Lagrange function can also be expressed as This quantity will be referred to later as the "free energy" of the vorticity distribution ω, a terminology that will be discussed in Section 1D below.
1D. Overview of our results.We are now able to describe more precisely the results of this paper.We consider a general family of radially symmetric vortices ω ∈ P with vorticity profile satisfying Hypotheses 2.1 below.Typical examples are the "algebraic vortex" ω(x) = (1 + |x| 2 ) −κ , where κ > 1 is a parameter, and the Oseen vortex for which ω(x) = e −|x| 2 /4 .
1D1.Arnold's quadratic forms with and without constraints.In Section 2, we study in detail the quadratic form (1-18) associated with the second variation of the Lagrange function (1-17) at the steady state ω ∈ P, paying some attention to the functional-analytic questions.First of all, while we know from the constrained maximization result that the restriction of that form to the tangent space T ωO ω is negative, it is not clear if this restriction is strictly negative definite, and if so in which function space.Our first main result is Theorem 2.5, where we show that, if two neutral directions corresponding to translational symmetry are disregarded, the restriction to T ωO ω of the quadratic form (1-18) is indeed strictly negative in an appropriate weighted L 2 space.The proof ultimately relies on a variant of the Krein-Rutman theorem.
We next investigate the index of the quadratic form (1-18) on a much larger subspace, corresponding to perturbations η ∈ T ωP satisfying ‫ޒ‬ 2 η(x) dx = 0.In other words, we relax all constraints given by the Casimir functions (1-10), except for the mass M 0 defined in (1-11), which is still supposed to be constant.A priori there is no reason why the form (1-18) should be negative definite in this larger sense, and indeed Theorem 2.8 shows that this is not always the case.More precisely, we show that negativity holds in the large sense if and only if the optimal constant in some weighted Hardy inequality (where the weight function depends on the vorticity profile ω) is smaller than 1.While that condition is not easy to check in general, we deduce from Corollary 2.11 that it is fulfilled at least for the Oseen vortex, as well as for the algebraic vortex ω(x) = (1 Although the mass constraint is rather natural, one may wonder if, for some vorticity profiles, the quadratic form (1-18) can be negative definite for all perturbations η ∈ T ωP; this question is briefly discussed in Section 2C.Finally, in Section 2D, we give a fairly explicit expression of the energy E( ω) in terms of the constraints h(a, ω) for all a > 0; see Proposition 2.18.One obtains in this way an infinite-dimensional analogue of the quantity M(c 1 , . . ., c n ) defined in (1-8).Among other things, we justify our claim that the index of the quadratic form (1-5) is related to the concavity of the function (1-8) (which is a well-known fact), and we discuss a similar link in the infinite-dimensional case.
As an aside, we mention here that the stability of radially symmetric vortices for the two-dimensional Euler equations can also be studied using other conserved quantities, such as the second-order symmetric moment I defined in (1-15); see, e.g., [Marchioro and Pulvirenti 1994, Chapter 3].
1D2.The global maximizers of the free energy.Let ψ be the stream function associated with the radially symmetric vortex ω.We have seen that the analogue of the Lagrange function (1-7) is given by the "free energy" (1-17), where the function is defined, up to an additive constant, by the relation ψ(x) = ′ ( ω(x)).The appellation "free energy" is partially justified by a (loose) analogy of formula (1-17) with the classical thermodynamical expression for the free energy (1-19) Here U is the internal energy (of a suitable system), T is the temperature, and S is the entropy.In (1-17), the energy E is analogous to U, the integral ‫ޒ‬ 2 (ω(x)) dx is analogous to S, and one can argue that it is reasonable to take T = −1.Of course, T has nothing to do with the real temperature of the fluid, but should roughly be thought of as the statistical mechanics temperature of our system in the sense of [Onsager 1949].
We have not attempted to make this connection rigorous, which would take us in a different direction.
In Section 3, we consider vortices ω which are global maximizers of the free energy F(ω) for all ω ∈ P satisfying ‫ޒ‬ 2 ω dx = ‫ޒ‬ 2 ω dx.Such equilibria can be expected to have strong stability properties, and may be useful for other purposes too.Using a direct approach, in the sense of the calculus of variations, we prove the existence of global maximizers under fairly general assumptions on the function ; see Theorem 3.4.However, we do not have any efficient method to determine if a given vortex ω is a global maximizer or not.A necessary condition is of course that the quadratic form (1-18) be negative on perturbations η with zero mean, see Theorem 2.8, but there is no reason to believe that this is sufficient.Numerical evidence indicates that the Oseen vortex is a global maximizer, and so are the algebraic vortices ω(x) = (1 + |x| 2 ) −κ for κ ≥ 2. In the particular case κ = 2, maximality can be deduced from the logarithmic Hardy-Littlewood-Sobolev inequality which holds for all ω ∈ P with M 0 (ω) = 1; see [Carlen and Loss 1992].We mention that (1-20) is related to Onofri's sharp version [1982] of the Moser-Trudinger inequality.
1D3.The effect of viscosity: application to Oseen vortices.In Section 4, we consider the stability of the Gaussian vortex under the evolution defined by the Navier-Stokes equation ∂ t ω + u • ∇ω = ν ω, where ν > 0 is the viscosity parameter.More precisely, we show that the quadratic form (1-18) can be used to give an alternative proof of the local stability results established in [Gallay and Wayne 2005].We believe that a proof relying on the second variation of the energy is of some interest, because the analogue of the form (1-18) can be defined for more complicated vortex structures as well, whereas the simpler approach in [Gallay and Wayne 2005] may be more difficult to adapt.The addition of the viscous term results in important new issues: the radial vortices are no longer steady states and the orbits (1-12) are no longer invariant under the evolution, so that much of the geometric picture underlying the Euler equation is destroyed.The first problem is settled by introducing self-similar variables and restricting ourselves to Oseen's vortex, which is a stationary solution of the Navier-Stokes equation in these new coordinates.Thanks to Theorem 2.8 and Corollary 2.11, the quadratic form (1-18) is positive definite for all perturbations with zero mean.This form is invariant under the evolution defined by the linearized Euler equation at the vortex, but not under the Navier-Stokes evolution due to the viscous term and the nonlinearity.The effect of viscosity is measured by a second quadratic form, which happens to have a favorable sign; see Theorem 4.2.We do not know if this is just a lucky coincidence, or if there are deeper reasons behind that.In any event, this nice structure allows us to recover the local stability result of [Gallay and Wayne 2005], except for a slight difference in the choice of the function space; see Theorem 4.5.Again, we emphasize that the functional setting used in that work relies in an essential way on the radial symmetry of Oseen's vortex, through the existence of conserved quantities such as the moment I in (1-15), whereas our new approach can, at least in principle, be adapted to more general situations, where other methods do not work.

ARNOLD'S VARIATIONAL PRINCIPLE AND ITS APPLICATION TO THE STABILITY OF PLANAR VORTICES 689
We introduce the weight function A : [0, +∞) → ‫ޒ‬ defined by A(0) = −ω * (0)/(2ω ′′ * (0)) and Let A : ‫ޒ‬ 2 → (0, ∞) be the radially symmetric extension of A to ‫ޒ‬ 2 , namely A(x) = A(|x|) for all x ∈ ‫ޒ‬ 2 .We introduce the weighted L 2 space X defined by ), and using Hölder's inequality we easily deduce that X → L 1 ‫ޒ(‬ 2 ).We also consider the closed subspaces X 1 ⊂ X 0 ⊂ X defined by We observe that, for any ω∈ X , the energy E(ω) introduced in (1-13) is well-defined.This a consequence of the following classical estimate, whose proof is reproduced in Section A1 for the reader's convenience. (2-6) Then the last member in (1-13) is well-defined, and the energy E(ω) satisfies the bound where log (2-8) Since any ω ∈ X obviously satisfies (2-6), we can consider the quadratic form J on X defined by (2-9) In the particular case where ω ∈ X 0 , namely when ω has zero average over ‫ޒ‬ 2 , Proposition 2.2 gives the alternative expression where u is the velocity field associated with ω via the Biot-Savart formula (2-8).In view of (1-18) and (2-3), we have J = − 1 2 F ′′ ( ω), where F ′′ ( ω) is the second variation of the free energy (1-17) at the equilibrium ω.It is clear that X is the largest function space on which this second variation is well-defined.
Our main goal in this section is to study the positivity and coercivity properties of the quadratic form J on the spaces X , X 0 , and X 1 defined in (2-4), (2-5).To formulate our results, it is useful to take the decomposition X = X rs ⊕ X ⊥ rs , where and X ⊥ rs is the orthogonal complement of X rs in the Hilbert space X .Referring to the geometric picture of Section 1B, we consider X ⊥ rs as the tangent space to the orbit O ω at ω.This interpretation can be formally justified as follows: if ω ∈ X is smooth, the tangent space T ωO ω is spanned by vorticities of the form v • ∇ ω, where v is a (smooth and localized) divergence-free vector field, and using polar coordinates as in Section 2A below one verifies that such vorticities are indeed orthogonal in X to all radially symmetric functions.A contrario, since there is a one-to-one correspondence in P between orbits and symmetric decreasing rearrangements, it is clear that any radially symmetric perturbation of the equilibrium ω is transverse to the orbit O ω.
It is easy to verify that J (ω 1 +ω 2 ) = J (ω 1 )+ J (ω 2 ) when ω 1 ∈ X rs and ω 2 ∈ X ⊥ rs , so that the restrictions of J to X rs and X ⊥ rs can be studied separately.We first consider the tangent space X ⊥ rs in Section 2A, and postpone the study of radially symmetric perturbations (with zero or nonzero mass) to Sections 2B and 2C.
Remark 2.4.Hypotheses 2.1 are sufficient for our results to hold, but can be relaxed in several ways.In particular, we can consider vortices that are not smooth at the origin, but the assumption that ω ′ * (r ) < 0 for all r > 0 seems essential.This excludes vortices with compact support from our considerations, but as our motivation comes from applications to the Navier-Stokes equations, Hypotheses 2.1 are good enough for our purposes here.Of course, extensions of the theory that would include compactly supported vortices might be relevant in other situations and can probably be constructed, although they may require additional work.
2A. Positivity of the quadratic form J on X ⊥ rs .Theorem 2.5.Under Hypotheses 2.1, the quadratic form J defined by (2-10) is nonnegative on the space X ⊥ rs ⊂ X 0 .Moreover, there exists a constant γ > 0 such that (2-13) Proof.We introduce polar coordinates (r, θ ) in ‫ޒ‬ 2 , and given any ω ∈ X ⊥ rs we use the Fourier decomposition where the sum runs over all nonzero integers k ∈ ‫ޚ‬ \ {0}.By Parseval's relation we have where B k is the integral operator on the half-line ‫ޒ‬ + defined by the formula which is regular at the origin and converges to zero at infinity.In view of (2-15), the proof of Theorem 2.5 reduces to the study of the one-dimensional inequality (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) which depends on the angular Fourier parameter k ∈ ‫ޚ‬ \ {0}.More precisely, the quadratic form J is nonnegative on X ⊥ rs if and only if, for all k ̸ = 0, inequality (2-18) holds with some constant C k ≤ 1.In addition, we have the lower bound (2-13) on the subspace X ⊥ rs ∩ X 1 if and only if inequality (2-18) holds with a better constant C k ≤ 1−γ for all k ̸ = 0, assuming when |k| = 1 that f satisfies the additional condition It remains to establish inequality (2-18) for all k ∈ ‫ޚ‬ \ {0}.We obviously have the pointwise bound , so that we can restrict ourselves to nonnegative functions f.Moreover the operator B k preserves positivity, and an inspection of the formula (2-16) reveals that 0 As a consequence, to show that J is nonnegative on X ⊥ rs , it is sufficient to prove inequality (2-18) in the particular case where |k| = 1 and f ≥ 0. Setting h = A 1/2 f, we write that inequality in the equivalent form where The following assertions play a crucial role in our argument: Claim 1: The operator B 1 is self-adjoint and compact in the (real) space Y = L 2 ‫ޒ(‬ + , r dr ).
Claim 2: The spectral radius of B 1 is equal to 1, and λ = 1 is a simple eigenvalue of B 1 .
To see that, we first observe that λ = 1 is an eigenvalue of B 1 with a positive eigenfunction.Indeed, using (2-2), it is straightforward to verify that the function g with the boundary conditions g(0) = g(+∞) = 0, where µ = 1/λ, We already observed that µ = 1 is the lowest eigenvalue of (2-23); see Remark 2.3.It follows that λ = 1 is the largest eigenvalue of the integral operator B 1 , whose spectral radius is therefore equal to 1.The argument above also shows that all positive eigenvalues of B 1 are simple, because (2-23) is a second-order differential equation.
It is now a simple task to conclude the proof of Theorem 2.5.Claims 1 and 2 imply the validity of inequality (2-20) with C 1 = 1.We deduce that (2-18) holds for |k| = 1 with C k = 1, and (since This possibility will be used in Section 4.
2B. Positivity of the quadratic form J on X rs ∩ X 0 .The quadratic form J is not necessarily positive when considered on the subspace X rs ∩ X 0 , which consists of radially symmetric functions with zero mean.This question is related to the optimal constant in the weighted Hardy inequality where f : [0, +∞) → ‫ޒ‬ is an absolutely continuous function with Both conditions in (2-26) are fulfilled in our case, since A(r Theorem 2.8.Under Hypotheses 2.1, the quadratic form J defined by (2-10) is coercive on X rs ∩ X 0 if and only if Hardy's inequality (2-25) holds for some C H < 1.In that case we have Proof.Given ω ∈ X rs ∩ X 0 , we write ω(x) = ω 0 (|x|) and we consider the stream function ψ 0 defined (up to an irrelevant additive constant) by Defining f (r ) = r ψ ′ 0 (r ), we see that f is absolutely continuous on ‫ޒ‬ + with f (0) = f (+∞) = 0. Moreover we have ω 0 (r ) = f ′ (r )/r and u 0 (r ) := ψ ′ 0 (r ) = f (r )/r by construction.Finally the assumption that ω 0 ∈ X rs ∩ X 0 ensures that A 1/2 ω 0 and u 0 belong to the space Y = L 2 ‫ޒ(‬ + , r dr ).We thus have (2-28) and using (2-25) we conclude that (2-27) holds with γ = 1 − C H .This proves that the quadratic form J is coercive on As is well known, the optimal constant in Hardy's inequality (2-25) is related to the lowest eigenvalue of a self-adjoint operator.A convenient way of seeing this is to apply the change of variables r = e x , h(x) = f (e x ), B(x) = e −2x A(e x ), which transforms (2-25) into the equivalent inequality (2-29) The integral in the right-hand side of (2-29) defines a closed quadratic form on the Hilbert space be the self-adjoint operator in H associated with the quadratic form (2-29) by Friedrich's representation theorem [Kato 1966].Since B(x) > 0 for all x ∈ ‫ޒ‬ we know that ‫ނ‬ is positive, and using the fact that x 2 B(x) −1 → 0 as |x| → ∞ it is easy to verify that ‫ނ‬ has compact resolvent in H , and hence purely discrete spectrum.The optimal constant in C H in (2-29) is precisely the inverse of the lowest eigenvalue of ‫:ނ‬ By Sturm-Liouville theory, if µ = C −1 H is the lowest eigenvalue of ‫,ނ‬ there exists a positive eigenfunction h ∈ D(‫)ނ‬ such that ‫ނ‬h = µh.Setting h(x) = f (e x ), we see that f is a positive solution of the ODE 2 dr/r < ∞ by construction.It is not easy to guess from (2-31) whether µ is smaller or larger than 1, but under additional assumptions on the vortex profile it is possible to make another change of variables which puts (2-31) into a form that allows for a comparison with (2-12).
Let L be the differential operator defined by where L 0 was introduced in (2-12).We know from (2-33) that Lg = µA −1 g, where µ = C −1 H and g is the positive function defined in Lemma 2.9.On the other hand, we observed in Remark 2.3 that L 0 φ = A −1 φ, where φ = ψ ′ * is also a positive function vanishing at the origin and at infinity.Using Sturm-Liouville theory, we easily deduce the following useful criterion: Corollary 2.11.Under assumptions (2-32), if the function V defined by (2-34) does not change sign, the optimal constant in Hardy's inequality Proof.With the notation above, we have L 0 φ − A −1 φ = 0 and ), we have for r 1 > r 0 > 0 the identity (2-37) Now, we let r 0 tend to 0 and r 1 to +∞ along appropriate sequences, in such a way that the right-hand side of (2-37) converges to zero.This possible, because we know that φ(r ) = O(r ) and φ ′ (r ) = O(1) as r → 0, while φ(r ) = O(1/r ) and φ ′ (r ) = O(1/r 2 ) as r → +∞; moreover, the behavior of g in these limits is given in Lemma 2.9 and Remark 2.10.We thus deduce from (2-37) that ∞ 0 Rφr dr = 0, which is impossible if the function R has a constant sign and is not identically zero.So, if V does not change sign, we must have H , this gives the desired conclusion.□ Remark 2.12.As is easily verified, the optimal constant C H in Hardy's inequality (2-25) is unchanged if the function A(r ) is replaced by λ −2 A(λr ) for some λ > 0. This corresponds to a rescaling of the vortex profile ω * .
We now give two important examples where the sign of C H − 1 can be determined.
Example 2.14 (Gaussian vortex).We next consider the Oseen vortex given by (2-39) In that case too, the potential V defined in (2-34) is positive; see Section A2.By Corollary 2.11, we conclude that C H < 1, so that the quadratic form J is coercive on X rs ∩ X 0 .A numerical calculation gives the approximate value C H ≈ 0.57, so that γ ≈ 0.43.
Remark 2.15.In a finite-dimensional situation, one can use statements such as Theorems 2.5 and 2.8 for showing the nonlinear Lyapunov stability of the corresponding steady solution, at least if the smoothness class of the relevant objects is C 2 .More precisely, if a flow ẋ = b(x) on a finite-dimensional manifold preserves a C 2 function f which attains a nondegenerate local maximum at x, then the sets { f (x) > f ( x) − ϵ} are invariant under the flow and for small ϵ are well-approximated by the small balls given by the quadratic form When f ′′ is continuous at x and x is close to x, the integral in this inequality is dominated by a small multiple of − 1 2 f ′′ ( x)[x − x, x − x] and the usual Lyapunov stability statements follow.In our situation here the set O ω is not a C 2 submanifold and the free energy functional ω → E(ω) + ‫ޒ‬ 2 (ω(x)) dx is not of class C 2 .It is not hard to see directly that the expression ′′ ( ω)(ω(x) − ω(x)) 2 dx in a suitable way.One may still use the invariance of the sets U ω,ϵ := {ω ∈ O ω ∩ X 1 : E(ω) > E( ω) − ϵ} under the Euler evolution, and possibly also the conservation of the second-order moment I (ω) defined in (1-15), to obtain Lyapunov-type stability statements.For results in this spirit when the domain occupied by the fluid is compact, the reader can consult [Burton 2005] and [Arnold and Khesin 1998, Section II.4].Our situation here is somewhat complicated by the noncompactness of our flow domain ‫ޒ‬ 2 , but under our assumptions one still has ϵ>0 U ω,ϵ = { ω} (by using the uniqueness of the maximizers discussed in [Carlen and Loss 1992], for example).This could be turned into Lyapunov-type stability statements, although not quite of the same form as in the C 2 case.The important point is that there are estimates for the proximity of "almost maximizers" to the exact maximizers, an issue that also appears in other problems, such as the stability of the isoperimetric inequality [Fusco et al. 2008], and of the Sobolev inequality [Bianchi and Egnell 1991].
In the present work our focus is on quadratic forms, due to their applicability to the viscous case.Of course, at the level of the linearized inviscid equation ω t + ū • ∇ω + u • ∇ ω = 0, the quadratic form J does provide Lyapunov stability in the space X 1 if inequality (2-25) holds with C H < 1.We note that the linearized analysis in other topologies can be more complicated; see for example [Bedrossian et al. 2019].
2C.The quadratic form J without mass constraint.In this short section we make a few remarks on the index of the quadratic form (2-9) when considered on the whole space X defined by (2-4), and not only on the subspace X 0 given by (2-5).Our first observation is that, due to lack of scale invariance in this context, the form J cannot be positive on X if the underlying steady state ω is sharply concentrated near the origin.To see this, we consider the rescaled vortex ωλ (x) = λ 2 ω(λx) and the associated weight function A λ (x) = λ −2 A(λx); see Remark 2.12.We denote by J λ the quadratic form on X corresponding to the steady state ωλ , namely the form (2-9) where A is replaced by A λ .If ω ∈ X and ω λ (x) = λ 2 ω(λx), a simple calculation shows that If M 0 ̸ = 0, it is clear that J λ (ω λ ) < 0 when λ > 0 is sufficiently large, so that the quadratic form J λ cannot be positive in this regime.
Remark 2.16.The negative direction arising by such a rescaling is related to a particular choice of the unit of length implicitly involved in the kernel 1 2π log |x|.In writing log |x|, we imply that x is dimensionless.When x is measured in some units of length, we should write the kernel as 1 2π log(|x|/r 0 ), where r 0 is a reference length.The choice of r 0 does not affect the behavior of the system, and in the stability analysis based on J it can be compensated for by adding to the quadratic form J a suitable multiple of the quantity ‫ޒ‬ 2 ω(x, t) dx 2 , which is preserved by the evolution.Hence, as one can expect, the stability analysis is independent of the choice of the reference length r 0 , or, equivalently, of the scaling parameter λ above.
We next argue that, for any vortex ω satisfying Hypotheses 2.1, the index of the quadratic form is welldefined in the sense that J has (at most) a finite number of negative directions.In view of Theorem 2.5, it is sufficient to evaluate J on radially symmetric functions ω ∈ X rs .The following expression will be useful: Lemma 2.17.For any ω ∈ X rs , we have (2-40) Proof.Here and below, with a slight abuse of notation, we consider any ω ∈ X rs as a function of the one-dimensional variable r = |x|.For such vorticities, the first integral in (2-9) obviously gives the first term in (2-40), so it remains to establish the following expression of the energy: (2-41) To this end, we introduce polar coordinates x = r e iθ , y = se iζ to compute the right-hand side of (1-13), and we use the identity (2-42) The formula (2-42) is well known and can be derived in many ways.For example, assuming that r is a fixed positive number, we interpret the last integral as a function of s ∈ ‫.ރ‬This expression obviously depends only on |s|, is continuous everywhere, and is analytic both inside and outside of the circle |s| = r .Inside the circle it has to be constant and outside the circle it coincides with the potential of a point particle of mass 2π located at the origin, which is 2π log |s|.This gives (2-42), and (2-41) follows.□ Applying the change of variables w(r ) = ω(r )A(r ) 1/2 , so that w ∈ Y = L 2 ‫ޒ(‬ + , r dr ) when ω ∈ X rs , the formula (2-40) becomes where k(r, s) = − log(max(r, s))A(r ) −1/2 A(s) −1/2 .Under Hypotheses 2.1, we have the lower bound This means that the right-hand side of (2-43) is the quadratic form in Y associated with a self-adjoint operator of the form 1 − K, where 1 is the identity and K is a Hilbert-Schmidt perturbation.By compactness, this operator has (at most) a finite number of negative eigenvalues, which means that the index of the quadratic form J on X is well-defined.
2D.The maximal energy as a function of the constraints.In Section 1A we considered the classical problem of maximizing a function f : ‫ޒ‬ n → ‫ޒ‬ under a family of constraints of the form g 1 = c 1 , . . ., g m = c m , where g 1 , . . ., g m : ‫ޒ‬ n → ‫.ޒ‬Given c = (c 1 , . . ., c m ) ∈ ‫ޒ‬ m , we recall the notation X c = {x ∈ ‫ޒ‬ n : g 1 (x) = c 1 , . . ., g m (x) = c m }.Assuming that f reaches a nondegenerate maximum on X c at some point x ∈ X c where the first-order derivatives g ′ 1 ( x), . . ., g ′ m ( x) are linearly independent, we introduced the quadratic form Q defined by (1-5), which is the second-order differential of the Lagrange function (1-7) at x.In the present section, we are interested in the index of the form Q on larger subspaces than T x X c .As was already mentioned, this question is closely related to concavity properties of the function M defined by (1-8) or, almost equivalently, to convexity properties of the set S = {(g 1 (x), . . ., g m (x), f (x)) : x ∈ ‫ޒ‬ n } ⊂ ‫ޒ‬ m+1 near its "upper boundary".
The situation becomes particularly transparent if we use adapted coordinates which, as it turns out, have a fairly complete analogy in the two-dimensional Euler case.Let us assume that we can introduce new coordinates (c 1 , . . ., c m , y 1 , . . ., y n−m ) in ‫ޒ‬ n such that, as before, c 1 , . . ., c m are the values of the constraints g 1 , . . ., g m , and the additional coordinates y 1 , . . ., y n−m are chosen so that the points having coordinates (c 1 , . . ., c m , 0, . . ., 0) are those where f attains its maximum on X c .5 Writing M(c 1 , . . ., c m ) = f (c 1 , . . ., c m , 0, . . ., 0) as in (1-8), one verifies that  where λ 1 , . . ., λ m are the Lagrange multipliers introduced in (1-4).Moreover the extremality condition on X c implies that We infer that  where all derivatives are evaluated at the point (c 1 , . . ., c m , 0, . . ., 0).The first submatrix in the right-hand side of (2-46) is precisely the Hessian of M, and the second submatrix is always negative definite, due to our assumption that f reaches a maximum at (y 1 , . . ., y n−m ) = (0, . . ., 0) for any fixed value of c 1 , . . ., c m .So we conclude that the quadratic form Q defined in (1-5) is negative definite at x if and only if the Hessian of M is negative definite at (c 1 , . . ., c m ), where c j = g j ( x) for j = 1, . . ., m.
Another interesting object is the function which is the Legendre transform of M.Under appropriate assumptions, the main one being the concavity of M, this quantity is well-defined and the relation (2-45) can be inverted (at least locally) via the formula (2-48) We now return to the infinite-dimensional framework of the two-dimensional Euler equation, with the manifold ‫ޒ‬ n replaced by the phase space P introduced in Section 1B, the function f replaced by the energy E in (1-13), the constraints g j replaced by the Casimir functionals h(a, ω) in (1-10), and the submanifolds X c replaced by the orbits O ω in (1-12).In that case we have (2-49) where, as before, ω * denotes the symmetric decreasing rearrangement of an element ω ∈ P. As O ω is characterized in terms of the functionals h(a, ω) defined in (1-10), the energy of the maximizer ω * in O ω can also be expressed in terms of the constraint function a → h(a, ω).It turns out that the representation formula is quite explicit.

ARNOLD'S VARIATIONAL PRINCIPLE AND ITS APPLICATION TO THE STABILITY OF PLANAR VORTICES 701
Integrating by parts in (2-41) and recalling that m = max ω, we can thus write where we have formally used the substitutions ω(r ) = a, ω(s) = b.This is straightforward when ω is strictly decreasing, and the general case where ω is nonincreasing can be treated by integrating only over the intervals where ω is strictly decreasing.□ We now make a more precise comparison with the finite-dimensional situation above.Let us assume that ω ∈ P is radially symmetric with ∂ r ω(r ) < 0 for all r > 0 and ∂ 2 r ω(0) < 0. To eliminate the translational symmetries, we work with the manifold where M 0 , M j are as in (1-11), (1-15).If η ∈ X 1 (see (2-24)) is smooth and compactly supported with sufficiently small C 2 norm, then ω + η ∈ P. Denoting by η rs the projection of η onto the subspace X rs defined in (2-11), we can take the quantities h(a, ω+η rs ) and η ⊥ rs := η−η rs as the (approximate) analogues of the coordinates c j and y k , respectively.The analogy is not perfect, due to the stronger-than-ideal assumptions on η, but it is sufficient for concluding that, when ω = ω * , the negative-definiteness of Arnold's form (1-18) on the tangent space T ω P is strongly related to the concavity of the energy E in the variable6 h at the function h(a) = π −1 h(a, ω).In some sense the expression (2-50) is "trying to be concave", although not quite achieving this: the function L(R, S) is separately concave, but not concave.The second variation on the space X 0 is given by the quadratic form which takes a function ξ(a) with Due to the separate concavity of L the first term and the third term are negative, but the second one can lead to the form being indefinite.In view of our previous considerations, the negativity of the form is equivalent to the validity of the Hardy inequality (2-25) with C H ≤ 1, and it is not hard to verify directly that this is indeed the case.As an analogue of (2-45), we also note that the variational derivative of E with respect to h is (2-54) We will not go into the details as we will not work with this expression.The reader can also derive the analogue of (2-48) (under appropriate assumptions).

Global maximization of the free energy
In the previous section we observed that some radially symmetric vortices ω, including the Gaussian vortex (2-39) and the algebraic vortex (2-38) with κ > 2, are nondegenerate local maxima of the associated free energy functional (1-17) once restricted to the manifold P defined in (2-53).This was established by showing that the second-order differential F ′′ ( ω) is strictly negative definite on the tangent space T ω P.
We now follow a different approach, which relies on the direct method in the calculus of variations: under appropriate assumptions on the function in (1-17), we show that the free energy F(ω) has a global maximum on the set of all vorticity distributions with a fixed mass M. By construction, if ω is any maximizer obtained in this way, the conclusion of Theorem 2.8 applies with γ ≥ 0, so that Hardy's inequality (2-25) holds with C H ≤ 1.Note also that, according to the discussion in Section 2D, prescribing amounts to fixing the "Lagrange multipliers" in our constrained maximization problem.We start with a preliminary result, which is probably well known.For the reader's convenience, the proof is reproduced in Section A1.
where the implicit constants only depend on the space dimension n.Moreover, if f is radially symmetric and nonincreasing in the radial direction, then the reverse inequalities also hold.
We next specify the function space in which we shall solve our maximization problem.
Definition 3.2.Given any M > 0, we denote by X M the set of all ω ∈ L 1 ‫ޒ(‬ 2 ) such that ω(x) ≥ 0 for almost all x ∈ ‫ޒ‬ 2 and For later use we observe that, if ω ∈ X M and if ω * denotes the symmetric nonincreasing rearrangement of ω, then This shows that the set X M ⊂ L 1 ‫ޒ(‬ 2 ) is invariant under the action of the symmetric nonincreasing rearrangement.

ARNOLD'S VARIATIONAL PRINCIPLE AND ITS APPLICATION TO THE STABILITY OF PLANAR VORTICES 703
We have shown in Proposition 2.2 that the energy E(ω) is finite for any ω ∈ X M .Unlike in Section 2, the function in the entropy term is not related here to any radially symmetric vortex, but is an arbitrary function satisfying the following properties: Hypotheses 3.3.The function : [0, +∞) → ‫ޒ‬ is continuous with (0) = 0.Moreover, there exist constants C 1 ∈ ‫,ޒ‬ C 2 < M/(8π ), and C 3 > M/(8π ) such that (3-4) Under Hypotheses 3.3, the positive part of satisfies + (ω) ≤ Cω(1+| log(ω/M)|) for some constant C > 0, and this implies in particular that the entropy S(ω) is well-defined in ‫ޒ‬ ∪ {−∞} for any ω ∈ X M .We are now in a position to state the main result of this section.
Theorem 3.4.Fix any M > 0. Under Hypotheses 3.3, there exists ω ∈ X M such that Moreover ω can be chosen to be radially symmetric and nonincreasing in the radial direction.
The proof of Theorem 3.4 is divided into two parts.The first one consists in showing that the free energy F is bounded from above on X M , and that there exists a maximizing sequence which is convergent in L 1 ‫ޒ(‬ 2 ).We formulate this in a separate statement: Proposition 3.5.Under Hypotheses 3.3, the free energy F = E +S is bounded from above on the space X M : Moreover, there exists a maximizing sequence (ω j ) j∈‫ގ‬ in X M which converges in L 1 ‫ޒ(‬ 2 ) to some limiting profile ω = ω * ∈ X M as j → +∞, and we have S( ω) > −∞.
Proof.Our starting point is the logarithmic Hardy-Littlewood-Sobolev inequality which holds for all ω ∈ X M ; see [Carlen and Loss 1992].In view of (3-4), we deduce from (3-5) that Now, let (ω j ) j∈‫ގ‬ be a sequence in X M such that E(ω j ) + S(ω j ) → F M as j → +∞.If we denote by (ω j ) * ∈ X M the symmetric nonincreasing rearrangement of ω j , we know that E((ω j ) * ) ≥ E(ω j ) and S((ω j ) * ) = S(ω j ) for all j ∈ ‫,ގ‬ so that ((ω j ) * ) j∈‫ގ‬ is a fortiori a maximizing sequence.So we assume henceforth that ω j = (ω j ) * ; i.e., ω j is radially symmetric and nonincreasing in the radial direction.In that case, there exists a constant C 0 > 0 such that for all j ∈ ‫.ގ‬ Indeed, the first inequality in (3-7) follows directly from (3-6), and the second one is a consequence of the first inequality and of Proposition 3.1, since ω j = (ω j ) * .It remains to verify that one can extract from (ω j ) j∈‫ގ‬ a convergent subsequence in L 1 ‫ޒ(‬ 2 ).We recall that ω j (x) is a nonincreasing function of the radial variable |x|, which satisfies the uniform pointwise estimate 0 ≤ ω j (x) ≤ M/(π|x| 2 ); see (A-3) below.By Helly's selection theorem [Rudin 1953], there exists a subsequence, still denoted by (ω j ) j∈‫ގ‬ , which converges pointwise to some limit ω : ‫ޒ‬ 2 → ‫ޒ‬ + as j → +∞.It is clear that ω is radially symmetric and nonincreasing, so that ω = ω * , and Fatou's lemma implies that ‫ޒ‬ 2 ω(x) dx ≤ M. Using in addition (3-7), we obtain similarly To prove the convergence in L 1 ‫ޒ(‬ 2 ) we take the decomposition, for any ϵ ∈ (0, 1), where The integral over A ϵ converges to zero as j → +∞ by the dominated convergence theorem, and in view of (3-7), (3-8) the integral over ‫ޒ‬ 2 \ A ϵ is bounded by 2C 0 /| log ϵ| uniformly in j.It thus follows from (3-9) that lim sup Finally, if we take the decomposition = + − − , where + , − denote the positive and negative parts of , we have the lower bound where the second inequality is again obtained by Fatou's lemma.But we have the identity where the first two terms in the right-hand side are bounded uniformly in j by (3-7), in view of Hypotheses 3.3 and Proposition 2.2, whereas F(ω j ) is bounded from below since (ω j ) is a maximizing sequence for F. We conclude that the right-hand side of (3-10) is finite, so that S( ω) > −∞.□ To prove (3-12), we first take the decomposition and we deduce that Here we used the convergence of ω j to ω in L 1 ‫ޒ(‬ 2 ), the a priori estimates (3-7), and the fact that log(r )M(r ) is bounded as r → 0, as a consequence of (3-8).On the other hand, since the function − is continuous and bounded from below, and since we integrate on the bounded domain {x ∈ ‫ޒ‬ 2 : |x| ≤ R}, we can apply Fatou's lemma to obtain (3-18) Combining (3-17) and (3-18), we obtain (3-12).We next prove (3-13).Recalling that R ≥ 1, we first observe that which means that the contribution of E(ω 2 j R ) can be disregarded since we only need an upper bound.The other terms in (3-13) have the expressions Since ω j is decreasing, we have ω j (r ) ≤ M j (r )/(πr 2 ) ≤ M for r ≥ R. So, using Hypotheses 3.3, we deduce that (ω j ) ≤ C 1 ω j + C 2 ω j log(M/ω j ), where C 1 ∈ ‫ޒ‬ and C 2 < M/(8π ).It follows that (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) where In view of (3-7), the first term in the right-hand side of (3-19) converges to zero uniformly in j as R → +∞, and can therefore be absorbed in the quantity δ 1 (R).To treat the second term, we fix a positive number α > 2 such that 4πC 2 α ≤ M, and we introduce the mutually disjoints sets (3-20) As M j (R) ≥ M/2, it follows from (3-20) that j (r ) ≤ 0 when r ∈ I (α, R), so the last integral in (3-19) can be restricted to the complement I (α, R) c .But on that set we have the upper bound ω j (r ) < Mr −α , where α > 2, and we easily deduce that I (α,R) c j (r )ω j (r )r dr converges to zero as R → +∞, uniformly in j.Altogether we obtain (3-13).
We do not have much information on the maximizer ω whose existence is established in Theorem 3.4.We expect that, if is as in Example 3.7, the maximizer is indeed the Gaussian vortex (2-39), but except for numerical evidence we have no proof so far.Similarly, we believe that the algebraic vortices (2-38) with κ ≥ 2 are global maximizers, but this is known only in the particular case κ = 2, where maximality follows from the logarithmic HLS inequality (3-5).
The examples above also suggest that the decay rate of the maximizer ω(x) as |x| → ∞ strongly depends on the behavior of the function (s) near s = 0. Extending the techniques in the proof of Theorem 3.4, one should be able to prove that, if is differentiable to the right at the origin, the corresponding maximizer ω is compactly supported.It is also worth mentioning that the entropy function associated with any radially symmetric decreasing vortex ω through the relation ψ(x) = ′ ( ω(x)) is necessarily concave on the range of ω, whereas no concavity assumption is included in Hypotheses 3.3.This suggests that the maximizer ω corresponding to a nonconcave function should be discontinuous, so that its range does not include the intervals where does not coincide with its concave hull.

Stability of viscous vortices
In this final section, we give a new proof of the nonlinear stability of the Oseen vortices, which are self-similar solutions of the Navier-Stokes equations in ‫ޒ‬ 2 .Our approach relies on the functional-analytic tools developed in Section 2, in connection with Arnold's variational principle, although we now consider a dissipative equation for which the Casimir functions (1-9) are no longer conserved quantities.Let w = w(y, τ ) ∈ ‫ޒ‬ denote the vorticity of the fluid at point y ∈ ‫ޒ‬ 2 and time τ > 0, and let φ = φ(y, τ ) ∈ ‫ޒ‬ be the associated stream function.The vorticity formulation of the Navier-Stokes equations is where {φ, w} = ∇ ⊥ φ •∇w is the Poisson bracket, ν > 0 is the viscosity parameter, and the Laplace operator acts on the space variable y ∈ ‫ޒ‬ 2 .As in [Gallay and Wayne 2002;2005], we introduce self-similar variables x = y/ √ ντ and t = log(τ/T ), where T > 0 is an arbitrary time scale.More precisely, we look for solutions of (4-1) in the form The evolution equation for the rescaled vorticity ω is where {ψ, ω} = ∇ ⊥ ψ • ∇ω and L is the Fokker-Planck operator Let ω be the vortex with Gaussian profile (2-39), namely ). (4-5) It is easy to verify that L ω = 0 and { ψ, ω} = 0.This implies that ω = α ω is a stationary solution of (4-3) for any α ∈ ‫.ޒ‬This family of equilibria is known to be stable with respect to perturbations in various weighted L 2 spaces; see [Gallay and Wayne 2005;Gallay 2012].We present here a new stability proof, which may be easier to adapt to more general situations.

ARNOLD'S VARIATIONAL PRINCIPLE AND ITS APPLICATION TO THE STABILITY OF PLANAR VORTICES 709
4A. Nonlinear stability of Oseen vortices.Given any α ∈ ‫,ޒ‬ we consider solutions of (4-3) of the form ω = α ω + ω, ψ = α ψ + ψ.The perturbation ω satisfies the modified equation where it is understood that the stream function ψ is expressed in terms of ω via the formula (1-14), so that ψ = ω.We assume henceforth that the perturbation ω satisfies the moment conditions ‫ޒ‬ 2 ω dx = 0 and ‫ޒ‬ 2 x j ω dx = 0 for j = 1, 2, (4-7) which are preserved under the evolution defined by (4-6).As is shown at the end of [Gallay and Wayne 2005], this hypothesis does not restrict the generality, in the sense that stability with respect to general perturbations (with no moment conditions) can then deduced by a simple argument.As for the existence of solutions to (4-6), we have the following standard result: Lemma 4.1.The Cauchy problem for (4-6) is globally well-posed in the weighted L 2 space X defined by (2-4), where A(x) = 4|x| −2 (e |x| 2 /4 − 1), and the subspace X 1 ⊂ X defined by (2-24) is invariant under the evolution.
Proof.It is known that the vorticity equation (4-3) or (4-6) is globally well-posed in various weighted L 2 spaces; see, e.g., [Gallay and Wayne 2002;Gallay 2012;2018].The nearly Gaussian weight A is not explicitly considered in those references, but the arguments therein can be easily modified to cover that case too.If A 1/2 ω ∈ L 2 ‫ޒ(‬ 2 ), then all moments of ω are well-defined, and a direct calculation shows that the conditions (4-7) are preserved under the evolution, so that (4-6) is globally well-posed in the subspace X 1 .□ Let ω0 ∈ X 1 , and let ω ∈ C 0 ([0, +∞), X 1 ) be the solution of (4-6) with initial data ω0 .By parabolic regularization, we have ω( • , t) ∈ Z 1 := Z ∩ X 1 for all t > 0, where Z is the weighted Sobolev space For later use, we introduce the following quadratic form on Z : where (4-10) We shall verify in Section A3 that A/2 ≤ B ≤ 2A, so that the form Q is well-defined.
The following coercivity result plays a crucial role in our argument.
Theorem 4.2.The quadratic form Q defined by (4-9) is coercive on the subspace Z 1 = Z ∩ X 1 : there exists a constant δ > 0 such that For the Gaussian vortex, we proved in Section 2 that Hardy's inequality (2-25) holds for some C H < 1.Thus, by Theorems 2.5 and 2.8, there exists a constant γ ∈ (0, 1) such that γ On the other hand, by Theorem 4.2, there exists δ > 0 such that where the second inequality follows from the definition (4-9) and the inequality B ≤ 2A.Taking a convex combination of both estimates in (4-22), we deduce where µ = δ/(3 + δ).Finally, it follows from Lemma 4.4 and Young's inequality that Now, as long as m 0 (t) ≤ ϵ 2 := µ 2 /(8C 2 0 ), we have by (4-13), (4-21), (4-23), (4-24) As a consequence, if we assume that ∥ ω0 ∥ 2 X = m 0 (0) ≤ ϵ 2 0 := γ ϵ 2 , we have m 0 (t) ≤ ϵ 2 for all t ≥ 0 and estimate (4-20) holds with C 1 = γ −1 .□ We briefly indicate here the meaning of our result for the Navier-Stokes equations in the original, unscaled variables.If ω = α ω + ω, where ω ∈ C 0 ([0, +∞), X 1 ) is as in Theorem 4.5, the vorticity w defined by (4-2) satisfies, in particular, the estimate which means that w( • , τ ) converges to a self-similar solution with Gaussian profile as τ → +∞.As is shown in [Gallay 2012, Theorem 1.2], that property holds in fact for all solutions of the vorticity equation (4-1) in L 1 ‫ޒ(‬ 2 ), although it is not possible to specify any decay rate in the general case.Note that the evolution defined by (4-1) in L 1 ‫ޒ(‬ 2 ) preserves the total mass, so that we necessarily have ‫ޒ‬ 2 w(y, τ ) dy = α for all τ > 0. Remark 4.6.Except for a slight difference in the definition of the function space X , Theorem 4.5 coincides with the well-known stability result [Gallay 2012, Proposition 4.5].The approach originally developed by C. E. Wayne and the first author relies on conserved quantities related to symmetries of the problem, such as the second-order moment I (ω) in (1-15).In many respects, it is simpler than ours, and it provides an estimate of the form (4-20) with explicit constants C 1 and µ.Note also that, in the limit of large circulation numbers |α| → ∞, the enhanced dissipation effect due to fast rotation can be used to improve both the decay rate of the perturbations and the size of the basin of attraction of the vortex; see [Gallay 2018].
4B. Coercivity of the diffusive quadratic form.This section is entirely devoted to the proof of Theorem 4.2, which is a key ingredient in Theorem 4.5.We first observe that the functions A(x), B(x) in (4-9) are both radially symmetric, with radial profiles A(r ), B(r ) given by the explicit expressions On can also verify that B/A is a decreasing function of r satisfying 1 2 ≤ B(r )/A(r ) ≤ 7 4 for all r > 0; see Section A3.
We next follow an approach similar to that in Section 2. If ω ∈ Z is decomposed in Fourier series like in (2-14), we have and we observe that ω ∈ Z 1 if and only if Introducing the new variables w k = A 1/2 ω k ≡ e χ ω k , where χ = 1 2 log(A), we obtain after straightforward calculations where the potential W is defined by r ) . (4-28) The coercivity estimate (4-11) is thus equivalent to the inequality (4-32) see Section A3.Our goal is to prove the lower bound L k ≥ δ in the entire space Y when |k| ≥ 2, and in the subspaces given by conditions (4-30) when k = 0 or k = ±1.We consider three cases separately.
Case 2: When |k| = 1, the lower bound (4-33) is of no use, but it is easy to verify that L k ≥ 0 in that case.Indeed, we claim that L k g 1 = 0, where g 1 (r ) = e χ (r ) r e −r 2 /4 .Since g 1 is a positive function vanishing at the origin and at infinity, this means that 0 is the lowest eigenvalue of L k in Y when k = ±1.To prove the above claim, we first observe that, for any (smooth) function f on ‫ޒ‬ + , we have the identity  because this is the property we used to go from (4-26) to (4-27).On the other hand, in view of (2-2) and (2-3), we have the identity which holds in fact for any vorticity profile ω * , if A is defined by (2-3).In the case of the Lamb-Oseen vortex, if we differentiate the equality (4-35) with respect to r , we find that the function f = −2ω ′ * =r e −r 2 /4 satisfies the relation But A ′′ + 2A ′ /r − r A ′ /2 = B − 1 by (4-10), so combining (4-34) and (4-36) we conclude that L k f = 0 if |k| = 1, which is the desired result.
To get coercivity, we now restrict ourselves to the subspace Y 1 ⊂ Y of all functions g satisfying ⟨g, h 1 ⟩ = 0, where h 1 (r ) = r e −χ (r ) ; see the second relation in (4-30).It is important to observe that h 1 is not proportional to g 1 , so that Y 1 is not the orthogonal complement in Y of the eigenspace spanned by g 1 .However, we have ⟨g 1 , h 1 ⟩ = 8 ̸ = 0, which means that the closed hyperplane Y 1 does not contain the eigenfunction g 1 .In view of Remark 4.8 below, we conclude that there exists some δ > 0 such that L k ≥ δ on Y 1 when |k| = 1.
Lemma 4.7.Let X be a Hilbert space and L : D(L) → X be a self-adjoint operator in X .We assume that there exist φ ∈ D(L) with ∥φ∥ = 1 and a, b ∈ ‫ޒ‬ with a + b ≥ 0 such that (i) Lφ = −aφ, and (ii) ⟨Lg, g⟩ ≥ b∥g∥ 2 for all g ∈ D(L) with g ⊥ φ.
It follows that where the integrals I 1 , I 2 , I 3 are defined and computed below.

e
−s(1+t) − e −s(2+t) s − 2e −s(2+t) + se −s(2+t) A 1/2 f is orthogonal in Y to the one-dimensional subspace Y 0 spanned by the positive function χ = A −1/2 .It is clear that Y ⊥ 0 does not contain any positive function, and in particular does not include the principal eigenfunction h 0 = −A 1/2 ω ′ * of the operator B 1 .So, applying Lemma 4.7 and Remark 4.8 below, we deduce that 1 − B 1 > 0 on Y ⊥ The Krein-Rutman theorem[Deimling 1985, Theorem 19.2] asserts that the spectral radius of the compact and positivity-preserving operator B 1 is an eigenvalue with positive eigenfunction.However, since the cone of positive functions has empty interior in Y, we cannot apply Theorem 19.3 in[Deimling 1985]to conclude that B 1 has a unique eigenvalue with positive eigenfunction, which is thus equal to the spectral radius.For this reason, we prefer invoking Sturm-Liouville theory to prove that 1 is the largest eigenvalue of B 1 .
This shows that the quadratic form J is nonnegative on X ⊥ rs .On the other hand, if we assume that ω ∈ X ⊥ rs ∩ X 1 , the function f = ω ±1 satisfies condition (2-19), which means that h = Remark 2.7.If β > 4 in Hypotheses 2.1, the conclusion of Theorem 2.5 remains valid, with the same proof, if the subspace X 1 is replaced by