Geometric Analysis of the Generalized Surface Quasi-Geostrophic Equations

We investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach.


Introduction
In the early 1990s Constantin et al. [6] investigated a 2D hydrodynamic model known as the surface quasi-geostrophic (SQG) equation.One motivation for this was a striking mathematical analogy with the 3D Euler equations exhibited by the authors via several key analytic similarities.These investigations led to the study of a whole family of inviscid SQG-type equations which can be viewed as interpolating between the standard SQG equation and the 2D Euler equations, cf.[5].
On the other hand, it is well known from the work of Arnold and his school that the motions of an ideal fluid have a beautiful geometric description as geodesic equations on the group of volume-preserving diffeomorphisms of the fluid domain [1,2].A key distinction between two and three dimensional hydrodynamics comes from examining the Fredholm properties of the associated Riemannian exponential M. Bauer was partially supported by NSF grant DMS-1953244 and by FWF grant FWF-P 35813-N.P. Heslin was supported by the National University of Ireland's Dr. Éamon de Valera Postdoctoral Fellowship.maps.In 2D the exponential map is a non-linear Fredholm map of index zero, but in 3D it is not [8,11,16].
It was shown by Washabaugh in [21] that the standard SQG equation can also be reformulated as a geodesic equation on the group of exact volume-preserving diffeomorphisms equipped with a metric given by the homogeneous Sobolev Ḣ− 1 2 inner product.He further illustrated via an explicit example that the exponential map in this setting fails to be Fredholm.Recently one of the authors and Vu showed that the entire generalized SQG family, given by can be viewed as geodesic equations [13].This opens up a possibility to investigate, using Riemannian geometric tools, the transition from 2D Euler to the surface quasi-geostrophic equations.
The central focus of this paper is to understand how the parameter β affects properties of the corresponding Riemannian exponential maps.In particular, we prove that for β < 1 these exponential maps are non-linear Fredholm of index 0, illustrating that the value β = 1 is critical.In the former case we establish, via a Morse theoretic argument, a finite upper bound on the number of conjugate points along an arbitrary geodesic segment.As the interpolation parameter approaches the critical value, this upper bound approaches infinity.We then provide an explicit example to show that this bound is attained.

Preliminaries
We will aim to keep this section reasonably general and concise.Full details of all the constructions can be found in [2,7,11] and the references therein.
2.1.Configuration Spaces.Let M be a compact surface without boundary equipped with a Riemannian metric g with volume form µ. Let ∇, ∇ ⊥ , div and ∆ denote the associated gradient, symplectic gradient, divergence and Laplacian operators.If E → M is a smooth vector bundle over M , we let H s (E) denote those sections of E with L 2 derivatives of order s.In particular, H s (T M ) is the space of H s vector fields on M which we equip with an inner product via We define the set H s (M, M ) to consist of all functions from M to itself which are of class H s in every chart.By the Sobolev Embedding Theorem we have that, for ) inherits a smooth submanifold structure as an open subset of H s (M, M ).Its tangent space at the identity, T e D s (M ) = H s (T M ), has an L 2 -orthogonal decomposition by the Hodge theorem: where H denotes the finite-dimensional subspace of harmonic vector fields on M .The subgroup D s µ,ex (M ) of exact diffeomorphisms is then defined to be the submanifold whose tangent space at the identity consists of those divergence-free vector fields with zero harmonic component, i.e. (2.2) We refer to ψ u as the stream function for u.This configuration space will be the central focus of the paper, however we will also consider its smooth counterpart It is worth noting at this point that the homogeneous Sobolev Ḣs inner product, given by u, v Ḣs = u, ∆ s v L 2 , when restricted to T e D s µ,ex (M ), induces a topology equivalent to the standard H s topology.
D s µ,ex (M ) is further a topological group under composition where right translation R η is smooth, but left translation L η is just continuous, in the H s topology.The group adjoint operator is given by: and the Lie algebra adjoint 1 by: 2.2.Right-Invariant Metrics and their Exponential Maps.We now recall the abstract geometric framework of V. Arnold [1,2].Let •, • be an inner product on T e D s µ,ex (M ) which we extend to a Riemannian metric on D s µ,ex (M ) via right translation.For a linear operator L : T e D s µ,ex (M ) → T e D s µ,ex (M ), we denote by L * its adjoint with respect to the metric, i.e, Lu, v = u, L * v for all u, v ∈ T e D s µ,ex (M ).As their definitions involve both the Lie and Riemannian structure, the geometry is in some sense encoded in the operators Ad * η u, v = u, Ad η v and ad * u v, w = v, ad u w associated with the Lie group and Lie algebra coadjoint representations.Geodesics t → γ(t) in D s µ,ex (M ) of the metric induced by •, • are critical paths for the energy functional.The corresponding geodesic equation, when reduced to the Lie algebra, is referred to as an Euler-Arnold equation which takes the form: If we further impose an initial condition u(0) = u 0 , then we obtain an initial value problem which enjoys the following (coadjoint) conservation law (2.6)Ad * γ(t) u(t) = u 0 .For a comprehensive list of notable examples of Euler-Arnold equations, cf.[2,11,20].The second equation in (2.5) is simply the flow equation, which provides a link between the (Eulerian) solution map u 0 → u and the (Lagrangian) solution 1 Strictly speaking, these groups of Sobolev diffeomorphisms are not Lie groups.For example, if u, v ∈ TeD s µ,ex (M ), their commutator is a priori only of Sobolev class H s−1 .However, the group structure they possess will be sufficient for our purposes.map u 0 → (γ, γ).The latter has a clear geometric interpretation as a Riemannian exponential map on D s µ,ex (M ).Namely, we have where γ(t) is the unique geodesic with γ(0) = e and γ(0) = u 0 .
2.3.The Jacobi Equation.Assume now that the metric induced by •, • has a well defined smooth exponential map on D s µ,ex (M ).For fixed u 0 ∈ T e D s µ,ex (M ) and T > 0, we have the geodesic segment, γ| [0,T ] , where γ(t) = exp e (tu 0 ).The Jacobi field along γ with initial conditions J(0) = 0 and J(0) = w 0 is then given by J(t) = d exp e (tu 0 )tw 0 .If we left-translate back to the Lie algebra via v L (t) = d e L η −1 (t) J(t), the field v L satisfies the system We denote the solution operator of this Cauchy problem by and recall the following result from [11].
Lemma 2.1.Let u 0 ∈ T e D µ,ex with Λ(t) and K 0 the operators defined by the formulas Then the solution operator w → Φ(t)w = td e L −1 γ(t) d exp e (tu 0 )w can be decomposed as where

The SQG Family of Equations
3.1.The Euler-Arnold Equations and Well-Posedness.We now equip the space of H s -Sobolev exact volume-preserving diffeomorphisms of a closed twodimensional Riemannian manifold M with a family of right-invariant metrics2 given at the identity map by: where again ψ u and ψ v are the stream functions for u and v. Using (2.3) and (2.4) we compute the group and Lie algebra coadjoints Hence, for the metrics (3.1) the Euler-Arnold equations (2.5) become the generalized SQG equations (1.1) Each of the β-metrics in (3.1) has an associated Riemannian exponential map as in (2.7).The following proposition is a direct consequence of the results in [12].
Proposition 3.1.For s > 3 and 0 ≤ β ≤ 1 the Riemannian exponential map on D s µ,ex (M ) of the metric (3.1) is C ∞ and, consequently, the differential is a bounded linear operator.
In particular, since a standard calculation shows that d exp e (0) is the identity, it follows from Proposition 3.1 that the exponential map is a local diffeomorphism near the identity map in D s µ,ex (M ).A further consequence of this is local well-posedness in the sense of Hadamard of the generalized SQG family (1.1).To the best of the authors' knowledge, for 0 < β ≤ 1, the situation for global well-posedness is not known.However, a sufficient criterion for global regularity is obtained in [5].Our aim now is to study how the parameter β affects the analytic and geometric properties of the corresponding exponential maps.

3.2.
Fredholm Properties of the Exponential Maps.Recall that a bounded linear operator F between Banach spaces is Fredholm if its range is closed and its kernel and cokernel are finite-dimensional.The difference of the dimensions of these two spaces is a topological invariant known as the index of F .The operator is semi-Fredholm if either of the finiteness requirements on its kernel or cokernel is dropped.Throughout the proceeding arguments, we will also use • X to denote the operator norm on the space of bounded linear operators on X, which should be clear from the context.
A map f between Banach manifolds is said to be Fredholm if its differential is a Fredholm operator on corresponding tangent spaces.If the source manifold is connected then we define the index of f to be the index of its differential df .Standard references for the above are [9,17].
The goal of this section is to establish the following theorem, which extends to fractional orders the results of [11].
Theorem 3.2.For s > 3 and 0 ≤ β < 1, the exponential map on the group of exact volume-preserving diffeomorphisms, D s µ,ex , induced by the metric (3.1) is a non-linear Fredholm map of index 0.
The value β = 1 is critical in the sense that the exponential map fails to be Fredholm.We will return to this in the following section.
Proof of Theorem 3.2.We begin by taking initial data u 0 ∈ T e D s ′ µ,ex for some s ′ ≫ s and letting γ(t) = exp e (tu 0 ) denote the corresponding geodesic in D s ′ µ,ex , which lives for some time T > 0. From Lemma 2.1, for any w ∈ T e D s µ,ex and t ∈ [0, T ], we have the decomposition d exp e (tu 0 )tw = Dγ(t) (Ω(t)w + Γ(t)w) where Ω(t) and Γ(t) are given by (2.12).Hence we estimate: d exp e (tu 0 )tw Ḣs Ω(t)w Ḣs − Γ(t)w Ḣs (3.4) where refers to some constant depending on the uniform (in time) Ḣs -norm of Dγ which we suppress for convenience.Focusing on the first term on the right side of (3.4) we have w, (−∆) From (2.3), (3.2) and (2.10) we acquire, for any v ∈ T e D s µ,ex Substituting the above and applying the Schwartz inequality, we have To address the first integral, notice that the adjoint Ad * γ with respect to the Ḣ−β/2 inner product can be expressed in terms of the L 2 adjoint Ad 0 γ = Dγ ⊤ R γ as 3 As γ(t) is of class H s ′ we have that Ad 0 γ(t) is a bounded operator on Ḣ β 2 .Using this we have where the suppressed constants above are uniform in time.Hence, for some constant c T > 0, we obtain the estimate for any w ∈ T e D s µ,ex .Regarding the second integral in (3.7), observe that R γ with a diffeomorphism γ is a pseudo differential operator of order s + β 2 , the commutator on the right-hand side above is 3 Where M ⊤ denotes the transpose of the matrix M .
a pseudo differential operator of order s − β 2 + 1, cf. [18].Hence there exists a constant C T > 0 such that w, Ω(t)w Ḣs c T w Ḣs − C T w Ḣs w Ḣs−1 .We next turn our attention to the operator Γ(t) in (3.4).From (2.10) and (3.3) we have w Ḣs for any w ∈ T e D s µ,ex where C ρ is the Hölder-Zygmund space with ρ > s, cf.[19].Hence, for 0 ≤ β < 1, K 0 is a compact operator by the Rellich-Kondrashov lemma.As Λ(τ ) −1 and Φ(τ ) are continuous families of bounded linear operators, the integrand in (2.12) is a compact operator.Hence, Γ(t) is itself compact.Combining (3.9) and (3.10), we arrive at the estimate With this estimate in hand, it now follows that d exp e (tu 0 ) has finite-dimensional kernel and closed range in T γ(t) D s µ,ex .The argument is standard.To see why this is true note that, if d exp e (tu 0 )w k converges in Ḣs with w k a bounded sequence in Ḣs , then w k must contain a subsequence which converges in Ḣs−1 as well as a subsequence for which Γ(t)w k converges in Ḣs .We may assume without loss of generality that both subsequences are simply w k .Substituting w N − w M into (3.12),we have Consequently w k is Cauchy in Ḣs and hence converges.Now, if the kernel were infinite-dimensional it would contain an orthonormal sequence with a convergent subsequence, which is a contradiction.Hence, dim ker d exp e (tu 0 ) < ∞.
Next we decompose T e D s µ,ex into a Ḣs orthogonal sum ker(d exp e (tu 0 )) ⊕ X s u0 .There exists a constant c > 0 such that, for all w in X s u0 , we have w Ḣs ≤ c d exp e (tu 0 )w Ḣs which implies closed range.If such a constant did not exist then there would be a sequence w k in X s u0 such that w k Ḣs = 1 and d exp e (tu 0 )w k Ḣs < 1 k containing a convergent subsequence.Passing to the limit and using the fact that X s u0 is closed, this would imply that there exists w ∞ ∈ X s u0 with w ∞ Ḣs = 1 and d exp e (tu 0 )w ∞ = 0, a contradiction.
The above shows that, for u 0 ∈ T e D s µ,ex , the operator d exp e (tu 0 ) is semi-Fredholm.Finally, as at t = 0 d exp e (0) is the identity on T e D s µ,ex , the theorem follows from the fact that the index function is continuous on the space of semi-Fredholm operators.⊠ 3.3.Conjugate Points and the Breakdown of Fredholmness.As mentioned in the Introduction, the distinction between two-and three-dimensional ideal hydrodynamics can be observed through the Fredholm properties of the L 2 exponential map.For example, Fredholmness in the 2D case precludes accumulation of conjugate points along finite geodesic segments, while such clustering is known to occur in the 3D setting, cf.[8,14].In this section we amplify Theorem 3.2 by establishing, via the Hydrodynamical Morse Index Theorem, cf.[11], an upper bound on the number of conjugate points along a finite geodesic segment for the β < 1 metrics; precluding clusters of such points.As the interpolation parameter approaches the critical value β = 1, this upper bound approaches infinity.We also provide an explicit example to illustrate that this bound is attained.This shows that the breakdown of Fredholmness occurs precisely at β = 1; in analogy with 3D ideal hydrodynamics.
Theorem 3.3.Let 0 ≤ β < 1.Given any initial velocity u 0 ∈ T e D s µ,ex let γ(t) be the corresponding geodesic of the metric (3.1), which exists for at least some time T > 0. The number of conjugate points along the segment γ| [0,T ] is bounded above by As in [11] consider the set of smooth (in time) vector fields along γ| [0,T ] which vanish at the endpoints Let Ḣ 1 β denote the completion of this space under the norm induced by the inner product where ∇ γ denotes the covariant derivative operator of the Ḣ−β/2 metric in D s µ,ex .The index form along γ is then a symmetric bilinear operator on Ḣ 1 β × Ḣ 1 β given by (3.14) where R γ (•) = R(•, γ) γ is the curvature tensor of the Ḣ−β/2 metric along γ in D s µ,ex .We recall the following result from [14].
Lemma 3.4.The index form (3.14) can be rewritten as where V = d e L γ v, W = d e L γ w with v, w ∈ T e D s µ,ex and K 0 is given in (2.10).
As in finite-dimensional Riemannian geometry, the Hydrodynamical Morse Index Theorem in [11] states that the number of conjugate points along the geodesic segment γ [0,T ] is equal to the dimension of the space on which this index form is negative definite.Hence, it will suffice to construct a finite-dimensional subspace, X β , of Ḣ 1 β such that the index form I β is non-negative on its Ḣ−β/2 -orthogonal complement, Y β .We proceed to the proof of the main theorem.
Proof of Theorem 3.3.Using Lemma 3.4, for W = d e L γ w ∈ Ḣ 1 β we have In order to simplify the above expression we let w(t) = e − t 2δ K0 v(t) for some vector field v(t).Clearly both fields vanish on the same subset of [0, T ] × M .Furthermore, since K 0 is anti-symmetric in the Ḣ−β/2 inner product, we have that e − t 2δ K0 is an orthogonal operator and hence preserves the Ḣ−β/2 norm.Hence from (3.16) we have where v k are time-independent fields we then have Expressing each v k = ∇ ⊥ ψ k and using an analogous estimate to (3.10), namely for any w ∈ T e D s µ,ex and ρ > 1 + β 2 , we have with C > 0 is a constant (independent of β) and u 0 = ∇ ⊥ ψ 0 ∈ T e D s µ,ex .
Finally, we expand each where a k,n are constants and φ n are non-constant eigenfunctions of the Laplacian with eigenvalues 0 < λ n ր ∞, normalized so that φ n L 2 = 1 and for any α ∈ R we have The estimate (3.18) becomes for each (k, n) such that a k,n = 0, then we have I β (W, W ) ≥ 0. This condition is equivalent to Notice that, for fixed β < 1, as λ n ր ∞ and since the right hand side of (3.19) is a decreasing function of k, there can be only finitely many k, say 1 ≤ k ≤ k max , for which (3.19) fails to hold for all n ≥ 1.If we consider such k, we can approximate the number of n for which (3.19) does not hold using the Weyl Law [22] which states that  Example 3.6.Following Washabaugh [21] we consider the case of M = S 2 with spherical coordinates (ϑ, ϕ).The rotation ∂ ϑ is then a steady state solution to (1.1) for any 0 ≤ β ≤ 1 with corresponding stream function − cos(ϕ).Note that this is in fact an eigenfunction of the Laplacian with eigenvalue 2. Letting w = ∇ ⊥ ψ, v L = ∇ ⊥ σ and noting that γ(t, ϑ, ϕ) = (ϑ + t, ϕ), (2.8) now becomes (3.20) ∂ t σ = ψ, ∂ t (−∆) 1− β 2 ψ + 2 1− β 2 ∂ ϑ ψ = 0 with initial condition σ(0) = 0.
Taking ψ(t, ϑ, ϕ) = ξ(t)φ n (ϑ, ϕ) where φ n is a spherical harmonic such that (−∆)φ n = n(n + 1)φ n and ∂ ϑ φ n = inφ n then gives: There are several natural continuations of this work.One is the study of degeneracy phenomena of the induced geodesic distance.For the L 2 -metric (β = 0), it has been shown that the geodesic distance is a true distance function [1,10].On the other hand, for the metric corresponding to the SQG equation (β = 1), it vanishes identically on the whole group [3].This raises the question: for which β does this change in behavior occur?Closely related to this is the notion of the corresponding diameter of the diffeomorphism group.This is well understood for the geodesic distance of the L 2 -metric [2,15,4], but the situation for general β has not yet been studied.

Finallyℵ
, since we have the Ḣ−β/2 orthogonal decomposition Ḣ 1 β = X A,β ⊕ Y A,β with I β Y A,β ≥ 0, the number of conjugate points along γ| [0,T ] is bounded above by ℵ β .⊠ An immediate consequence of the above is Corollary 3.5.We have lim β→1We conclude with an example illustrating the number of conjugate points along a finite geodesic segment approaching infinity as β → 1.
19)does not hold and X A = span sin kπt