One-parameter solutions of the Euler-Arnold equation on the contactomorphism group

We study solutions of the equation $$ g_t-g_{tyy} + 4g^2 - 4gg_{yy} = y gg_{yyy}-yg_yg_{yy}, \qquad y\in\mathbb{R},$$ which arises by considering solutions of the Euler-Arnold equation on a contactomorphism group when the stream function is of the form $f(t,x,y,z) = zg(t,y)$. The equation is analogous to both the Camassa-Holm equation and the Proudman-Johnson equation. We write the equation as an ODE in a Banach space to establish local existence, and we describe conditions leading to global existence and conditions leading to blowup in finite time.

Main Theorem. Let φ 0 (y) = g 0 (y) − g 0 (y). Suppose g 0 is C 2 and φ 0 satisfies the decay condition φ 0 (y) = O(1/y 2 ) as |y| → ∞. Then there is a T > 0 such that there exists a unique solution of (1.1) on [0, T ) × R with g(t) ∈ C 2 for each t. If φ 0 (and hence g 0 ) is nonnegative, then solutions exist globally. If g 0 is even and negative, then solutions blow up at some T in the sense that g(t, y) → −∞ as t T for every y ∈ R.
Equation (1.1) is a special case of the Euler-Arnold equation on the contactomorphism group Diff θ (M): m t + u(m) + (n + 2)λm = 0, (1.2) where M is a Riemannian manifold of odd dimension 2n+1 with a 1-form θ satisfying θ∧(dθ) n 0. Here f : M → R is a stream function, while u = S θ f is a contact vector field (satisfying the condition that L u θ is proportional to θ); the field u is uniquely determined by f via the condition f = θ(u). We denote by λ the function such that L u θ = λθ, and we write S θ f = (S θ f, λ). A Riemannian metric on M determines a right-invariant Riemannian metric on Diff(M) × C ∞ (M), which allows us to define m = S θ * S θ f , which is called the contact Laplacian. See [EP] for the derivation and local well-posedness theory of this equation when M is compact, along with other examples.
When n = 0 (so that M is R or S 1 ) and θ = dx, we have u = f ∂ x , λ = f x , and m = f − f xx , and equation (1.2) becomes the Camassa-Holm (CH) equation [CH] f (1.3) Equation (1.2) can thus be considered a generalization of (1.3) to higher dimensions; it shares some of the same conservation laws and also has features in common with hydrodynamics. The fact that (1.3) is the Euler-Arnold equation of Diff(S 1 ) with right-invariant H 1 metric is due to Misiołek [Mi] and Kouranbaeva [K]. Equation (1.1) arises in the case where n = 1 and M = R 3 (viewed as the Heisenberg group) with the "standard" contact form θ = dz − y dx. Here u = − f y ∂ x + ( f x + y f z ) ∂ y + ( f − y f y ) ∂ z and λ = f z . If the Riemannian metric is ds 2 = dx 2 + dy 2 + (dz − y dx) 2 , the natural left-invariant metric on the Heisenberg group which makes M a Sasakian manifold (see Boyer [Bo]), we will gives m = z(g−g yy ), u = −zg y ∂ x +yg ∂ y +z(g−yg y ) ∂ z , and λ = zg, and equation (1.2) reduces to (1.1). A similar ansatz in ideal hydrodynamics leads to the Proudman-Johnson equation, which has been studied in [CISY, PJ, ST, CW, S, CINT]. The Proudman-Johnson equation was originally derived from the incompressible Euler equations by considering velocity fields of the form u(t, x, y) = ( f (t, x), −y f x (t, x)), also known as stagnation-point similitude, which arise from a stream function ψ(t, x, y) = y f (t, x) on an infinitely long 2D channel (x, y) ∈ [0, L] × R. The outline of the paper is as follows. In §2 we establish some conservation laws and a local existence result for (1.1); the results of [EP] do not apply here since our M is not compact, so we give an independent proof. In §3 we prove global existence of solutions to (1.1) for a class of initial data satisfying a particular sign condition. Finally in §4 we demonstrate the existence of solutions of (1.1) which blow up in finite time from smooth initial data.

Local Existence
First we derive some preliminary results. Set φ(t, y) = g(t, y) − g yy (t, y) (2.1) and φ 0 (y) = φ(0, y) = g 0 (y) − g 0 (y), g 0 (y) = g(0, y). (2.2) Analogous to the Camassa-Holm equation, we may refer to (2.1) as the momentum associated to the velocity g. In terms of the momentum, equation (1.1) may be rewritten in the form Note that we may determine g from φ using the explicit solution formula The characteristics are given by the solution of the flow equation ∂γ ∂t (t, y) = γ(t, y)g t, γ(t, y) , γ(0, y) = y, (2.5) and in terms of the flow γ we obtain a convenient formula for the momentum conservation law. This formula should be considered analogous to the momentum transport law for Camassa-Holm and for the vorticity conservation law for the Euler equation of ideal hydrodynamics, in the sense that they all express the Noetherian conservation law arising from right-invariance of a Riemannian metric on a diffeomorphism group; see for example Arnold-Khesin [AK].
A simple consequence of Proposition 2.1 is the conservation of the sign of the momentum, which is important for our global existence results.
Another consequence of Proposition 2.1 is the local existence theorem, which we establish by writing everything in terms of γ as a "particle trajectory equation" and using Picard iteration, as in Chapter 4 of Majda-Bertozzi [MB]. (2.10) Then there is a unique solution g of equation ( Proof. By formula (2.4) we have (2.11) Using formula (2.6) and plugging this into (2.5), we obtain the differential equation (2.12) We now view this as the equation on a certain open subset of a Banach space, where the function F is given by (2.14) Define a Banach space B by Clearly U contains the identity, which is the initial condition for (2.13). Our goal is to show that F is Lipschitz on U, and we do this by showing that F has a uniformly bounded derivative on U.
Next we establish the non-existence of breaking wave solutions to (1.1) for initial data g 0 such that φ 0 0 does not change sign. Recall that the wave breaking phenomenon for a nonlinear wave equation is the existence of a blowup time T such that |g y (t, y * )| → ∞ as t → T for some y * while |g(t, y * )| remains bounded.
Lemma 2.4. Let T > 0 denote the maximal life-span of g. If g 0 (y) is such that φ 0 (y) = g 0 (y) − g 0 (y) does not change sign, then |g y (t, y)| ≤ |g(t, y)| for all t ∈ [0, T ) and y ∈ R. Hence no singularity in the form of wave breaking can occur on (0, T ]. Proof. Writing formula (2.4) in the form it is easy to check that As a result, if φ 0 never changes sign, then by Lemma 2.2 we know that φ is either nonnegative or nonpositive for all (t, y) ∈ [0, T ) × R. Hence the right side of (2.18) is nonnegative and we have the bound |g y (t, y)| ≤ |g(t, y)| for all y.
For the Camassa-Holm equation (1.3), it is known [CE1,Mc] that if the initial momentum does not change sign, then the solution of the equation is global in time. For equation (1.1), we will see that even when the sign of the momentum is assumed constant, the behavior may be very different depending on whether it is positive or negative.

Global Existence
In this section we study global existence of certain solutions to (1.1). Theorem 3.2 below establishes global existence in time of solutions to (1.1) arising from initial data g 0 such that φ 0 is nonnegative.
Proposition 3.1. Suppose g is a solution of (1.1) with initial condition φ 0 satisfying the decay condition (2.10). Then Proof. We first observe that by equation (2.6), if φ 0 satisfies the decay condition then so does φ(t, y) for any y. Using (2.1) we may write (1.1) as φ t + 4gφ = y∂ y gg yy − g 2 y . (3.2) Integrating the right side of (3.2) using the decay condition, we obtain R y∂ y gg yy − g 2 y dy = 2 R g 2 y dy, The result follows using the fact that R φ dy = R g dy.
Amongst other conserved quantities, the integral of the momentum φ associated to the Camassa-Holm equation is known to be conserved [CE1]. The same is true in general for equation (1.2) on a compact manifold, but not in this case since the equation really "lives" on R 3 . Essentially what is happening is that the solutions of the form (1.4) are not well-behaved since for example they have infinite energy; the same type of phenomenon appears when solving the Euler equation of ideal hydrodynamics: in two dimensions finite-energy solutions exist globally [Ba], but there are infinite-energy solutions of the form (1.4) which can blow up in finite time; see for instance [CISY, S] and references therein.
Theorem 3.2. Suppose φ 0 satisfies the decay condition (2.10) and φ 0 (y) ≥ 0 for all y ∈ R. Then the solution g of (1.1) with initial data g 0 exists globally in time.
Proof. By the general theory of ODEs in Banach spaces (see e.g. [H]), the only way a solution of γ (t) = F(γ(t)) can blow up in finite time is if it leaves all of the bounded sets on which F is Lipschitz. Recall that we established the Lipschitz property under the condition that a ≤ γ y (t, y) ≤ b for some constants a and b satisfying 0 < a < 1 < b, so we will have global existence as long as γ y (t, y) does not approach either zero or infinity in finite time.
From (2.9) we see that a uniform bound on both g and g y is sufficient to control γ y . By Lemma 2.2 we know that g(t, y) ≥ 0 and φ(t, y) ≥ 0 for all t and y. Using this and the decay condition (2.10) to ensure that φ 0 is in L 1 , (3.1) implies that Using (2.4) we obtain from which we conclude by Lemma 2.4 that g y is uniformly bounded, and thus no blowup can occur.

Blowup
Constantin-Escher [CE1,CE2] showed that solutions of the Camassa-Holm equation (1.3) cannot persist globally in time if the initial data f 0 is odd and satisfies f 0 (0) < 0. Further, they derived an upper bound, T ( f 0 ) = 1/2| f 0 (0)|, for the maximal time of existence of solutions. Theorem 4.1 below establishes the existence of solutions to (1.1) which blow up in finite time from initial data g 0 that is both symmetric about y = 0 and satisfies g 0 (0) < 0. Although singularities may form in (1.1) from nonpositive initial data 1 , we note that solutions of (1.1) seem to retain a few properties that are inherent to the blowup mechanism of (1.3). For instance, an upper bound for the maximal time of existence of a solution to (1.1) is, analogous to that of (1.3), given by 1/ √ 6|g 0 (0)|. Here g 0 (0) < 0 serves as analogue to the Camassa-Holm condition f 0 (0) < 0 and blowup, in both cases, is to negative infinity. Moreover, in [CE2] it was shown that if the Camassa-Holm initial profile f 0 is even instead of odd, with f 0 (0) negative enough, then f x (t, 0) can still diverge to negative infinity. A main difference between the qualitative behavior of blowup solutions to (1.1) and (1.3) is established in the second part of Theorem 4.1. More particularly, since (1.1) preserves the symmetry of the initial condition, for φ 0 both symmetric and nonpositive we show that solutions of (1.1) will actually diverge everywhere on R. Theorem 4.1. Suppose g is a solution of (1.1) with initial condition φ 0 = g 0 − g 0 satisfying the decay condition (2.10). Furthermore, assume g 0 is even through y = 0 and g 0 (0) < 0. Then g(t, 0) → −∞ as t T ≤ 1/ √ 6|g 0 (0)| . Additionally, if g 0 is such that φ 0 ≤ 0, then as t T we have g(t, y) → −∞ for all y ∈ R.