A geometric view on the generalized Proudman-Johnson and r-Hunter-Saxton equations
aa r X i v : . [ m a t h . DG ] J a n A GEOMETRIC VIEW ON THE GENERALIZEDPROUDMAN–JOHNSON AND r -HUNTER–SAXTONEQUATIONS MARTIN BAUER, YUXIU LU, AND CY MAOR
Abstract.
We show that two families of equations, the generalized in-viscid Proudman–Johnson equation, and the r -Hunter–Saxton equation(recently introduced by Cotter et al.) coincide for a certain range of pa-rameters. This gives a new geometric interpretation of these Proudman–Johnson equations as geodesic equations of right invariant homogeneous W ,r -Finsler metrics on the diffeomorphism group. Generalizing a con-struction of Lenells for the Hunter–Saxton equation, we analyze theseequations using an isometry from the diffeomorphism group to an ap-propriate subset of real-valued functions. Thereby we show that theperiodic case is equivalent to the geodesic equations on the L r -spherein the space of functions, and the non-periodic case is equivalent to ageodesic flow on a flat space. This allows us to give explicit solutions tothese equations in the non-periodic case, and answer several questionsof Cotter et al. regarding their limiting behavior. Contents
1. Introduction and main results 12. The non-periodic case 33. The periodic case 12References 141.
Introduction and main results
In his seminal article [1] Arnold found a geometric interpretation of theincompressible Euler equation as the geodesic equation of a right-invariantRiemannian metric on the group of diffeomorphisms. Since then an ana-logues geometric picture has been constructed for several other equationsin mathematical hydrodynamics and equations that admit such an inter-pretation are referred to as Euler–Arnold equations. Examples include theCamassa–Holm [6, 20, 13], the Hunter–Saxton [12, 15], the Burgers, theKdV [23], or the modified Constantin–Lax–Majda equation [7, 26, 10], seealso [25, 2, 4] for further examples.In this article we study the family of inviscid, generalized Proudman–Johnson equations with parameter λ (henceforth λ -PJ equation), which are Mathematics Subject Classification. given by(1.1) u txx + (1 + 2 λ ) u x u xx + uu xxx = 0 , where u is either a function on the real line (non-periodic λ -PJ) or on thecircle (periodic λ -PJ). The original Proudman–Johnson equation, in which λ = −
1, corresponds to axisymmetric Navier–Stokes equations in R ; thegeneralized equations, which were first proposed in [21], contain several otherimportant special cases, in particular the Hunter–Saxton equation for λ =1 /
2, the µ -Burgers equation λ = 1, and self-similar axisymmetric Navier–Stokes equations in higher dimensions. See [27] for further information aboutthe equation and its motivation.In the article [17] Lenells and Misio lek constructed a geometric interpre-tation of the λ -PJ equations for λ ∈ [0 ,
1] by interpreting them as geodesicequations of an affine connection ∇ α on the homogenous space of all dif-feomorphism of the circle modulo the group of rotations (in their notation α = 1 − λ ). The connection reduces to a Levi-Civita connection of the ho-mogenous W , Riemannian metric for λ = 1 /
2. In this case this recasts thegeometric interpretation of the Hunter–Saxton equation as an Euler–Arnoldequation, as described in [15, 16].
Contributions of the article.
In very recent work Cotter et al. [8] in-troduced a family of PDEs, entitled the r -Hunter–Saxton equations ( r -HSequations). This one-parameter family of PDEs is derived as the geodesicequation, in Eulerian coordinates, of the right-invariant, homogenous W ,r –Finsler metric (henceforth the ˙ W ,r metric). To emphasize the relation toEuler–Arnold equations, which are defined as geodesic equations of a right-invariant Riemannian metrics, we will refer such equations as Finsler–Euler–Arnold equations.The starting point of our investigations is the observation that the fam-ily of r -HS equations is formally equivalent to the λ -PJ equations for any λ ∈ (0 , λ = 1 /r . Using the derivation of the r -HS equation asgeodesic equation of a right-invariant Finsler metric, one has thus found aninterpretation of the λ -PJ equations as Finsler–Euler–Arnold equations andthereby complemented the geometric picture of Lenells and Misio lek [17].This observation positively answers a recent question by Gibilisco as pro-posed in [11, Problem 5].Using this geometric picture, as geodesic equation of the right-invariant,˙ W ,r -metric, allows one to investigate the properties of the λ -PJ equationsby studying the geometry of the corresponding infinite dimensional group. Inparticular, we extend a construction, originally found by Lenells for the pe-riodic Hunter–Saxton equation [15], and later extended to the non-periodiccase in [3], to the whole family of ˙ W ,r -metric, for 1 ≤ r < ∞ . This con-struction, which isometrically maps the diffeomorphism group to a (subsetof a) vector space of functions, implies the following: • The non-periodic- λ -PJ equation is equivalent to geodesics with re-spect to the L r norm in an open and convex subset of W ∞ , ( R ) = T k ∈ N W k, ( R ) (Section 2.2). See also [9] for a similar interpretation of the b -equations, which include the Camassa–Holm and the Degasperis–Procesi equation. ENERALIZED PROUDMAN–JOHNSON AND r -HUNTER–SAXTON EQUATIONS 3 • The periodic- λ -PJ equation is equivalent to geodesics on the L r -sphere on C ∞ ( S ) (Section 3.2).In this article we mainly focus in the non-periodic case, where this equiva-lence turns out to be utmost advantageous for studying the equations. Since W ∞ , ( R ) (with the L r -norm) is a vector space, geodesics in it are given bystraight lines; from this we obtain explicit formulas for the solutions of thenon-periodic- λ -PJ equation for any λ ∈ (0 , u . In particular, thisgives a rigorous interpretation of the piecewise-linear solutions studied inCotter et al. [8], and shows that the space of piecewise-linear velocities is to-tally geodesic. We use this geometric approach to easily retrieve some of theresults of Cotter et al. [8], such as blow-up time of solutions (Section 2.4), aswell as answer several questions stated in [8] regarding the limiting behaviorof solutions as r → ∞ ( λ →
0, resp.): we show that the solutions converge,as r → ∞ , to a solution of the λ -PJ equation for λ = 0, and that this solu-tion preserves the ˙ W , ∞ -norm of the velocity along the flow (Section 2.8).We also show existence of the boundary-value problem in Lagrangian coor-dinates (Section 2.6). Finally, in Section 2.10, we show that large parts ofour analysis continue to hold for λ / ∈ [0 ,
1] ( r <
1, resp.): while the geometricinterpretation of the PDEs as Finsler–Euler–Arnold equations breaks down,the solution formula is still valid for this wider range of parameters.The periodic case is considered in Section 3. There, the transformation tothe L r -sphere does not immediately yields explicit solutions (when r = 2),but it is still very useful: in fact, it was already used in the analysis ofblow-up of the solutions by Sarria and Saxton [24]. Our contribution hereis unveiling the geometry behind this transformation. Also, similar to thenon-periodic case, the geometric picture yields a framework for discussingthese equations in lower regularity (Lipschitz velocities).We note that for the non-periodic case our construction directly providesa change of coordinates that linearizes the flow of the λ -PJ equation. In thespirit of [17], this can be interpreted as an analogue of the inverse scatter-ing transform formalism, and thus the integrability of these equations forany λ ∈ (0 ,
1) follows, and in fact to any λ ∈ R in view of the results ofSection 2.10. To the best of our knowledge, integrability of the λ -PJ equa-tions was previously only known for the case λ = 1 / λ = 1 (the Burger’s equation) and λ = 0. Our results thereforeconfirm the analogue of [11, Problem 6] for the non-periodic λ -PJ equations. Acknowledgements.
The authors are grateful to S. Preston and G. Mi-sio lek for various discussions during the preparation of the manuscript.2.
The non-periodic case
In this section we will study the geometric picture for the non-periodic r -Hunter–Saxton equation (generalized Proudman–Johnson equation resp.), For λ = 0 the integrability of the periodic λ -PJ equation is shown in the article [17].This proof translates directly to the non-periodic case. MARTIN BAUER, YUXIU LU, AND CY MAOR as the Finsler–Euler–Arnold equation on a group of diffeomorphisms equippedwith a right-invariant Finsler metric. The unboundedness of R will requireus to specify appropriate decay conditions for the elements of the diffeomor-phisms group, which we introduce below in Section 2.1. We then show thatthe r -HS equations are equivalent to the λ -PJ equations, for λ = r ∈ (0 , Diffeomorphisms on the circle and the r -HS ( λ -PJ) equation. We start by introducing an appropriate functional setting for studying the r -Hunter–Saxton equation on the real line. To this end, we need to identifya group of smooth, orientation preserving diffeomorphims on the real line,on which the flow will be defined. Note that the group of all smooth, ori-entation preserving diffeomorphisms of the real line is not an open subsetof the space C ∞ ( R , R ) and consequently it is not a smooth Fr´echet mani-fold. To overcome this difficulty one usually only consider diffeomorphismsthat satisfy certain decay conditions, which then allows to retain a manifoldstructure for the corresponding space. While several different types of de-cay conditions have been considered in the literature [19, 14], we will restrictour analysis in this paper to groups related to the function space W ∞ , ( R ).Specifically, we will consider the following diffeomorphism group:Diff −∞ ( R ) = (cid:26) ϕ = id + f : f ′ ∈ W ∞ , ( R ) , f ′ > − , and lim x →−∞ f ( x ) = 0 (cid:27) , where W ∞ , ( R ) = T W k, ( R ) is defined as the intersection of all Sobolevspaces of order k ≥
0. It has been shown in [3] that this space is a smoothFr´echet Lie-groups with Lie-algebra: g −∞ = (cid:26) u : u ′ ∈ W ∞ , ( R ) and lim x →−∞ u ( x ) = 0 (cid:27) . The reason for working with this group and not the smaller, more commonlyknown one Diff( R ) := (cid:8) ϕ = id + f : f ∈ W ∞ , ( R ) and f ′ > − (cid:9) will be clear from Theorem 2.1.1 below, where we show that the r -Hunter–Saxton equation is not consistent with two-sided decay conditions on f , andthus the geodesic equation is not even locally well-defined on Diff( R ). Itswell-definiteness on Diff −∞ ( R ) will follow from Theorem 2.3.1.We now define the right-invariant ˙ W ,r -Finsler metric. To this end, wewrite any tangent vector h ∈ T ϕ Diff −∞ ( R ) as X ◦ ϕ with X ∈ g −∞ . Thisallows us to define the right-invariant, homogenous, W ,r -Finsler metric onDiff −∞ ( R ) via(2.1) F ϕ ( h ) = (cid:18)Z R | ( h ◦ ϕ − ) ′ | r dx (cid:19) /r = (cid:18)Z R | X ′ | r dx (cid:19) /r . Note that for any r ≥ g −∞ ⊂ ˙ W ,r , and thus equa-tion (2.1) is well-defined. Furthermore, we remark that the Finsler norm is ENERALIZED PROUDMAN–JOHNSON AND r -HUNTER–SAXTON EQUATIONS 5 non-degenerate on this space, as the only constant vector fields in g −∞ arethe zero vector fields.The r -HS equation has been recently derived by Cotter et al. [8] as theFinsler–Euler–Arnold equation on the group of diffeomorphism with respectto the Finsler metric F . As the following theorem shows, in these non-compact setting one has to pay careful attention to choose the appropriatedecay conditions (c.f. [3] for the case r = 2). As a byproduct we will observethe equivalence of the λ -PJ and the r -HS equations. Theorem 2.1.1.
For r ∈ (1 , ∞ ) , the geodesic equation of the right-invariantFinsler metric F on the Lie group Diff −∞ ( R ) is given by (2.2) r ddt ϕ tx ϕ x (cid:12)(cid:12)(cid:12)(cid:12) ϕ tx ϕ x (cid:12)(cid:12)(cid:12)(cid:12) r − ! + ( r − (cid:12)(cid:12)(cid:12)(cid:12) ϕ tx ϕ x (cid:12)(cid:12)(cid:12)(cid:12) r = 0 . The corresponding Finsler–Euler–Arnold equation — the geodesic equationin Eulerian coordinates u = ϕ t ◦ ϕ − — is the r -Hunter–Saxton equation: (2.3) (cid:0) u x | u x | r − (cid:1) xt + (cid:0) u x | u x | r − (cid:1) x u x + (cid:0) ( u x | u x | r − ) x u (cid:1) x = 0 . This equation is formally equivalent to the non-periodic λ -PJ equations (2.4) u txx + (1 + 2 λ ) u x u xx + uu xxx = 0 , λ = r − ∈ (0 , . On the smaller group
Diff( R ) both the geodesic equations and the Finsler–Euler–Arnold equations do not exist.Proof. The length functional of a Finsler-metric F on a (possibly infinitedimensional) manifold M is defined as(2.5) L ( ϕ ) = Z F ϕ ( ϕ t ) dt, where ϕ : [0 , → M is a path in the manifold and where ϕ t denotes itsderivative. A geodesic is a path that locally minimize the length functional;since L is invariant to reparametrization, we can restrict ourselves to pathsof constant speed. By H¨older inequality, it is immediate that constant speedgeodesics are exactly the local minimizers of the q -energy E q ( ϕ ) = Z F qϕ ( ϕ t ) dt, for any q >
1. In our case, for the ˙ W ,r -Finsler metric the most convenientchoice is to consider the q -Energy with q = r . This leads us to the sameLagrangian as in [8]:(2.6) E r ( ϕ ) = Z Z R (cid:12)(cid:12)(cid:12)(cid:12) ϕ tx ϕ x (cid:12)(cid:12)(cid:12)(cid:12) r ϕ x dxdt. Calculating the variation of E r in direction δϕ and using integration by partsone obtains δE r ( ϕ )( δϕ ) = Z Z R r (cid:12)(cid:12)(cid:12)(cid:12) ϕ tx ϕ x (cid:12)(cid:12)(cid:12)(cid:12) r − ϕ tx ϕ x δϕ tx − ( r − (cid:12)(cid:12)(cid:12)(cid:12) ϕ tx ϕ x (cid:12)(cid:12)(cid:12)(cid:12) r δϕ x dxdt = − Z Z R r ddt (cid:12)(cid:12)(cid:12)(cid:12) ϕ tx ϕ x (cid:12)(cid:12)(cid:12)(cid:12) r − ϕ tx ϕ x ! + ( r − (cid:12)(cid:12)(cid:12)(cid:12) ϕ tx ϕ x (cid:12)(cid:12)(cid:12)(cid:12) r ! δϕ x dxdt. (2.7) MARTIN BAUER, YUXIU LU, AND CY MAOR
Thus we can read off the geodesic equation:(2.8) r ddt ϕ tx ϕ x (cid:12)(cid:12)(cid:12)(cid:12) ϕ tx ϕ x (cid:12)(cid:12)(cid:12)(cid:12) r − ! + ( r − (cid:12)(cid:12)(cid:12)(cid:12) ϕ tx ϕ x (cid:12)(cid:12)(cid:12)(cid:12) r = 0 . A straight-forward calculation shows that, in Eulerian coordinate u = ϕ t ◦ ϕ − , this equation reduces to:(2.9) r | u x | r − ( u tx + u xx u ) + | u x | r = 0 . To see that this is equivalent to the λ -PJ equation (2.4), one integrates thelatter in the variable x . From this the equivalence of the λ -PJ and (2.9)follows by choosing λ = 1 /r .Taking another derivative in the variable x one obtains equation (2.3),which is the r -HS equation as defined in [8, Definition 2.1].To see that these equations do not exist on the smaller group Diff( R ) wedivide equation (2.9) by | u x | r − and integrate it again in the variable x toobtain the formally equivalent equation(2.10) u t = − uu x + (1 − r ) Z x −∞ u x ( t, z ) dz . Note, that the constant of integration is zero, due to the decay assumptionson the the vector fields u . Now the non-existence follows as for any non-trivial initial conditions u ∈ W ∞ , ( R ) (the Lie-algebra of Diff( R )), the term R x −∞ ( u ) x ( z ) dz dominates u ( u ) x for x large enough. This implies that u t (0 , x ) does not decay as x → ∞ , hence that the corresponding solution u ( t, x ) / ∈ W ∞ , ( R ) for any t > r = 2). Alternatively, this can be seen from thesolution formula as presented in Theorem 2.3.1 below. (cid:3) Remark 2.1.2.
Note that F is only a weak Finsler metric, as the the W ,r topology is weaker than the original C ∞ -manifold topology. As aconsequence several results of finite dimensional Riemannian geometry donot hold in this setting. In particular, it is not guaranteed that the geodesicdistance function defines a true metric, as it can be degenerate or even vanishidentically. As a byproduct of the analysis in the following sections, we willobtain an explicit formula for this distance function. This will in particularimply that the geodesic distance of the ˙ W ,r -metric does not admit thismisbehavior and is indeed inducing a true distance function.2.2. An isometry to a flat space.
In this section we introduce an isom-etry that will map the diffeomorphism group with the right-invariant ˙ W ,r -Finsler metric to an open subset of a vector space. As mentioned in theintroduction, this construction, which is a generalization of [15, 16, 3] forthe r = 2 case, will allow us to obtain explicit formulas for solutions to thegeodesic equation on the diffeomorphism group and consequently also forthe r -HS ( λ -PJ, resp.) equations: Theorem 2.2.1.
For r ∈ [1 , ∞ ) , the mapping (2.11) Φ : (Diff −∞ ( R ) , F ) → (cid:0) W ∞ , ( R ) , L r (cid:1) ϕ r (cid:18) ϕ r x − (cid:19) ENERALIZED PROUDMAN–JOHNSON AND r -HUNTER–SAXTON EQUATIONS 7 is an isometric embedding. Furthermore, the image U = Φ(Diff −∞ ( R )) isthe set of all positive functions in W ∞ , ( R ) , i.e., (2.12) U = { f ∈ W ∞ , ( R ) : f > − r } . The inverse of Φ is given by (2.13) Φ − : ( W ∞ , ( R ) → Diff −∞ ( R ) f x + R x −∞ (cid:16)(cid:16) f (˜ x ) r + 1 (cid:17) r − (cid:17) d ˜ x. Proof.
First, note that since ϕ x − ∈ W ∞ , ( R ) and non-negative, a straight-forward calculation shows that ϕ /rx − W ∞ , ( R ) (since (1+ α ) /r < r α ). Similarly, it is easy to see that the image of the inverse is indeedin Diff −∞ ( R ). Next, we calculate the variation formula of the mapping Φ.We have: D ϕ,h Φ = ϕ x r − h x (2.14)and thus k D ϕ,h Φ k rL r = Z R | ϕ x r − h x | r dx = Z R ϕ x − r | h x | r dx = F ϕ ( h ) r . (2.15)It remains to prove the statement on the image. Let f = r ( ϕ x r −
1) =Φ( ϕ ) for some ϕ ∈ Diff −∞ ( R ). Since elements of Diff −∞ ( R ) are orientationpreserving diffeomorphisms, this implies that ϕ x >
0. Thus it follows that f > − r , which concludes the characterization of the image. The statementon the inverse follows by direct calculation. (cid:3) A solution formula for the r -HS ( λ -PJ) equation. As a conse-quence of Theorem 2.2.1 and the simple form of the image of Φ we obtainan explicit formula for geodesics on (Diff −∞ ( R ) , F ). Theorem 2.3.1.
Let r ∈ (1 , ∞ ) . Given initial conditions ϕ (0) = id ∈ Diff −∞ ( R ) , ϕ t (0) = u ∈ T id Diff −∞ ( R ) = g −∞ the unique solution to the geodesic equation of F is given by: (2.16) ϕ ( t, x ) = x + Z x −∞ (cid:18)(cid:18) tu ′ ( y ) r + 1 (cid:19) r − (cid:19) dy. Consequently, the solution to the r -HS (equivalently, λ -PJ with λ = r − )equation with initial condition u (0) = u ∈ g −∞ is given by u = ϕ ( t, ϕ − ( t, x )) with ϕ given by (2.16) .In particular, this implies that the equations are well-defined on Diff −∞ ( R ) ,and that these solutions are length minimizing paths with respect to the dis-tance function induced by the ˙ W ,r -metic. Remark 2.3.2.
As we will discuss in Section 2.10 below, formula (2.16),with r = λ − , provides a solution to the λ -PJ equation for every λ = 0;however for λ / ∈ (0 ,
1) we do not have an interpretation of the equation as ageodesic equation. The case λ = 0, which is equivalent to r = ∞ , is treatedin Section 2.8. MARTIN BAUER, YUXIU LU, AND CY MAOR
Proof.
By Theorem 2.2.1 the mapping Φ is an isometry from (Diff −∞ ( R ) , F )to an open subset of the vector space ( W ∞ , ( R ) , L r ). Thus geodesics onthe former space are the pre-images of Φ of geodesics in the image, i.e.,pre-images of straight lines and thus the formula follows directly from theinversion formula (2.13).Since straight lines are length minimizing in ( W ∞ , ( R ) , L r ), so are theirpre-images in Diff −∞ ( R ). (cid:3) Geodesic incompleteness and blowup of the r -HS ( λ -PJ) equa-tion. As a direct consequence of the geometric interpretation in Theo-rem 2.3.1 we obtain the following result concerning geodesic incompletenessof (Diff −∞ ( R ) , F ). This, in turn, implies blow-up for the r -HS ( λ -PJ resp.)equation. The following theorem is in correspondence with the blow-upresult in [8, Theorem 3.3]. Corollary 2.4.1.
Let r ∈ (1 , ∞ ) . The space (Diff −∞ ( R ) , F ) is geodesicallyincomplete. More precisely, given any initial conditions ϕ (0) = id ∈ Diff −∞ ( R ) , ϕ t (0) = u ∈ T id Diff −∞ ( R ) = g −∞ the geodesic as given by formula (2.16) exists for all time t > if and onlyif u ′ ( x ) ≥ for all x ∈ R . If there exists a point x with u ′ ( x ) < thenthe geodesic only exists for finite time T ∗ ( u ) , with (2.17) T ∗ ( u ) = − r inf x ∈ R u ′ ( x ) At time T ∗ ( u ) we have min x ∈ S ϕ x ( T ∗ ( u ) , x ) = 0 and thus the corre-sponding solution u = ϕ t ◦ ϕ − to the r -HS (equivalently, λ -PJ with λ = r − )equation blows up at the same time. Remark 2.4.2.
Note that this shows that every geodesics blows up either infinite positive or finite negative time. Furthermore, we see that the maximalexistence time of geodesics grows linearly with the parameter r . Proof.
To obtain this result, we only need to observe that a geodesics ϕ ( t, x )in (Diff −∞ ( R ) , F ) ceases to exists if and only if ϕ x ( t, x ) approaches zero(as there is no loss of regularity). By the solution formula (2.16) this isequivalent to:(2.18) (cid:18) tu ′ ( x ) r + 1 (cid:19) r = 0 , which directly leads to the desired statement. Note that ϕ x ( t, x ) = 0 impliesthat u = ϕ t ◦ ϕ − blows up. (cid:3) The metric completion.
In this section we will calculate the metriccompletion. Note that the solution formula (2.16) is no longer well-definedon this larger space, as u ∈ AC ( R ) does not imply that u ′ ∈ L r ( R ). Wewill study a slightly smaller functions space such that the solution formulais still well-defined later in Section 2.7. Proposition 2.5.1.
Let r ∈ (1 , ∞ ) . The metric completion of (cid:0) Diff −∞ ( R ) , dist F (cid:1) is the monoid of all absolutely contionous surjective maps: Mon( R ) = (cid:26) ϕ ∈ AC ( R ) : ϕ is surj., ϕ ′ ≥ a.e. and lim x →−∞ ( ϕ ( x ) − x ) = 0 (cid:27) . ENERALIZED PROUDMAN–JOHNSON AND r -HUNTER–SAXTON EQUATIONS 9 Proof.
To calculate the metric completion of (cid:0)
Diff −∞ ( R ) , dist F (cid:1) it is suffi-cient to study the metric completion of the image U of Φ, i.e., the L r closureof U , which is given by(2.19) U = { f ∈ L r ( R ) : f ≥ − r a.e. } . Using the inversion formula for Φ it is easy to show that the elements in thepre-image of Φ are precisely the elements of Mon( S ). (cid:3) Geodesic convexity.
In the previous sections we have seen that theinitial value problem is locally well-posed, but that solutions might blow-upin finite time. In this section we show the geodesic boundary value problemis better behaved. Namely, we show that for any any boundary conditions ϕ , ϕ ∈ Diff −∞ ( R ) there exists a minimizing geodesic. This is, again, aconsequence of the fact that our space is isometric to a convex open subsetof a vector space. Corollary 2.6.1.
For every r ∈ (1 , ∞ ) , the space (Diff −∞ ( R ) , F ) is geodesi-cally convex. More precisely, given any boundary conditions ϕ (0) = id ∈ Diff −∞ ( R ) , ϕ (1) = ϕ ∈ Diff −∞ ( R ) there exists a unique minimizing geodesic connecting id to ϕ .Proof. This result follows immediately from the convexity of the set U as asubset of an infinite dimensional vector space. (cid:3) Extensions to low regularity.
The analysis above holds for spacesof lower regularity: all the constructions hold without any major change onany space of diffeomorphisms of R for which the following holds: (i) it isclosed under taking inverse; (ii) composition from the right is smooth; (iii)the ˙ W ,r Finsler norms make sense on it. A natural space to consider inthis case, which includes Diff −∞ ( R ), is the space of integrable bi-Lipschitzhomeomorphisms:(2.20) biLip , −∞ ( R ) := n ϕ = id + f : ϕ is invertible, ϕ − = id + g,f, g ∈ ˙ W , ( R ) ∩ W , ∞ ( R ) , lim x →−∞ f ( x ) = 0 o . Here ˙ W , ( R ) is the space of functions with an integrable derivative. Thisspace is a manifold and a topological group, in which composition from theright is smooth (thus it is a half Lie-group in the terminology of [14, 18]).Its Lie-algebra is the space lip , −∞ ( R ) := { u ∈ W , ∞ ( R ) ∩ ˙ W , ( R ) : lim x →−∞ u ( x ) = 0 } , on which all the ˙ W ,r Finsler norms are well defined and also the r → ∞ limit make sense (see below). Moreover, the basic mapping and the solutionformula from Theorem 2.3.1 naturally extend to this space; extensions toany larger space of functions seem difficult.The space biLip , −∞ ( R ) includes, in particular, all bi-Lipschitz piecewise-linear homeomorphisms of R that decay at −∞ :Diff PL −∞ ( R ) := n ϕ = id + f ∈ biLip , −∞ ( R ) : f ′ is piecewise constant o . Using the solution map (2.16) we immediately obtain the following resultconcerning this submanifold:
Corollary 2.7.1.
For every r ∈ (1 , ∞ ) , the space Diff PL −∞ ( R ) is totally ge-odesic in biLip , −∞ ( R ) with respect to the flow of the r -Hunter–Saxton equa-tion. That is, any solution whose initial velocity u ∈ lip , −∞ ( R ) is piecewise-linear, remains in Diff PL −∞ ( R ) as long as the flow exists. These piecewise linear solutions were studied in detail in [8], and thiscorollary provides a geometric framework for them.2.8.
The limit r → ∞ . In the following we study the limiting behaviorof r -HS as r → ∞ . In particular, we show that the limiting solutions aresolutions of the λ -PJ equation for λ = 0, and that the ˙ W , ∞ -norm of thevelocity is preserved along the flow, thus answering several questions posedby Cotter et al. [8]. Theorem 2.8.1 (Lagrangian viewpoint) . Let ϕ r ( t, x ) be the solution of the r -Hunter–Saxton equation with initial conditions ϕ r (0 , x ) = x and ϕ rt (0 , x ) = u ∈ g −∞ . Then we have the pointwise limit (2.21) lim r →∞ ϕ r ( t, x ) = x + Z x −∞ (cid:16) e tu ′ ( y ) − (cid:17) dy =: ϕ ∞ ( t, x ) , which exists for all time. The function ϕ ∞ ( t, x ) ∈ Diff −∞ ( R ) satisfies thedifferential equation (2.22) (cid:18) ϕ tx ϕ x (cid:19) t = 0 , which is the formal limit of equation (2.2) as r → ∞ . Corollary 2.8.2 (Eulerian viewpoint) . The vector field associated with (2.21) satisfies the equation (2.23) u xt + u xx u = 0 , which is the Eulerian version of (2.22) . By differentiating, this equation isequivalent to the λ -PJ equation with λ = 0 .The derivative of the vector field satisfies (2.24) u ∞ x ( t, x ) = u ′ (( ϕ ∞ ) − ( t, x )) , and thus the ˙ W , ∞ -norm of u is preserved along the flow.Proof of Theorem 2.8.1. Taking the limit r → ∞ of ϕ r as given by thesolution formula (2.16), we immediately obtain (2.21) by the monotone con-vergence theorem. A straight-forward calculation shows that (2.21) satisfiesthe equation (2.22).Writing (2.2) as (cid:18) r − (cid:12)(cid:12)(cid:12)(cid:12) ϕ tx ϕ x (cid:12)(cid:12)(cid:12)(cid:12) + ϕ tx ϕ x (cid:19) (cid:18) ϕ tx ϕ x (cid:19) t + r − r ( r − (cid:12)(cid:12)(cid:12)(cid:12) ϕ tx ϕ x (cid:12)(cid:12)(cid:12)(cid:12) = 0 , and taking the formal limit r → ∞ , we obtain (2.22) (similarly, (2.23) isformally obtained from (2.10) by differentiating and taking r → ∞ ). (cid:3) ENERALIZED PROUDMAN–JOHNSON AND r -HUNTER–SAXTON EQUATIONS 11 Proof of Corollary 2.8.2.
From (2.21) we have that ϕ ∞ x ( t, x ) = e tu ′ ( x ) and ϕ ∞ xt ( t, x ) = u ′ ( x ) e tu ′ ( x ) , and thus u ∞ x ( t, ϕ ( t, x )) = ϕ ∞ xt ( t, x ) ϕ ∞ x ( t, x ) = u ′ ( x ) , from which (2.24) follows, and thus k u ∞ x ( t, · ) k ∞ = k u ′ k ∞ , which proves the preservation of the ˙ W , ∞ -norm. Differentiating (2.24) andusing (2.22) immediately shows that u ∞ satisfies (2.23). (cid:3) r = 1 and Burgers’ equation. In [8, Remark 3.4] it was observed thatthe 1-HS equation is formally equivalent to the inviscid Burgers’ equation.In the following we will give a geometric interpretation for this phenomenon.The inviscid Burgers’ equation u t + uu x = 0 , in Lagrangian coordinates, is transformed to the straight line equation ϕ tt =0, where ϕ t = u ◦ ϕ . Thus, the Burgers’ equation can be interpreted as (isequivalent to) the geodesic equation of any Finsler metric on Diff( R ), wheregeodesics are given by straight lines. This happens, as is widely known, forthe flat L metric, but also, in fact, for any flat (non-invariant) W k,r -Finslermetric, ¯ F ϕ ( h ) = k h k W k,r , as well as their homogeneous counterparts. Here, by flat we mean that thenorm of h is independent of the footpoint ϕ , unlike invariant metrics whichare the focus of this paper, which correspond to taking the appropriate normof h ◦ ϕ − .However, for the specific case of the homogeneous W , -Finsler metric,the flat (non-invariant) metric and the invariant metric coincide: F ϕ ( h ) = Z R | ( h ◦ ϕ − ) ′ | dx = Z R (cid:12)(cid:12)(cid:12)(cid:16) h ′ ϕ ′ (cid:17) ◦ ϕ − (cid:12)(cid:12)(cid:12) dx = Z R | h ′ | dx = ¯ F ϕ ( h ) , and thus the geodesic equation of the ˙ W , -Finsler metric, which formallycorresponds to the 1-HS equation, is equivalent to the Burgers’ equation.Note that this is the only value for r and k such, that the non-invariant andinvariant W k,r -metrics are equal.2.10. The range r < . In this section we discuss the parameter range r < λ ∈ ( −∞ , ∪ (1 , ∞ )). This is of particularimportance as it includes the original PJ equations — r = λ = − L r is only a normed space for r ≥ F does not define a Finsler structure if r <
1. Consequently we do not havea geometric interpretation of the λ -PJ ( r -HS, resp.) equation as a geodesicequation of a Finsler metric on a diffeomorphism group. However, muchof our analysis, including the explicit solution formula does go through, atleast formally. First, we note that for any λ ∈ R , the λ -PJ equation, in Lagrangiancoordinates, can be written as(2.25) ddt (cid:18) ϕ xt ϕ x (cid:19) + λ (cid:18) ϕ xt ϕ x (cid:19) = 0 , which is equivalent to (2.2) (with r = λ − ) as long as r = 0 ,
1. Thus, theproof of Theorem 2.1.1 shows, that at least formally (without worrying aboutintegrability or differentiability of the functions involved), the λ -PJ equationis equivalent to the Euler-Lagrange equations of the invariant energy E λ − ( ϕ ) = Z Z R (cid:12)(cid:12)(cid:12)(cid:12) ϕ xt ϕ x (cid:12)(cid:12)(cid:12)(cid:12) /λ ϕ x dx dt for any λ = 0 ,
1, and not only for λ ∈ (0 ,
1) as in Theorem 2.1.1. However,for λ / ∈ (0 , λ < r < Z Z R f (cid:18) ϕ xt ϕ x (cid:19) ϕ x dx dt, that yield the λ -PJ equation, as a straightforward calculation shows that f must satisfy λs f ′′ ( s ) − sf ′ ( s ) + f ( s ) = 0 , f ′′ , the solutions of which are f ( s ) = s /λ as above (in the case λ = r = 1 wealso get f ( s ) = s log s ).Finally, a straightforward calculation shows that (2.16) indeed solves(2.25), for 0 = r = λ − , which leads to the following corollary: Corollary 2.10.1.
Let r ∈ R \ { } . Given initial conditions u ∈ g −∞ the unique solution to the r -HS equation ( r − -PJ, resp.) is given by u = ϕ ( t, ϕ − ( t, x )) , where ϕ is given by (2.16) . Note that formula (2.17) for the blowup time still holds whenever r > r < λ < ϕ if and only if u ′ ( x ) > x (the exact converse of the case r > T ∗ ( u ) = | r | sup x ∈ R u ′ ( x ) . The nature of the blowup is alsodifferent between r > r <
0: in the former, ϕ loses the immersionproperty at the blowup time (that is, we have ϕ x ( T ∗ , x ) = 0 at some point);in the latter, ϕ x blows up at some point at the blowup time. For a detailedstudy of blowup in this range for the periodic λ -PJ equation, see Sarriaand Saxton [24] and the references therein (note that their parameter λ corresponds to our − λ ). 3. The periodic case
In this section we briefly discuss the periodic situation. From a functionalanalytic point of view the compactness of the domain simplifies the situa-tion — there are no decay conditions required to equip the diffeomorphismgroup with a manifold structure. The geometric picture, however, is morecomplicated: we will show that the diffeomorphism group with the ˙ W ,r metric is isometric to an open subset of an L r -sphere. ENERALIZED PROUDMAN–JOHNSON AND r -HUNTER–SAXTON EQUATIONS 13 The derivation of the equations themselves is similar to the non-periodiccase, as well as the generalization to lower regularity, and thus we do notrepeat them here.3.1.
Diffeomorphisms on the circle and the r -HS ( λ -PJ) equation. We start by introducing the group of smooth, orientation preserving diffeo-morphims on the circle, i.e., we consider the space(3.1) Diff( S ) = (cid:8) ϕ ∈ C ∞ ( S , S ) : ϕ ′ > , ϕ − ∈ C ∞ ( S , S ) (cid:9) . It is well known (see, e.g., [22]) that the space Diff( S ) is a smooth, infinitedimensional Fr´echet Lie-group with Lie-algebra g = C ∞ ( S ) the space ofvector fields on S . The right-invariant ˙ W ,r -Finsler metric, as introducedin (2.1) with integration over R replaced by integration over S , is only adegenerate Finsler metric on this group (constant vector fields are in thekernel). This leads us to consider the metric on the homogenous spaceRot \ Diff( S ) of Sobolev diffeomorphisms modulo rotations, which we willidentify with the section(3.2) Diff ( S ) = (cid:8) ϕ ∈ Diff( S ) : ϕ (0) = 0 (cid:9) , where we identified the circle S with the interval [0 , ( S ) consists of the space C ∞ ( S ) := (cid:8) u ∈ C ∞ ( S ) : u (0) = 0 (cid:9) . As the kernel of the norm F consists exactly of all constant vector fields, It isnow easy to see that the norm F as defined in (2.1) defines a right-invariantFinsler metric on Diff ( S ) (recall that the only constant tangent vector toDiff ( S ) is the zero vector field). The periodic r -HS (periodic λ -PJ resp.)equation can now be interpreted as the Finsler–Euler–Arnold equation onDiff ( S ), i.e., the analogue of the existence part of Theorem 2.1.1 holds inthe periodic case.Also, similarly to Section 2.7, this formulation allows us to consider theseequations in lower regularity, namely on the space of bi-Lipschitz, orientationpreserving homeomorphisms (modulo rotations):biLip ( S ) := n ϕ ∈ W , ∞ ( S ) : ϕ ′ > , ϕ − ∈ W , ∞ ( S ) , ϕ (0) = 0 o , whose Lie-algebra is the space of Lipschitz velocities lip ( S ) := { u ∈ W , ∞ ( S ) : u (0) = 0 } . An isometry to the sphere.
We will now construct an isometry thatwill map the diffeomorphism group with the right-invariant ˙ W ,r -Finslermetric to an infinite dimensional L r -sphere of radius r . As mentioned ear-lier, this transformation was already used by Sarria and Saxton to studythese equations [24]. The merit of this section is in revealing the geometricmeaning of it. Also, we hope that this viewpoint will provide further toolsto study the λ -PJ equations by studying the L r -sphere (in a similar way tothe Hunter–Saxton case [15]). It was also used as a tool for studying the diameter of diffeomorphism groups [5].
Theorem 3.2.1.
Let r ∈ [1 , ∞ ) . The mapping (3.3) Φ : ((cid:0) Diff ( S ) , F (cid:1) → (cid:0) C ∞ ( S ) , L r (cid:1) ϕ r ϕ x r is an isometric embedding. Furthermore, the image U = Φ(Diff ( S )) is anopen subset of the L r -sphere of radius r given by (3.4) U = { f ∈ C ∞ ( S ) : f > , k f k L r = r } . The inverse of Φ is given by (3.5) Φ − : ( U →
Diff ( S ) f r r R x | f (˜ x ) | r d ˜ x. Proof.
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Martin Bauer: Department of Mathematics, Florida State University
Email address : [email protected] Yuxiu Lu: Department of Mathematics, Florida State University
Email address : [email protected] Cy Maor: Einstein Institute of Mathematics, The Hebrew University ofJerusalem
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