Global results for a Cauchy problem related to biharmonic wave maps
aa r X i v : . [ m a t h . A P ] F e b GLOBAL RESULTS FOR A CAUCHY PROBLEMRELATED TO BIHARMONIC WAVE MAPS
TOBIAS SCHMIDAbstract. We prove global existence of a derivative bi-harmonic wave equation with a non-generic quadratic nonlinearity and small initial data in the scaling critical space ˙ B , d ( R d ) × ˙ B , d − ( R d ) for d ≥ . Since the solution persists higher regularity of the initial data, we obtain a smalldata global regularity result for the biharmonic wave maps equation for a certain class of targetmanifolds including the sphere. Introduction
In the following we consider critical points of an extrinsic (rigid) action functional Φ( u ) = 12 Z R Z R d | ∂ t u | − | ∆ u | dx dt, (1.1)for smooth maps u : R × R d → S L − into the round sphere S L − ⊂ R L . Taking smoothvariations u δ : R × R d → S L − with u δ − u having compact support and vanishing at δ = 0 ,the critcal points satisfy ∂ t u + ∆ u ⊥ T u S L − (1.2)pointwise on R × R d . The variation of (1.1) thus gives rise to Hamiltonian equations with elastic energy functional E ( u ( t )) = 12 Z R d | ∂ t u ( t ) | + | ∆ u ( t ) | dx. (1.3)Evaluating the Euler-Lagrange equation (1.2), we infer that critical maps u of (1.1) are solutionsof the following biharmonic wave maps equation ∂ t u + ∆ u = −| ∂ t u | u − ∆( |∇ u | ) u (1.4) − ( ∇ · h ∆ u, ∇ u i ) u − h∇ ∆ u, ∇ u i u = − ( | ∂ t u | + | ∆ u | + 4 h∇ u, ∇ ∆ u i + 2 h∇ u, ∇ u i ) u, where ∆ denotes the bi-Laplacien ∆ = ∆(∆ · ) = ∂ ij ∂ ij and h∇ u, ∇ u i = h ∂ i ∂ j u, ∂ j ∂ i u i .Hence (1.4) is considered to be a fourth order analogue of the (spherical) wave maps equation − ∂ t u + ∆ u = ( | ∂ t u | − |∇ u | ) u, which has been studied intensively in the past concerning wellposedness, regularity and gaugeinvariance, see e.g. the surveys [26], [15]. For the general wave maps equation of the form(1.5) (cid:3) u = Γ( u )( ∂ α u, ∂ α u )) = ˜Γ( u )( (cid:3) ( u · u ) − u · (cid:3) u ) local wellposedness holds almost optimal for (scaling) subcritical regularity in H s ( R d ) × H s − ( R d ) with s > d . This relies on the null condition for (1.5) as seen in the proof ofKlainerman-Machedon in [12] for d ≥ (and Klainerman-Selberg in [13] for d = 2 ). In fact,a counterexample of Lindblad in [18] shows that if this condition is absent in a generic waveequation, the sharp regularity for local existence is strictly above d/ .Many advances towards (critical) regularity s = d/ lead to insights for the wave maps equationwith impact on related equations. For instance, global solutions with small initial data in the Mathematics Subject Classification.
Primary: 35A01. Secondary: 35G50.
Key words and phrases. biharmonic wave map, biharmonic map, fourth order, wave equation. space ˙ H d × ˙ H d − were constructed by Tao in [22] ( d ≥ ) for sphere targets using a novelmicrolocal renormalization procedure. The (2 + 1) dimensional case, i.e. small data in theenergy space ˙ H ( R ) × L ( R ) , was for example treated by Krieger in [16] with H targetspace, Tao in [23] for the sphere target and Tataru [27] for more general targets. Global wavemaps with large initial energy were considered by Krieger-Schlag in [17] (for the H target) andin the analysis of Sterbenz-Tataru in [20], [21]. Since the literature is vast and the list is notexhaustive, we refer e.g. to [6] for a general overview.In this article, we study the analogue of the division problem for wave maps with small datain ˙ B , d ( R d ) × ˙ B , d − ( R d ) , which has been solved by Tataru in [24] (for d ≥ ) and in lowdimension [25] (i.e. for d = 2 , ) by the use of null-frame estimates . More recently, the divisionproblem for wave maps (in dimension d ≥ ) has also been solved in a U based space byCandy-Herr in [4]. For (1.4), we achieve to solve the division problem in dimension d ≥ usingspaces Z, W = L ( Z ) which are the analogues of Tataru’s F, (cid:3) F spaces in [24]. Especially L : Z → W is a continuous operator. The results in this article are part of the authors PhDthesis [19].1.1. The Cauchy problem and outline.
We consider the following generalized Cauchy prob-lem ∂ t u + ∆ u = Q u ( u t , u t ) + Q u (∆ u, ∆ u ) + 2 Q u ( ∇ u, ∇ ∆ u )+2 Q u ( ∇ ∆ u, ∇ u ) + 2 Q u ( ∇ u, ∇ u ) =: Q ( u ) , ( u (0) , ∂ t u (0)) = ( u , u ) (1.6)where Q J ( u ) = [ Q u ] JK,M ( ∂ t u K ∂ t u M ) + [ Q u ] JK,M (∆ u K ∆ u M ) + 2[ Q u ] JK,M ( ∂ i u K ∂ i ∆ u M )+ 2[ Q u ] JK,M ( ∂ i ∆ u K ∂ i u M ) + 2[ Q u ] JK,M ( ∂ i ∂ j u K ∂ j ∂ i u M ) , and { Q x | x ∈ R L } is a smooth family of bilinear forms (in fact required to be analytic atthe origin). Here we contract the derivatives over i = 1 , . . . , d and the components of u over K, M, J ∈ { , . . . , L } . The bilinear term Q ( u ) in (1.6) is non-generic for our results, in thesense that for bilinear interactions, the set of resonances (cid:8) (( τ , ξ ) , ( τ , ξ )) | ( τ + τ ) − | ξ + ξ | = τ + τ − | ξ | − | ξ | (cid:9) , is canceled by Q ( u ) . We use this fact in the form of the following commutator identity for theoperator L = ∂ t + ∆ Q ( u ) = 12 Q u ( L ( u · u ) − u · Lu − Lu · u ) (1.7) = 12 [ Q u ] K,M ( L ( u K · u M ) − u K · Lu M − u M · Lu K ) . This will then be exploited following the work of Tataru in [24], [25] for wave maps. To beprecise, the idea used in Tataru’s F, (cid:3) F spaces from [24] allow to treat Q ( u ) by continuity of L . As a consequence, we find a simple way to solve the disvision problem for (1.6) even in lowdimensions compared to the energy scaling (of (1.3)) for biharmonic wave maps (1.4), see e.g.the remark 1.4 below. However, we do not obtain scattering at t → ±∞ from this approach.The main difference to [24] is that we have to use the control of a lateral Strichartz space and amaximal function bound in order to exploit a smoothing effect for the Schrödinger group. Moredetails are given below.The following second Cauchy problem will be solved with the same approach (presented inthe following Sections) and further (in Section 4) applies to solve (1.2) for more general targetmanifolds.The general biharmonic wave maps equation for maps u : [0 , T ) × R d → N into a generalembedded manifold N ⊂ R L reads similarly as to the spherical case Lu ⊥ T u N on (0 , T ) × R d . LOBAL RESULTS FOR BIHARMONIC WAVE MAPS 3
Via the smooth family of orthogonal tangent projector P u : R L → T u N , this is equivalent to Lu = ( I − P u )( Lu ) = ( I − d Π u )( Lu ) , (1.8)where Π is nearest point projector Π : V ε ( N ) → N, | Π( p ) − p | = inf q ∈ N | q − p | withthe identity d Π u = P u in case u maps to N and V ε ( N ) = { p | dist ( p, N ) < ε } . Using ( ∂ t , ∇ u ) ∈ T u N (1.8) is expanded into a semilinear equation, where in contrast to (1.4), tri-linear and quadri-linear terms appear on the RHS. For the sake of readability, we give theexpansion of (1.8) in Section 4. We hence consider the Cauchy problem Lu = L (Π( u )) − d Π u ( Lu )( u (0) , ∂ t u (0)) = ( u , u ) , (1.9)where Π : R L → R L is smooth and real analytic at x = 0 . Calculating the series expansion of Π in the RHS of (1.9), we infer formally L (Π( u )) − d Π u ( Lu ) = X k ≥ C k d k Π ( L ( u k ) − ku k − Lu ) . Thus the nonlinearity on the RHS reduces the same non-resonant form (1.7), where it is laterjustified, by the spaces we use, to commute L with the series expansion. The resulting theoremis given below in Corollary 1.2.At least formally, Duhamel’s formula is given by (cid:18) u ( t ) u t ( t ) (cid:19) = S ( t ) · (cid:18) u u (cid:19) + Z t S ( t − s ) · (cid:18) Lu ( s ) (cid:19) ds, (1.10)where S ( t ) = (cid:18) cos(( − ∆) t ) ( − ∆) − sin(( − ∆) t )∆ sin(( − ∆) t ) cos(( − ∆) t ) (cid:19) = 12 Q − (cid:18) e − it ∆ e it ∆ (cid:19) Q with Q = (cid:18) − i ∆ 1 i ∆ 1 (cid:19) . (1.11)Thus, in the analysis for biharmonic wave maps (1.4), it is in principle possible to exploitmethods developed for derivative Schrödinger equations, which will become apparent below.Results on the division problem for Schrödinger maps, see e.g. [1], [10], involve versions oflateral Strichartz estimates in the norm ( x x e e + x e ⊥ , e ∈ S d − ) k f k pL pe L qt,e ⊥ = Z ∞−∞ Z [ e ] ⊥ Z ∞−∞ | f ( t, re + x ) | q dt dx ! pq dr, in order to exploit smoothing effects for Schrödinger equations, see the Appendix A. Especially,we likewise rely on factoring L ∞ e L t,e ⊥ · L e L ∞ t,e ⊥ ⊂ L t,x , where the (lateral) energy L ∞ e L t,e ⊥ gives additional regularity of order |∇| and the maximalfunction bound L e L ∞ t,e ⊥ is controlled uniform in e ∈ S d − . Apart from the usual Strichartzspace S λ , this will be essential (in one particular frequency interaction) in Section 3.The operator Lu = ∂ t u + ∆ u appears in the Euler-Bernoulli beam model ( d = 1 ) and ineffective thin-plate equations ( d = 2 ) such as the Kirchoff- and Von Kármán elastic plate modelswith small plate deflections if rotational forces are neglected. As a reference we mention e.g. [5],where this situation has been considered explicitly with (nonlinear) boundary dissipation. Outline of the article
TOBIAS SCHMID
In Section 2.2, we provide (lateral) Strichartz estimates L pe L qt,e ⊥ and the L e L ∞ t,e ⊥ estimate for thelinear Cauchy problem of the operator L = ∂ t +∆ . This is a consequence of the correspondingestimates for e ± it ∆ which orginally appeared in [9], [10] and [1]. In the Appendix A, we brieflyoutline proofs of the Strichartz estimates we need for e ± it ∆ based on the calculation by Bejenaruin [1].In Section 2.3, we construct spaces Z d , W d such that Z d ⊂ C ( R , ˙ B , d ( R d )) ∩ ˙ C ( R , ˙ B , d − ( R d )) , (1.12) k u k Z d . k ( u , u ) k ˙ B , d × ˙ B , d − + k Lu k W d , (1.13)and similar Z s , W s for s > d with data in ˙ H s ( R d ) × ˙ H s − ( R d ) .Further, we prove the algebra properties Z d · Z d ⊂ Z d , (1.14) W d · Z d ⊂ W d , (1.15)in Section 3. For the higher regularity, we need to provide the following embeddings ( Z d ∩ Z s ) · ( Z d ∩ Z s ) ⊂ Z d ∩ Z s , (1.16) ( W d ∩ W s ) · ( Z d ∩ Z s ) ⊂ W d ∩ W s . (1.17)To be more precise it suffices, as in [24] and [1], to conclude from the dyadic estimates k uv k Z s . k u k Z s k v k Z d + k v k Z s k u k Z d , u, v ∈ Z d ∩ Z s , (1.18) k uv k W s . k u k W s k v k Z d + k v k Z s k u k W d , u ∈ W d ∩ W s , v ∈ Z d ∩ Z s , (1.19)which is outlined in Section 3 for s > d . Finally, we sketch the fixed point argument from [24]and the application to biharmonic wave maps stated in Corollary 1.2 in Section 4.We emphasize that the construction of the dyadic blocks Z λ , W λ are the analogues of Tataru’s F λ , (cid:3) F λ spaces in [24], since we globally bound Lu in the spaces L t L x . In particular, theoperator L : Z λ → W λ is continuous by construction of Z λ and W λ . Combining this with (1.14) and (1.15), it sufficesto estimate Q ( u ) in (1.6) with the identity (1.7).As mentioned above, we can not fully rely on the usual Strichartz norm and have to usethe control of the lateral Strichartz norm, which exploits additional smoothing in the proofof (1.14). This idea has been used in the similar context of the Schrödinger maps flow byIonescu-Kenig [9], [10], Bejenaru [1] and Bejenaru-Ionescu-Kenig [2].1.2.
The main results.
The system (1.6) is largely motivated by biharmonic wave maps,however the results for (1.6) are based on the structural extension of evolution equtions with anonlinearity that, due to (1.7), can be considered non-generic .We turn to general systems (1.6) and (1.9) for functions u , . . . , u L with L ∈ N , where weassume that x Q x , x Π( x ) are real analytic in the point x = 0 . Theorem 1.1. (i)
For d ≥ there exists δ > sufficiently small such that the followingholds. Let ( u , u ) ∈ ˙ B , d ( R d ) × ˙ B , nd − ( R d ) with k u k ˙ B , d ( R d ) + k u k ˙ B , d − ( R d ) ≤ δ. (1.20) Then (1.6) and (1.9) have a global solution u ∈ C ( R , ˙ B , d ( R d )) ∩ ˙ C ( R , ˙ B , d − ( R d )) with sup t ≥ (cid:0) k u ( t ) k ˙ B , d ( R d ) + k ∂ t u ( t ) k ˙ B , d − ( R d ) (cid:1) ≤ Cδ, (1.21)
LOBAL RESULTS FOR BIHARMONIC WAVE MAPS 5 for some
C > . Further, the solution depends Lipschitz on the initial data. (ii) If additionally ( u , u ) ∈ ˙ H s ( R d ) × ˙ H s − ( R d ) for some s > d , then also ( u ( t ) , ∂ t u ( t )) ∈ ˙ H s ( R d ) × ˙ H s − ( R d ) for all t ∈ R and in fact sup t ≥ (cid:0) k u ( t ) k ˙ H s ( R d ) + k ∂ t u ( t ) k ˙ H s − ( R d ) (cid:1) ≤ C ( k u k ˙ H s ( R d ) + k u k ˙ H s − ( R d ) ) . This theorem applies to (1.4), however it is not clear if the solution maps to S L − for all times.This is proven within the following (slightly more general) setup. Let N ⊂ R L be an embeddedmanifold and such that the nearest point projector Π : V ε ( N ) → N is analytic on N with auniform lower bound on the radius of convergence. An explicit example is a uniformly analyticpertubation of the round sphere S L − . Corollary 1.2.
Let ( u , u ) : R d → T N , i.e. u ∈ N, u ∈ T u N , be a smooth map suchthat supp ( ∇ u , u ) is compact, d ≥ . Then if k u k ˙ B , d ( R d ) + k u k ˙ B , d − ( R d ) ≤ δ, where δ = δ ( d, N ) > is sufficienty small, then (1.8) , i.e. ∂ t u + ∆ u ⊥ T u N has a global smooth solution u : R × R d → N with ( u (0) , ∂ t u (0)) = ( u , u ) . Remark 1.3.
The statement of Corollary 1.2 has to be rigorously corrected to u − p ∈ ˙ B , d ( R d ) for p = lim x →∞ u ( x ) since u : R d → N has no decay.In [8] the authors proved local wellposedness of the Cauchy problem for (1.2), resp. theexpansion of (1.8) for general compact target manifolds N , in the Sobolev space H k × H k − for k ∈ Z with k > ⌊ d ⌋ + 2 . Here, the term involving ∇ u in (1.4) did not allow for a directenergy estimate, but instead required a parabolic regularization similar as to the classical workof e.g. Bona-Smith.This approach, however, uses priori energy estimates that rely on the geometric condition(1.2). In a recent preprint the author proved in a similar mannar via energy estimates that if ( u (0) , ∂ t u (0)) are smooth with compact support (ie. u (0) is constant outside of a compactsubset of R d ) and d ∈ { , } , then (1.8) has a global smooth solution.We further mention that Herr, Lamm and Schnaubelt proved the existence of a global weaksolution, see [7], for the case N = S L − by a Ginzburg-Landau approximation and the use ofNoether’s law for the sphere. Remark 1.4.
The equation (1.4) has parabolic scaling u λ ( t, x ) = u ( λ t, λx ) , x ∈ R d , t ∈ R . Thus it holds λ − d E ( u ( λ t )) = E ( u λ ( t )) and d = 4 , the (energy) critical dimension, is included in our results Theorem 1.1 and Corollary1.2. This is due to the larger Strichartz range for the dispersion rate d/ , whereas the lowdimensional case for wave maps is more involved than [24] and has first been solved by Tataruin [25]. 2. Linear estimates and function spaces
Notation.
For real
A, B ≥ we write A . B short for A ≤ cB , where c > is a constant.Likewise we write A ∼ B if there holds A . B and B . A . The space of Schwartz functionswill be denoted by S and the Fourier transform for u ∈ S ( R d ) will be F ( u )( ξ ) = Z R d e − ix · ξ u ( x ) dx, (2.1) TOBIAS SCHMID for which we write ˆ u ( ξ ) = F ( u )( ξ ) . We indicate by F x ′ ( ξ ) that the Fourier transform is takenover x ′ where x = ( x ′ , ˜ x ) if necessary and let ϕ ∈ C ∞ ( R ) be a Littlewood-Paley function, i.e.such that supp ( ϕ ) ⊂ ( 12 , , ϕ ∈ [0 , , and X j ∈ Z ϕ (2 − j s ) = 1 , for s > . (2.2)We define the multiplier P, Q for u ∈ S ′ ( R d ) , v ∈ S ′ ( R d ) and dyadic numbers λ, µ by \ P λ ( ∇ ) u ( ξ ) = ϕ ( | ξ | /λ )ˆ u ( ξ ) , \ P λ ( D ) v ( τ, ξ ) = ϕ (( τ + | ξ | ) /λ )ˆ v ( τ, ξ ) , \ Q µ ( D ) v ( τ, ξ ) = ϕ ( w ( τ, ξ ) /µ )ˆ v ( τ, ξ ) ,P ≤ λ = X ˜ λ ≤ λ P ˜ λ , Q ≤ µ = X ˜ µ ≤ µ Q ˜ µ , where w ( τ, ξ ) = | τ − | ξ | | ( τ + | ξ | ) ∼ || τ | − ξ | , ( τ + | ξ | ) ∼ ( | τ | + ξ ) . Further, we write v λ = P λ v = P λ ( D ) v, P λ, ≤ µ = P λ Q ≤ µ ( D ) for short and define A λ = { ( τ, ξ ) | λ/ ≤ ( τ + ξ ) ≤ λ } , A dλ = { ξ | λ/ ≤ | ξ | ≤ λ } . For a distribution f ∈ S ′ ( R d +1 ) we say f is localized at frequency λ ∈ Z if ˆ f has supportin the set A λ and a similar notation is used for g ∈ S ′ ( R d ) and A dλ . In addition, we need tolocalize in the sets A e := (cid:26) ξ | ξ · e ≥ | ξ |√ (cid:27) , e ∈ S d − , in order to exploit the smoothing effect for the linear equation. Thus, as in [1], we choose M ⊂ S d − with e ∈ M ⇒ − e ∈ M such that R d = [ e ∈M A e , ∀ e ∈ M : { ˜ e ∈ M | A e ∩ A ˜ e = ∅ } ≤ K, (2.3)with a constant K = K d > . Further we require a smooth partition of unity { h e } e ∈M subordinate to { A e } e ∈M , i.e. h e ∈ C ∞ ( R d ) , supp ( h e ) ⊂ A e , h e ∈ [0 , (2.4) X e ∈M h e ( ξ ) = 1 , ξ ∈ R d \{ } . (2.5)We note that this is possible since in particular for x ∈ R d \{ } we have x ∈ A e if and only if ∡ ( x, e ) ≤ π . We define the respective Fourier multiplier by \ P e ( ∇ ) v ( τ, ξ ) = h e ( ξ )ˆ v ( τ, ξ ) , v ∈ S ′ ( R d +1 ) . (2.6)Finally, we choose χ ∈ C ∞ ( R d +1 ) such that χ ( τ, ξ ) = | τ − | ξ | | < τ + | ξ | , | τ − | ξ | | > τ + | ξ | . In order to have χ invariant under parabolic scaling, we choose χ ( τ, ξ ) = η ( | τ −| ξ | τ + | ξ | | ) , where η ∈ C ∞ ( R ) with ≤ η ≤ and such hat η ( x ) = 1 if | x | < / and η ( x ) = 0 if | x | > / .We then define d P v ( τ, ξ ) = χ ( τ, ξ )ˆ v ( τ, ξ ) , \ (1 − P ) v ( τ, ξ ) = (1 − χ ( τ, ξ ))ˆ v ( τ, ξ ) . (2.7)Thus, we have supp ( d P v ) ⊂ (cid:26) ( τ, ξ ) | || τ | − ξ | ≤ | τ | + ξ (cid:27) , (2.8) supp ( \ (1 − P ) v ) ⊂ (cid:26) ( τ, ξ ) | || τ | − ξ | ≥ | τ | + ξ (cid:27) . (2.9) LOBAL RESULTS FOR BIHARMONIC WAVE MAPS 7
Especially, measuring the distance to the characteristic surface P ,dist (( τ, ξ ) , P ) ∼ || τ | − ξ | ( | τ | + ξ ) , P = (cid:8) ( τ, ξ ) | τ = ξ (cid:9) , we infer that (1 − P ) v (with v being localized at frequency λ ) is localized wheredist (( τ, ξ ) , P ) ∼ λ, such that frequency ( τ + | ξ | ) ∼ λ and modulation || τ | − ξ | ∼ µ are of comparable size µ ∼ λ . For P v we have localization wheredist (( τ, ξ ) , P ) = O ( λ ) , with a small constant that suffices to obtain additional smoothing in the linear estimates of thefollowing sections. Further, we use the homogeneous spaces ˙ B ,ps ( R d ) , ≤ p < ∞ given bythe closure of k u k p ˙ B ,ps = X λ ∈ Z λ sp k P λ u k pL x , u ∈ S ( R d ) . Hence we have ˙ H s ( R d ) ∼ ˙ B , s ( R d ) for s > d and ˙ B , d ( R d ) ⊂ L ∞ ( R d ) is a well-definedBanach space.Let λ be a fixed dyadic number. Then for b ≤ and p ∈ [1 , ∞ ) we set k f k pX b,pλ = X µ ∈ Z µ pb k Q µ ( D ) f k pL t,x , (2.10)and denote by X b,pλ the closure of the (semi-)norm in S restricted to functions f localized atfrequency λ . This definition is extended as usual to the case p = ∞ . We observe that f ∈ X b,pλ has the representation f = X µ . λ h µ + h, (2.11)where h is a solution of Lh = 0 (with initial data localized at frequency λ ). Thus f isonly well-defined up to homogeneous solutions Lh = 0 . In the following, we will correct(2.10) by k h k L ∞ t L x + k ∂ t h k L ∞ t ˙ H − x as a limiting dyadic block ( µ ց ), where Lh = 0 and h (0) = f (0) , ∂ t h (0) = ∂ t f (0) .More precisely, the atomic decomposition (2.11) has the form h µ ( t, x ) = Z ∞−∞ e its | s | b h µ ( s, x ) ds, (2.12) k f k p ˙ X b,pλ ∼ X µ . λ (cid:18)Z ∞−∞ k h µ ( s, x ) k L x ds (cid:19) p . (2.13)Here the h µ ( s, · ) solves Lh µ ( s, · ) = 0 for some L × ˙ H − initial data and is localized where s ∼ µ . Further (2.13) only holds up to µ = 0 as mentioned above. We infer (2.12) and (2.13) byfoliation, which also shows that the sum in (2.11) is well-defined distributionally for the cases b < and p ≥ or b = and p = 1 . We will use the foliation explicitly in the proof of Lemma2.7.2.2. Linear estimates.
The goal of this section is to develope estimates for the linear equation ∂ t u ( t, x ) + ∆ u ( t, x ) = F ( t, x ) ( t, x ) ∈ R × R d u [0] = ( u (0) , ∂ t u (0)) = ( u , u ) , on R d , (2.14)with data F, u , u . In the following we provide lateral Strichartz estimates and a maximalfunction estimate for the Cauchy problem (2.14) in case F ∈ L t L x . The main results of thissection summarize all necessary homogeneous bounds in Lemma 2.5 and the inhomogeneousbounds in Lemma 2.6. Further we give a proof of the trace estimate in Lemma 2.7.First, we start by recalling the classical Strichartz estimate, which follows similar as for the TOBIAS SCHMID linear wave equation. Since we did not find it in the literature for (2.14), we briefly state theestimate.
Definition 2.1.
We say that a pair ( p, q ) with ≤ p, q ≤ ∞ is admissible in dimension d ∈ N , d ≥ if there holds p + dq ≤ d . (2.15)and ( p, q ) = (2 , ∞ ) in the case of d = 2 . Lemma 2.2 (Strichartz) . Let u be a weak solution of (2.14) for data u , u , F . Then there holds k u k C ( R , ˙ H γ ) + k u k L pt L qx . k u k ˙ H γ + k u k ˙ H γ − + k F k L ˜ p ′ t L ˜ q ′ x , (2.16) where ( p, q ) , (˜ p, ˜ q ) are admissible pairs with q, ˜ q < ∞ and γ ∈ [0 , satisfies p + dq = d − γ = 2˜ p ′ + d ˜ q ′ − (2.17) Proof.
We prove the inequality for P λ ( ∇ ) u, P λ ( ∇ ) F , where λ is a dyadic number. Then(2.16) follows by the Littlewood-Paley theorem since q, ˜ q < ∞ . Further, (2.16) is invariantunder scaling u λ ( t, x ) = u ( λ t, λx ) , F λ = λ F ( λ t, λx ) , which follows from (2.17). Especially, since ( P λ u ) λ − = P u λ − , we assume λ ∼ . ByDuhamels formula we obtain u ( t ) = cos( − t ∆) u + sin( − t ∆)( − ∆) u + Z t sin( − ( t − s )∆)( − ∆) F ( s ) ds. Therefore, as used above already, we decompose sin( − t ∆) f = 12 i ( e − it ∆ f − e it ∆ f ) , cos( − t ∆) f = 12 ( e − it ∆ f + e it ∆ f ) , and by λ ∼ this can hence be estimated via \ U ± ( t ) f ( ξ ) = χ { t ≥ } e ∓ itξ ψ ( | ξ | ) ˆ f ( ξ ) , f ∈ S ( R d ) , where ψ ∈ C ∞ c ((0 , ∞ )) with ψ ( x ) = 1 for x ∈ supp ( ϕ ) and ϕ is a Littlewood-Paley function.Clearly U ± ( t ) extends to L ( R d ) ∩ L ( R d ) satisfying the energy bound and for the dispersiveestimate, we use U ± ( t ) f = K ± ( t, · ) ∗ d f, K ± ( t, x ) = χ { t ≥ } π ) d Z R d e ix · ξ ∓ it | ξ | ψ ( | ξ | ) dξ. The kernel then applies to the classical theorem for the decay of the Fourier transform of surfacecarried measures on P ± = { ( τ, ξ ) | ± τ + ξ = 0 } . Especially all principle curvature functionson P are non-vanishing ( ( τ, ξ ) = (0 , ) and thus k U ± ( t ) f k L ∞ ( R d ) . (1 + | t | ) − d k f k L ( R d ) . In particular this implies the homogeneous and the inhomogeneous estimate by the
T T ∗ principleand interpolation (respectively Keel-Tao’s endpoint argument) . Thus (2.16) holds on [0 , ∞ ) andwe apply this inequality to u − ( t, x ) = u ( − t, x ) , F − ( t, x ) = F ( − t, x ) , which in turn impliesthe full estimate. It remains to prove u ∈ C t L x , which follows analogously as for the waveequation. (cid:3) We note that due to ( − ∆) − in Duhamel’s formula, the frequency localization in the proof isevitable in contrast to the direct decay estimate for the kernel of the Schrödinger group. Corollary 2.3.
Let u have Fourier support in A λ . Then k u k S λ . k u (0) k L + k ∂ t u (0) k ˙ H − + λ − k Lu k L t L x , (2.18)where S λ = (cid:8) f ∈ C t L x | supp ( ˆ f ) ⊂ A λ , k f k S λ = sup ( p,q ) (cid:0) λ p + dq − d k f k L pt L qx (cid:1) < ∞ (cid:9) and the supremum is taken over admissible pairs ( p, q ) . LOBAL RESULTS FOR BIHARMONIC WAVE MAPS 9
Proof.
From Bernstein’s estimate k u k ˙ H γ + k u k ˙ H γ − . λ γ ( k u k L + k u k ˙ H − ) , which by (2.16) and the gap (2.17) implies the desired estimate for all admissible pairs ( p, q ) with q < ∞ . For the case q = ∞ in d ≥ we estimate by Soblev embedding (or Bernstein’sbound) for any q ≥ dd − λ − d − k u k L t L ∞ x . λ − d +1+ dq k u k L t L qx . k u (0) k L + k ∂ t u (0) k ˙ H − + λ − k Lu k L t L x . (cid:3) The Corollary 2.3 is not sufficient for our proof of bilinear estimates in Section 3 and weadditionally need to apply a well known smoothing estimate for the Schrödinger group.For this reason, we define the following norm (see also A) k u k pL pe L qt,e ⊥ = Z ∞−∞ Z [ e ] ⊥ Z ∞−∞ | u ( t, re + x ) | q dt dx ! pq dr, e ∈ S d − . (2.19)In order to introduce the necessary notation, we recall Bejenaru’s calculus from [1] (see also theAppendix A) for the Cauchy problem (2.14). In the case F = 0 we havesupp (ˆ u ) ⊂ P = { ( τ, ξ ) | τ − | ξ | = 0 } , which is a paraboloid in the variables ( τ, ξ ) . More precisely, denoting by Ξ = ( τ, ξ ) the Fouriervariables, we split the symbol (in case of general F ) b F ( τ, ξ ) = L (Ξ)ˆ u ( τ, ξ ) = − ( τ − ξ )( τ + ξ )ˆ u ( τ, ξ ) . (2.20)Hence, we further split in the Fourier space into − ( τ + ξ ) − b F ( τ, ξ ) χ { τ > } = ( τ − ξ )ˆ u ( τ, ξ ) χ { τ > } , (2.21) ( − τ + ξ ) − b F ( τ, ξ ) χ { τ ≤ } = ( τ + ξ )ˆ u ( τ, ξ ) χ { τ ≤ } , | ξ | 6 = 0 , (2.22)and introduce coordinates adapted to a characteristic unit normal e ∈ S d − . That means we usethe change of coordinates Ξ ( τ, ξ · e, ξ − ( ξ · e ) e ) =: ( τ, ξ e , ξ e ⊥ ) =: Ξ e , and the sets A e = (cid:26) ξ | ξ e ≥ | ξ |√ (cid:27) , B e := (cid:26) ( τ, ξ ) | || τ | − ξ | ≤ | τ | + ξ , ξ ∈ A e (cid:27) (2.23) B ± e := (cid:26) ( τ, ξ ) | | ± τ − ξ | ≤ | τ | + ξ , ξ ∈ A e (cid:27) = B e ∩ {± τ > } ∪ { (0 , } . (2.24)Then for any ( τ, ξ ) ∈ B e , we clearly have | τ | − ξ e ⊥ ≥ , ξ e ∼ ( | τ | + ξ ) , ξ e + q | τ | − ξ e ⊥ ∼ ( | τ | + ξ ) , (2.25)and similar for ± τ on B ± e .Especially, the latter two quantities in (2.25) are controlled by frequency. Also, if we assumethat supp (ˆ u ) ⊂ B e , then for | τ | + ξ > , we have from (2.21) and (2.22) − ( | τ | + ξ ) − (cid:18) ξ e + q | τ | − ξ e ⊥ (cid:19) − b F ( τ, ξ ) = (cid:18)q | τ | − ξ e ⊥ − ξ e (cid:19) ˆ u ( τ, ξ ) , (2.26)Now, taking the FT in the variable Ξ e , we obtain that (2.14) is equivalent to (cid:0) i∂ x e + D t,e ⊥ (cid:1) ˜ u ( t, x e , x e ⊥ ) = ˜ F ( t, x e , x e ⊥ ) , (2.27) where \ D t,e ⊥ u ( τ, ξ e , ξ e ⊥ ) = (cid:18)q | τ | − ξ e ⊥ (cid:19) ˆ u ( τ, ξ ) , (2.28) F ( ˜ F )( τ, ξ e , ξ e ⊥ ) = − ( | τ | + ξ ) − (cid:18) ξ e + q | τ | − ξ e ⊥ (cid:19) − b F ( τ, eξ e + ξ e ⊥ ) (2.29) F (˜ u )( τ, ξ e , ξ e ⊥ ) = ˆ u ( τ, ξ e e + ξ e ⊥ ) , (2.30) Remark 2.4.
The calculations above apply to prove inhomogeneous linear estimates for (2.14)with F ∈ L e L t,e ⊥ that are based on on the reduction to (2.27). However, using the abovenotation for the sets B e and A e , we only need estimates for F ∈ L t L x localized on B e ∩ A λ .These estimates follow directly from Corollary A.3 ( a ) and Lemma A.5 ( a ) in the Appendix A.We now state the homogeneous estimates which follow from the Appendix A. Lemma 2.5 (Linear estimates I) . Let u , u ∈ L ( R d ) , e ∈ M , λ > dyadic withsupp (ˆ u ) , supp (ˆ u ) ⊂ A dλ ∩ A e . Then the solution u of (2.14) with F = 0 satisfies k u k L pe L qt,e ⊥ ≤ Cλ d − p − ( d +1) q ( k u k L + k u k ˙ H − ) , (2.31) where ( p, q ) is an admissible pair. Further if d ≥ and ˆ u , ˆ u have Fourier support in A dλ ,then the solution u of (2.14) with F = 0 satisfies sup e ∈M k u k L e L ∞ t,e ⊥ ≤ Cλ d − ( k u k L x + k u k ˙ H − x ) . (2.32) k u k L pt L qx ≤ Cλ d − p − dq ( k u k L + k u k ˙ H − ) . (2.33) Proof.
By (1.10), we note u ( t ) = 12 e − it ∆ ( u − i ( − ∆) − u ) + 12 e it ∆ ( u + i ( − ∆) − u ) , hence (2.31) follows from Corollary A.3 and (2.32) follows from Lemma A.5. Estimate (2.33)is the classical Strichartz estimate for the Schrödinger group, for which we refer to Corollary2.3. (cid:3) Lemma 2.6 (Linear estimates II) . For e ∈ M and λ > a dyadic number let F ∈ L t L x belocalized in A λ ∩ B e . Then the solution u of (2.14) with u = u = 0 satisfies k u k L pe L qt,e ⊥ . λ ( d +1)( − q ) − p − k F k L t L x , (2.34) sup ˜ e ∈M (cid:0) k u k L e L ∞ t, ˜ e ⊥ (cid:1) . λ d − k F k L t L x , (2.35) where ( p, q ) is an admissible pair. If ˆ F has support in A λ , then the solution u of (2.14) with u = u = 0 satisfies k u k L pt L qx . λ d ( − q ) − p − k F k L t L x , (2.36) where ( p, q ) is an admissible pair.Proof. The estimate (2.36) is the classical Strichartz estimate, which is stated in Corollary (2.3).For the remaining bounds (2.34), (2.35), we decompose the solution u ( t ) = Z t sin( − ( t − s )∆)( − ∆) F ( s ) ds = 12 i Z t e − i ( t − s )∆ ( − ∆) − F ( s ) ds + 12 i Z t e i ( t − s )∆ ( − ∆) − F ( s ) ds. Especially, we have the pointwise bound (cid:12)(cid:12)(cid:12)(cid:12)Z t e ± i ( t − s )∆ ( − ∆) − F ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ∞ | e ± i ( t − s )∆ ( − ∆) − F ( s ) | ds, LOBAL RESULTS FOR BIHARMONIC WAVE MAPS 11 and observe (2.34) and (2.35) by Corollary A.3 ( a ) , Lemma A.5 ( a ) . If X denotes either oneof the spaces on the LHS of (2.36) and (2.35), we estimate (cid:13)(cid:13)(cid:13)(cid:13)Z t −∞ e ± i ( t − s )∆ ( − ∆) − F ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) X ≤ Z ∞−∞ (cid:13)(cid:13) e −± i ( t − s )∆ ( − ∆) − F ( s ) (cid:13)(cid:13) X ds . Z ∞−∞ k ( − ∆) − F ( s ) k L x ds. Here we note that in order to use the Corollary and the Lemma,we verify that e ∓ is ∆ ( − ∆) − F ( s ) has Fourier support (in ξ ) in ( A dλ ∪ A dλ/ ) ∩ A e for all s ∈ R . This follows since F is localizedon B e ∩ A λ and hence also implies for normalized frequencies λ ∼ (cid:13)(cid:13) ( − ∆) − F (cid:13)(cid:13) L t L x . k F k L t L x . (cid:3) The next lemma follows from the homogeneous estimates in Lemma 2.5, resp. CorollaryA.3 and Lemma A.5.
Lemma 2.7 (Trace estimate) . Let F ∈ X , λ for a dyadic number λ . (a) There holds sup e ∈M (cid:0) k F k L e L ∞ te ⊥ (cid:1) . λ d − k F k X , λ (2.37) k F k L pt L qx . λ d − dq − p k F k X , λ , (2.38) for any admissible pair ( p, q ) . (b) We additionally assume ˆ F ( τ, · ) has support in A e for some e ∈ M and all τ ∈ R .Then there holds k F k L pe L qt,e ⊥ . λ d − d +1 q − p k F k X , λ , (2.39) where ( p, q ) is an admissible pair, p ≥ . Remark 2.8.
In the following, we often use the dual estimates of (2.38) - (2.39), i.e. k F k X − , ∞ λ . λ d − dq − p k F k L p ′ t L q ′ x , k F k X − , ∞ λ . λ d − d +1 q − p k F k L p ′ e L q ′ t,e ⊥ , Proof of Lemma 2.7.
For F ∈ X , λ , we have the representation F = X µ ≤ λ Q µ F + h, where Lh = 0 as mentioned in the previous section. We want to use (2.12) and (2.13). However,here we split over sign ( τ ) and write X µ ∈ Z Q µ F = X µ ∈ Z Z Z e ix · ξ + itτ ϕ ( w ( τ, ξ ) /µ ) ˆ F ( τ, ξ ) dτ dξ = X µ ∈ Z Z Z χ ( s + ξ > e ix · ξ + it ( s + ξ ) ϕ ( w ( s + ξ , ξ ) /µ ) ˆ F ( s + ξ , ξ ) ds dξ + X µ ∈ Z Z Z χ ( − s + ξ > e ix · ξ + it ( s − ξ ) ϕ ( w ( s − ξ , ξ ) /µ ) ˆ F ( s − ξ , ξ ) ds dξ = X µ ∈ Z Z e it ( s − ∆) h + µ ( s ) ds + X µ ∈ Z Z e it ( s +∆) h − µ ( s ) ds, where h ± µ ( s ) = χ (cid:8) µ/ ≤ | s | ≤ µ (cid:9) Z e ix · ξ χ ( ± s + ξ > ϕ ( w ( s ± ξ , ξ ) /µ ) ˆ F ( s ± ξ , ξ ) dξ, and we used µ/ ≤ w ( τ, ξ ) = || τ | − ξ | | τ | + ξ ( τ + | ξ | ) ≤ √ || τ | − ξ | ≤ √ w ( τ, ξ ) ≤ √ µ. Now we assume there holds (cid:13)(cid:13) e iθ e ± it ∆ f (cid:13)(cid:13) X . k f k L x for some space X and all θ, t ∈ R , then X µ X ± (cid:13)(cid:13)(cid:13)(cid:13)Z e it ( s ∓ ∆) h ± µ ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) X . X µ X ± µ (cid:18)Z (cid:13)(cid:13) h ± µ ( s ) (cid:13)(cid:13) L x ds (cid:19) ∼ X µ X ± µ (cid:13)(cid:13)(cid:13) χ ( ± τ > ϕ ( w ( ± τ, ξ ) /µ ) ˆ F ( τ, ξ ) (cid:13)(cid:13)(cid:13) L ξ,τ . X µ µ k Q µ F k L t,x . Hence (2.38) follows from the Strichartz estimate for Schrödinger groups and Lemma A.5, since(for the limiting dyadic block µ = 0 with Lh = 0 ) we have (see Lemma 2.3, resp. Lemma 2.5) k h k X . k h (0) k L + k ∂ t h (0) k ˙ H − , (2.40)where X = λ d − p − dq L pt L qx . For (2.37), (2.39), we use the decomposition F = Q ≤ λ F + (1 − Q ≤ λ ) F. Then we check that calculating h ± µ ( s ) in the above argument for Q ≤ λ F , the function ξ \ h ± µ ( s )( ξ ) has support in A dλ/ ∪ A dλ for all s ∈ R . Hence, following the argument with X = λ d − p − ( d +1) q L pe L qt,e ⊥ , X = \ e λ d − L e L ∞ t,e ⊥ , we obtain (2.37), (2.39) by Corollary A.3 and Lemma A.5 for Q ≤ λ F on the LHS. For (2.39),we further note that by assumption h ± µ ( s ) localizes in A e for all s ∈ R . The remaining estimatesfor (1 − Q ≤ λ ) F are equivalent to k (1 − Q ≤ λ ) F k X . λ k (1 − Q ≤ λ ) F k L t,x , which follow from Sobelev embedding (thus the restriction to p ≥ ). As above, we obtain theestimates for the Lh = 0 part of the limiting dyadic block µ = 0 by Lemma 2.5. (cid:3) Function spaces.
We now define the dyadic building blocks of the function spaces Z d , W d and use the convention k·k λB λ = λ − k·k B λ . We set Z λ = X , λ + Y λ , (2.41)where Y λ is the closure of { f ∈ S | supp ( ˆ f ) ⊂ A λ , k f k Y λ < ∞ } , k f k Y λ = λ − k Lf k L t L x + k f k L ∞ t L x , and the norm of Z λ is given by k u k Z λ = inf u + u = u (cid:0) k u k X , λ + k u k Y λ (cid:1) . For the nonlinearity, we construct W λ = L ( Z λ ) , i.e. W λ = λ (cid:0) X − , λ + ( L t L x ) λ (cid:1) (2.42)where ( L t L x ) λ is the closure of { F ∈ S | supp ( ˆ F ) ⊂ A λ , k F k L t L x < ∞ } , LOBAL RESULTS FOR BIHARMONIC WAVE MAPS 13 and k F k W λ = λ − inf F + F = F (cid:0) k F k X − , λ + k F k L t L x (cid:1) . Then, we define k u k Z = X λ ∈ Z λ d k P λ ( D ) u k Z λ , (2.43) k F k W = X λ ∈ Z λ d k P λ ( D ) F k W λ , (2.44)and k u k Z s = X λ ∈ Z λ s k P λ ( D ) u k Z λ for s > d . (2.45) k F k W s = X λ ∈ Z λ s k P λ ( D ) F k W λ for s > d . (2.46)2.3.1. Embeddings, linear estimates and continuous operator.
In this section, we provide someuseful embeddings and multiplier theorems concerning Z λ and W λ . We also show that thesolution of (2.14) satisfies u ∈ Z s if Lu ∈ W s with the correct initial regularity ˙ B , d × ˙ B , d − or ˙ H s × ˙ H s − , respectively for s > d .At the end of this section, we show that Z λ bounds λ − L ∞ e L t,e ⊥ , \ e λ d − L e L ∞ t,e ⊥ , in a suitable sense. Therefore, we apply the following heurstic argument, used similarly in [1]for Schrödinger maps. For solving (2.14) by u = V ( F ) with u = u = 0 , we rely on theinhomogeneous Strichartz estimate Lemma 2.6 k V ( F ) k λ − L ∞ e L t,e ⊥ . λ − k F k L t L x , ( τ, ξ ) ∈ supp ( d P F ) and otherwise on inverting the symbol of LV ( F ) = F − (cid:0) b F ( τ, ξ ) τ − | ξ | (cid:1) , ( τ, ξ ) ∈ supp ( \ (1 − P ) F ) . We first consider the following Lemma, which clearifies how the the spaces Z λ , W λ behaveunder modulation cut-off and is essentially from [24] (adapted to the paraboloid τ = | ξ | ). Lemma 2.9.
The following operator are continuous for ≤ p < ∞ with norms that areuniformly bounded in µ ≤ λ . ( a ) P λ, ≤ µ , P λ P : L pt L x → L pt L x , µ ≤ λ ( b ) (1 − Q ≤ µ ) P λ : Y λ → µ − L t L x , µ ≤ λ . Proof.
We first recall the following version of Tataru’s argument in [24], stated in [6, chapter2.4, Lemma 2.8].Let
C > and M = F − ( m ( τ, ξ ) F ( · )) be a Fourier multiplier such that the following holds.(i) For any ξ , there holds supp ( τ m ( τ, ξ )) ⊂ A ξ , where A ξ has measure ≤ C .(ii) For N ≥ there exists C N > such that k m k L ∞ τ,ξ + C N (cid:13)(cid:13) ∂ Nτ m ( τ, ξ ) (cid:13)(cid:13) L ∞ τ,ξ ≤ C N . Then the operator M : L pt L x → L pt L x , ≤ p ≤ ∞ , (2.47)is continuous and k M k . C N .By Plancherel (in ξ ) and Young’s inequality (in t ), it suffices to proof K ∈ L t L ξ , where K ( t, ξ ) = Z e itτ m ( τ, ξ ) dτ. However by ( i ) , ( ii ) it follows k K k L t L ξ . C N C and by ( ii ) and integration by parts | K ( t, ξ ) | = (cid:12)(cid:12)(cid:12)(cid:12) ( − N | t | N i N Z e itτ ∂ Nτ m ( τ, ξ ) dτ (cid:12)(cid:12)(cid:12)(cid:12) . C N C N | t | N , by which | K ( t, ξ ) | . C N C (1 + C | t | ) N . This argument applies, similar to [24], to P λ, ≤ µ with multiplier m µ,λ ( τ, ξ ) = X ˜ µ ≤ µ ϕ (( τ + ξ ) /λ ) ϕ ( w ( τ, ξ ) / ˜ µ ) , since there holds for N ∈ N and ξ ∈ R d fixed | ∂ Nτ m µ,λ ( τ, ξ ) | . N µ − N , supp ( m µ,λ ) ⊂ { ( τ, ξ ) | || τ | − ξ | ≤ µ } . (2.48)For the second operator P λ P u = F − ( ϕ (( τ + ξ ) /λ )) χ ( τ, ξ )ˆ u ( τ, ξ )) in ( a ) , we note that χ is invariant under scaling and hence the claim reduces to continuity of P P : L pt L x → L pt L x . This follows directly from the above argument.Now for part ( b ) , we write F (1 − Q ≤ µ ) P λ u )( τ, ξ ) = (cid:0) − X ˜ µ ≤ µ ϕ ( w ( τ, ξ ) / ˜ µ ) (cid:1) ϕ (( τ + ξ ) /λ )ˆ u ( τ, ξ )= µ − λ − (cid:0) − X ˜ µ ≤ µ ϕ ( w ( τ, ξ ) / ˜ µ ) (cid:1) ϕ (( τ + ξ ) /λ ) µλ w ( τ, ξ )( τ + ξ ) c Lu ( τ, ξ )=: µ − λ − ˜ m µ,λ ( τ, ξ ) c Lu ( τ, ξ ) . It hence suffices to prove continuity of the operator F − ( ˜ m µ,λ F ( · )) on L t L x . This followssimilarly as in the proof for the cone in [24]. We sketch the argument following the proofin [6, chapter 2.4]. There holds || τ | − ξ | ˜ m µ,λ ( τ, ξ ) + || τ | − ξ | ∂ τ ˜ m µ,λ ( τ, ξ ) . µ. (2.49)Hence, considering the support { ( τ, ξ ) | || τ | − ξ | ≥ µ/ √ , | τ | + ξ ≤ √ λ } , we infer (cid:12)(cid:12)(cid:12)(cid:12)Z e itτ ˜ m µ,λ ( τ, ξ ) dτ (cid:12)(cid:12)(cid:12)(cid:12) . µ log( λ /µ ) , (cid:12)(cid:12)(cid:12)(cid:12) t Z e itτ ˜ m µ,λ ( τ, ξ ) dτ (cid:12)(cid:12)(cid:12)(cid:12) . µ − . Integration gives boundedness of the following terms (uniform in µ, λ ) k K k L t L ξ . Z | t |≤ √ λ k K ( t, · ) k L ∞ dt + Z | t |≥ √ µ k K ( t, · ) k L ∞ dt + Z √ λ ≤| t |≤ √ µ k K ( t, · ) k L ∞ dt. For the last term, we estimate k K ( t, · ) k L ∞ . Z µ √ ≤|| τ |− ξ |≤ | t | µ || τ | − ξ | dτ + 1 t Z || τ |− ξ |≥ | t | µ || τ | − ξ | dτ . µ (1 − log( | t | µ )) , and hence Z √ λ ≤| t |≤ √ µ k K ( t, · ) k L ∞ dt . . (cid:3) Lemma 2.10.
We have W λ ⊂ λ L t,x , Z λ ⊂ λ d L ∞ t,x (2.50) X , λ ⊂ Z λ ⊂ X , ∞ λ . (2.51) LOBAL RESULTS FOR BIHARMONIC WAVE MAPS 15
Proof.
For (2.51), we note that X , λ ⊂ Z λ follows by definiton and Z λ ⊂ X , ∞ λ is proven asfollows.The norm of X , ∞ λ is estimated against the norm of the X , λ part and further, for the L t L x part in Y λ , we deduce from Lemma 2.7 k u λ k X , ∞ λ . λ − k Lu λ k X − , ∞ λ + k u (0) k L x + k ∂ t u (0) k H − x . λ − k Lu λ k L t L x + k u λ k L ∞ t L x , which reads as k u λ k X , ∞ λ . k u λ k Y λ , u λ ∈ Y λ . Concerning (2.50) in the Lemma, we note k u λ k L t,x . λ k u λ k L t L x ∼ λ k u λ k λ L t L x , u λ ∈ L t L x , where we used that ˆ u λ ( · , ξ ) is localized (in τ ) on an interval on length ∼ λ . Hence, since also, k u λ k L t,x . λ X µ . λ µ − k Q µ ( u λ ) k L t,x , u λ ∈ X − , λ , we obtain the first claim. For the L ∞ t,x embedding, we estimate similarly by Lemma 2.7 k u λ k L ∞ t,x . λ d k u λ k X , λ . For the Y λ part, we obtain by a direct application of the classical Strichartz estimate k u λ k L ∞ t,x . k u (0) k ˙ H d + k ∂ t u (0) k ˙ H d − + λ d − k Lu k L t L x . λ d k u λ k Y λ . from Lemma 2.5 and Lemma 2.6 for p = q = ∞ . (cid:3) Proposition 2.11.
There holds Z d ⊂ C ( R , ˙ B , d ) ∩ ˙ C ( R , ˙ B , d − ) (2.52) Z s ⊂ C ( R , ˙ H s ) ∩ ˙ C ( R , ˙ H s − ) (2.53)Further we have k u k Z d . k ( u (0) , ∂ t u (0)) k ˙ B , d × ˙ B , d − + k Lu k W d , (2.54) k u k Z s . k ( u (0) , ∂ t u (0)) k ˙ H s × ˙ H s − + k Lu k W s , s > d , (2.55) k Lu k W d . k u k Z d , k Lu k W s . k u k Z s , s > d . (2.56) Proof.
The claim (2.56) follows from the definition of Z λ , W λ since λ L t L x = LY λ and forthe X ,pλ part, we use k Lu k X − , λ . λ k u k X , λ . For (2.52) and (2.53), if suffices to show k P λ ( D ) u k L ∞ t ˙ B , d + k P λ ( D ) ∂ t u ( t ) k L ∞ t ˙ B , d − . λ d k P λ ( D ) u k Z λ , where by Bernstein k P λ ( D ) ∂ t u ( t ) k L ∞ t ˙ B , d − . k P λ ( D ) u k L ∞ t ˙ B , d . (2.57)Then, since k P λ ( D ) u k L ∞ t ˙ B , d ≤ X ˜ λ ≤ λ (cid:0) ˜ λ/λ (cid:1) d λ d (cid:13)(cid:13) P λ ( D ) P ˜ λ ( ∇ ) u (cid:13)(cid:13) L ∞ t L x , (2.58)the embedding and the continuity in time follow from Z λ ⊂ S λ ⊂ C t L x and we proceedsimilarly for the emdedding of Z s using square sums. Now for (2.54) and (2.55), we useDuhamel’s formula u = S ( u (0) , ∂ t u (0)) + V ( Lu ) , where S ( u , u ) solves (2.14) for F = 0 and V F solves (2.14) for u = u = 0 . Thehomogeneous solution is estimated by the Strichartz bound in Lemma 2.5 in the energy case p = ∞ , q = 2 . This is also directly verified by F x ( P λ ( D ) S ( u (0) , ∂ t u (0)))( t, ξ ) = ϕ (2 | ξ | /λ )(cos( | ξ | t ) d u (0)( ξ )+ | ξ | − sin( | ξ | t ) \ ∂ t u (0)( ξ )) , and hence k P λ S ( u (0) , u t (0)) k Z λ . k P λ S ( u (0) , u t (0)) k L ∞ t L x . k u λ (0) k L + k ∂ t u λ (0) k H − . For the inhomogeneous solution V ( Lu ) we estimate the X ,pλ part by k V ( Lu λ ) k X , λ . λ − k Lu λ k X − , λ , (2.59)and for Y λ , we use Lemma 2.6 in order to conclude k V ( Lu λ ) k Y λ = λ − k Lu λ k L t L x + k u λ k L ∞ t L x . λ − k Lu λ k L t L x . (cid:3) We further estimate the lateral Strichartz norm and establish the maximal function estimate.
Proposition 2.12.
For any dyadic number λ ∈ Z we have Z λ ⊂ S λ ∩ X e ∈M S eλ , (2.60) Z λ ⊂ \ e ∈M λ n − L e L ∞ t,e ⊥ . (2.61)where S eλ is the closure of (cid:26) f ∈ S | supp ( ˆ f ) ⊂ A λ , k f k S eλ = sup ( p,q ) (cid:0) λ p + ( d +1) q − d k f k L pe L qt,e ⊥ (cid:1) < ∞ (cid:27) with ( p, q ) ranging over all admissible pairs with p ≥ . Proof.
For (2.60), we first consider the embedding Z λ ⊂ S λ . Thus, the X , λ part satisfies forany admissible pair ( p, q ) λ p + dq − d k u λ k L pt L qx . k u λ k X , λ , by Lemma 2.7. Likewise, we obtain the same bound against the Y λ part by Lemma 2.5 andLemma 2.6. For the S eλ embedding, we decompose as follows u λ = X e ∈M u eλ , u eλ = P e ( ∇ ) u λ , (2.62)which suffices to obtain (2.60) for the X , λ part directly from Lemma 2.7. Now, consideringthe Y λ part of Z λ , we further write u eλ = P u eλ + (1 − P ) u eλ . Then, P u eλ is localized in B e and (by definition of P , − P ) P u eλ = S ( u eλ (0) , ∂ t u eλ (0)) + V ( P L ( u eλ )) , (1 − P ) u eλ = V ((1 − P ) L ( u eλ )) . Hence by Lemma 2.5 and Lemma 2.6 we have λ p + ( d +1) q − d k P u eλ k L pe L qt,e ⊥ . λ − k P Lu eλ k L t L x + k u eλ (0) k L + k ∂ t u eλ (0) k H − (2.63) . k u λ k Y eλ , by Lemma 2.9 and continuity of P e ( ∇ ) on L t L x . Similarly, by Lemma 2.7, we infer λ p + ( d +1) q − d k (1 − P ) u eλ k L pe L qt,e ⊥ . k V (1 − P )( Lu eλ ) k X , λ . λ − k (1 − P ) Lu eλ k X − , λ . λ − k (1 − P ) Lu eλ k X − , ∞ λ . k u λ k Y eλ , LOBAL RESULTS FOR BIHARMONIC WAVE MAPS 17 where we used (1 − P ) X , λ ∼ (1 − P ) X , ∞ λ uniform in the frequency λ ∈ Z and the dualtrace inequelity from Lemma 2.7 in the last step. Hence we sum over e ∈ M and take theinfimum over u λ = P e u eλ with u eλ ∈ S eλ . The L e L ∞ t,e ⊥ embedding (2.61) follows similarlyusing Lemma 2.7, the decomposition (2.62) and Lemma 2.5, 2.6. Especially sup ˜ e (cid:0) λ − d k P u eλ k L e L ∞ t, ˜ e ⊥ (cid:1) . k u λ k Y λ , (2.64) sup ˜ e (cid:0) λ − d k (1 − P ) u eλ k L e L ∞ t, ˜ e ⊥ (cid:1) . λ − k Lu λ k X − , ∞ λ . k u λ k Y λ (2.65)Again the estimate for the X , λ part follows directly by Lemma 2.7. (cid:3) Bilinear estimates
For the bilinear interaction, we write u · v = X λ ,λ ,λ ( u λ v λ ) λ = X λ ≫ λ (cid:2) ( u λ v λ ) λ / + ( u λ v λ ) λ + ( u λ v λ ) λ (cid:3) (3.1) + X λ ≫ λ (cid:2) ( u λ v λ ) lambda / + ( u λ v λ ) λ + ( u λ v λ ) λ (cid:3) (3.2) + X | log ( λ /λ ) |∼ X λ . max { λ ,λ } ( u λ v λ ) λ . (3.3)Due to symmetry, we restrict (3.1) - (3.3) to X λ ≫ λ (cid:2) ( u λ v λ ) λ / + ( u λ v λ ) λ + ( u λ v λ ) λ (cid:3) + X λ ∼ λ X λ . λ ( u λ v λ ) λ , and thus further reduce to the interactions λ ≪ λ : ( u λ v λ ) λ , and λ ≤ λ : ( u λ v λ ) λ . Lemma 3.1. ( a ) k u λ v λ k Z λ . λ d k u λ k Z λ k v λ k Z λ , λ ≪ λ (3.4) ( b ) k ( u λ v λ ) λ k Z λ . λ d k u λ k Z λ k v λ k Z λ , λ ≤ λ . (3.5) Proof.
For part ( a ) , we decompose ( u λ v λ ) λ = Q ≤ λ λ ( u λ v λ ) λ + (1 − Q ≤ λ λ )( u λ v λ ) λ , (3.6)Here splitting the modulation by µ = λ λ (instead of e.g. the natural choice λ ) is necessaryin order to handle L ( u λ (1 − Q ≤ λ λ ) v λ ) λ ∈ L t L x and specifically the worst interaction ∇ x u λ · ∇ x v λ which is done below. The smoothing isthen exploited via L e L ∞ t,e ⊥ · L ∞ e L t,e ⊥ ⊂ L for the term Q ≤ λ λ ( u λ v λ ) λ as follows . First, we place Q ≤ λ λ ( u λ v λ ) λ ∈ X , λ byestimating k ( u λ v λ ) λ k L t,x . λ d − λ − k u λ k Z λ k v λ k Z λ . (3.7)Then, from X , λ ⊂ Z λ , (3.7) gives k Q ≤ λ λ ( u λ v λ ) λ k Z λ . k Q ≤ λ λ ( u λ v λ ) λ k X , λ . (cid:0) X µ ≤ λ λ µ ( λ λ ) − (cid:1) λ d k u λ k Z λ k v λ k Z λ . For (3.7), we write u λ v λ = P e ∈M u λ v eλ where v eλ ∈ S eλ . Hence (cid:13)(cid:13) ( u λ v eλ ) λ (cid:13)(cid:13) L t,x ≤ k u λ k L e L ∞ t,e ⊥ (cid:13)(cid:13) v eλ (cid:13)(cid:13) L ∞ e L t,e ⊥ (3.8) ≤ λ d − k u λ k λ d − T ˜ e L e L ∞ t, ˜ e ⊥ λ − (cid:13)(cid:13) v eλ (cid:13)(cid:13) λ − L ∞ e L t,e ⊥ . Summing over e ∈ M , the claim follows from Proposition 2.12. Secondly, we note for λ ≪ µQ µ ( u λ v λ ) λ = Q µ (cid:0) u λ X | j |≤ Q j µ v λ (cid:1) (3.9)Hence we write (1 − Q ≤ λ λ )( u λ v λ ) λ = (1 − Q ≤ λ λ )( u λ (1 − Q ≤ λ λ ) v λ ) λ In order to estimate the remaining part in (3.6), using Lemma 2.9, it thus suffices to prove k ( u λ (1 − Q ≤ λ λ ) v λ ) λ k X , λ . k u λ k Z λ k v λ k X , λ (3.10) k ( u λ (1 − Q ≤ λ λ ) v λ ) λ k Y λ . k u λ k Z λ k v λ k Y λ . (3.11)The estimate (3.10) and the L ∞ t L x summand of (3.11) follow from the embedding Z λ ⊂ λ d L ∞ t,x by factoring off the L ∞ t,x norm of u λ . For the second estimate (3.11), we furthercalculate L ( u λ (1 − Q ≤ λ λ ) v λ ) λ = u λ L (1 − Q ≤ λ λ ) v λ + ∂ t u λ ∂ t (1 − Q ≤ λ λ ) v λ + ∂ t u λ (1 − Q ≤ λ λ ) v λ + ∆ ( u λ (1 − Q ≤ λ λ ) v λ ) − u λ ∆ (1 − Q ≤ λ λ ) v λ ) , hence we estimate k L ( u λ (1 − Q ≤ λ λ ) v λ ) λ k L t L x . k u λ L (1 − Q ≤ λ λ ) v λ k L t L x + k ∂ t u λ ∂ t (1 − Q ≤ λ λ ) v λ k L t L x + (cid:13)(cid:13) ∂ t u λ (1 − Q ≤ λ λ ) v λ (cid:13)(cid:13) L t L x + (cid:13)(cid:13) ∆ ( u λ (1 − Q ≤ λ λ ) v λ ) − u λ ∆ (1 − Q ≤ λ λ ) v λ ) (cid:13)(cid:13) L t L x . Calculating the expression in the latter norm and factoring off the derivatives of u λ in L ∞ , weinfer (using Bernstein’s inequality) k L ( u λ (1 − Q ≤ λ λ ) v λ ) λ k L t L x . k u λ L (1 − Q ≤ λ λ ) v λ k L t L x + λ k u λ k L ∞ λ k (1 − Q ≤ λ λ ) v λ k L t L x ≈ k u λ L (1 − Q ≤ λ λ ) v λ k L t L x + k u λ k L ∞ λ λ ( λ λ ) − k (1 − Q ≤ λ λ ) v λ k ( λ λ ) − L t L x where we note λ ≪ λ . We now proceed by Lemma 2.9 ( b ) (for µ = λ λ ) λ − k L ( u λ (1 − Q ≤ λ λ ) v λ ) λ k L t L x . λ d k u λ k Z λ ( λ − k Lv λ k L t L x + k v λ k Y λ ) , which gives the claim. The proof part ( b ) follows similarly, in fact easier, since we can directlyplace ( u λ v λ ) λ ∈ X , λ by estimating k u λ v λ k L t,x . λ d − k u λ k Z λ k v λ k Z λ . (3.12)Then, from X , λ ⊂ Z λ , (3.12) gives k ( u λ v λ ) λ k Z λ . X µ ≤ λ (cid:18) µλ (cid:19) λ k u λ v λ k L t,x . λ ( λ d − k u λ k Z λ k v λ k Z λ ) . LOBAL RESULTS FOR BIHARMONIC WAVE MAPS 19
For (3.12), we write u λ v λ = P e ∈M u eλ v λ where u eλ ∈ S eλ . Hence (cid:13)(cid:13) u eλ v λ (cid:13)(cid:13) L t,x ≤ (cid:13)(cid:13) u eλ (cid:13)(cid:13) L ∞ e L t,e ⊥ k v λ k L e L ∞ t,e ⊥ ≤ λ − (cid:13)(cid:13) u eλ (cid:13)(cid:13) λ − L ∞ e L t,e ⊥ λ d − k v λ k λ d − T ˜ e L e L ∞ t, ˜ e ⊥ . Summing over e ∈ M , we infer the claim. (cid:3) From Lemma 3.1, we obtain (1.14) as outlined above by summation according to the definitonof Z d and W d . Note that the estimates for the remaining frequency interactions in (3.1) and(3.3) follow the same arguments provided in Lemma 3.1.Similarly, for the embedding (1.15) we prove the subsequent estimates. Lemma 3.2. k u λ v λ k W d . λ d λ d k u λ k Z λ k v λ k W λ , λ ≤ λ (3.13) k u λ v λ k W λ . λ d k u λ k W λ k v λ k Z λ , λ ≪ λ (3.14) Proof.
We first estimate by Sobolev embedding λ − k u λ v λ k L t L x . λ − k u λ k L t L dd − x k v λ k L t L dx . λ − k u λ k L t L dd − x λ d − k v λ k L t,x . k u λ k S λ λ d − k v λ k L t,x . k u λ k Z λ λ d k v λ k W λ where we used Lemma 2.10 for W λ ⊂ λ L t,x . Thus from (cid:13)(cid:13) u eλ v λ (cid:13)(cid:13) W d . λ d − k u λ v λ k L t L x . λ d λ d k u λ k Z λ k v λ k W λ , we obtain the claim (3.13). Estimate (3.14) is implied by λ − d Z λ · L t L x ⊂ L t L x , (3.15) λ − d Z λ · X − , λ ⊂ W λ λ − , (3.16)where the first embedding follows from Z λ ⊂ λ d L ∞ t,x . For (3.16), we note that since we restrictto λ ≪ λ , we only consider λ ≤ λ C for a large, fixed constant C > . We thus decompose u λ = Q ≤ C λ u λ + (1 − Q ≤ C λ ) u λ . In particular, each dyadic piece Q µ u λ in (1 − Q ≤ C λ ) X − , λ satisfies λ ≪ µ ≤ λ . Wethen estimate (note that we use (3.9)) (cid:13)(cid:13)(cid:13) v λ (1 − Q ≤ C λ ) u λ (cid:13)(cid:13)(cid:13) X − , λ ∼ X C λ ≤ µ ≤ λ µ − k v λ Q µ u λ k L t,x . X C λ ≤ µ ≤ λ µ − k v λ k L ∞ t,x k Q µ u λ k L t,x . λ d k v λ k Z λ k u λ k X − , λ . Further λ (cid:13)(cid:13)(cid:13) v λ Q ≤ C λ u λ (cid:13)(cid:13)(cid:13) W λ . (cid:13)(cid:13)(cid:13) v λ Q ≤ C λ u λ (cid:13)(cid:13)(cid:13) L t L x . k v λ k L t L ∞ x (cid:13)(cid:13)(cid:13) Q ≤ C λ u λ (cid:13)(cid:13)(cid:13) L t,x . λ d k v λ k λ d − L t L ∞ x X µ ≤ C λ µ − k Q µ u λ k L t,x . λ d k v λ k Z λ k u λ k X − , λ , which follows from Z λ ⊂ S λ . (cid:3) We now infer (1.14) and (1.15) by the summation argument provided in the beginning of thesection.3.1.
Higher regularity.
The percisteny of higher regularity of the ˙ B , d × ˙ B , d − solution asstated in Theorem 1.1 follows as in [24] and [1] from (1.18) and (1.19). We will briefly outlinehow to employ these estimates for the proof of Theorem 1.1 and Corollary 1.2 in the next Section4.For (1.18), we rely again on Lemma 3.1 and the decomposition uv = X λ ≪ λ u λ v λ + X λ ≪ λ u λ v λ + X λ ∼ λ u λ v λ , from the beginning of the Section 3. However, we now sum as follows (cid:18) X λ λ s k ( uv ) λ k Z λ (cid:19) . X λ (cid:18) X λ λ s (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X λ ≪ λ u λ v λ (cid:19) λ (cid:13)(cid:13)(cid:13)(cid:13) Z λ (cid:19) + X λ (cid:18) X λ λ s (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X λ ≪ λ u λ v λ (cid:19) λ (cid:13)(cid:13)(cid:13)(cid:13) Z λ (cid:19) + X λ ∼ λ k u λ v λ k Z s . Hence, we need to estimate the three terms X λ (cid:18) X λ ≪ λ λ s k ( u λ v λ ) λ k Z λ (cid:19) , X λ (cid:18) X λ ≪ λ λ s k ( u λ v λ ) λ k Z λ (cid:19) , X λ ∼ λ k u λ v λ k Z s , where for s > d , the latter sum is treated by Lemma 3.1 ( b ) similar as before via (note that weidentify λ and λ for simplicity) X λ X λ . λ λ s k ( u λ v λ ) λ k Z λ . X λ ( λ s k u λ k Z λ ) λ d k v λ k Z λ . k u k Z s k v k Z d . The LHS of this inequality now bounds the l ( Z ) norm (wrt λ ) and for the first two sums abovewe directly estimate the squares via Lemma 3.1 ( a ) . For (1.19), we sum in the same way anduse the following dyadic estimates X λ . λ λ s k ( u λ v λ ) λ k W λ . λ s + d k u λ k Z λ k v λ k W λ , k ( u λ v λ ) λ k W λ . λ d k u λ k W λ k v λ k Z λ , λ ≪ λ k ( u λ v λ ) λ k W λ . λ d k u λ k Z λ k v λ k W λ , λ ≪ λ which are the same as in (or follow from) Lemma 3.2. LOBAL RESULTS FOR BIHARMONIC WAVE MAPS 21 Proof of the main theorem
The proof of Theorem 1.1 follows straight forward perturbatively by convergence of u k +1 = Su [0] + V ( Q ( u k )) , k ≥ , u ( t, x ) = 0 (4.1)in the space Z d , where Su [0] = S ( u , u ) solves (2.14) with F = 0 and V F solves (2.14) withvanishing initial data. To be more precise, we combine Lemma 2.11, i.e. k u k +1 k Z d . k u k ˙ B , d + k u k ˙ B , d − + kQ ( u k ) k W d (4.2) sup t ∈ R (cid:0) k u ( t ) k ˙ B , d + k ∂ t u ( t ) k ˙ B , d − (cid:1) . k u k Z d , (4.3)with the Lipschitz estimate kQ ( u k ) − Q ( v k ) k W d . C ( k u k k Z d , k v k k Z d ) k u k − v k k Z d . (4.4)This is a direct consequence of (1.14), (1.15) combined with the identity (1.7), i.e. Q ( u ) = 12 Q u ( L ( u · u ) − u · Lu − Lu · u ) and Lemma 2.11 provided Q is analytic (at x = 0 ), δ > is small enough and (1.20) holds.This is necessary to expand the coefficients of Q , which then converge uniformely near x = 0 hence in Z d . Especially, for δ > sufficiently small (4.1) converges to a solution of (1.6) inthe δ -ball of Z d centered at u = 0 . For higher regularity, we proceed as in [24] and construct asolution for (1.6) in the space Z d ∩ Z s with norm k u k Z d ∩ Z s = 1 M k u k Z s + 1˜ δ k u k Z d , where aditionally ( u , u ) ∈ ˙ H s ( R d ) × ˙ H s − ( R d ) for some s > d . Then, provided ˜ δ < δ , theestimates (1.18) and (1.19) imply a Lipschitz estimate similar to (4.4) for W s on the LHS and Z n ∩ Z s on the RHS. More precisely, there holds kQ ( u ) − Q ( v ) k W s . k u − v k Z s ( k u k Z d + k v k Z d ) + k u − v k Z d ( k u k Z s + k v k Z s ) . Especially, with the corresponding linear estimates as above, (4.1) converges in the unit ball of Z d ∩ Z s (in the above norm), where we take M ∼ k u k ˙ H s ( R d ) + k u k ˙ H s − ( R d ) . In order to obtain the Lipschitz estimate and the fact that the fixed point operator maps the unitball into itself, we note that from (1.18) (combined with (1.14)) there holds by induction over k ∈ N for u, v ∈ Z d ∩ Z s (cid:13)(cid:13) u k (cid:13)(cid:13) Z s . k k u k k − Z d k u k Z s , (cid:13)(cid:13) ( u − v ) u k − (cid:13)(cid:13) Z s . k u − v k Z s k u k k − Z d + ( k − k u k k − Z d k u − v k Z d k u k Z s . In particular, the smallness assumption is only necessary in Z d in order to estimate the seriesexpansion of Q ( u ) , Q ( v ) in (1.4). Thus from (1.19) and (1.15) we infer k V ( Q ( u )) k Z s . k u k Z s k u k Z d . ˜ δ k u k Z s , k V ( Q ( u )) k Z d . (cid:13)(cid:13) u (cid:13)(cid:13) Z d . ˜ δ k u k Z d . Similarly, for the difference u − v of u, v ∈ Z s ∩ Z d , we infer the Lipschitz estimate. Since ˜ δ < δ , any such solution also lies in the δ -ball in Z d and thus coincides with the solution in thisspace.The second problem (1.9) is treated similarly, i.e. we expand Π( x ) = ∞ X k =0 k ! d k Π( x ) | x =0 ( x k ) , (4.5) where d k Π( x ) are k -tensors with the notation for l = 1 , . . . , L . Especially we have for any v ∈ R L d Π x ( v ) = ∞ X k =1 k − L X l =1 d k − ∂ x l Π( x ) | x =0 ( x k − ) v l (4.6) = ∞ X k =1 k ! L X l =1 d k − ∂ x l Π( x ) | x =0 ( x k − ) kv l . Since now consider N ( u ) = L (Π( u )) − d Π u ( Lu ) we note that by continuity of L : Z d → W d ,i.e. k Lv k W d . k v k Z d . δ, and by convergence of the series in B Z d (0 , δ ) , we justify to pull L into the series expansionand all terms in the series expression of N ( u ) are at least quadratic. More precisely N ( u ) = X k ≥ k ! ( d k Π( x )) | x =0 ( L ( u k ) − ku k − Lu ) , converges absolutely in W d if u ∈ B Z d (0 , δ ) and δ > is small enough. Similarly for thedifference we have N ( u ) − N ( v ) = X k ≥ k − X l =0 k ! ( d k Π( x )) | x =0 ( L ( v l wu k − − l ) − kv l wu k − − l Lu − kv k − Lw ) , where for the middle term, we only sum l = 0 , . . . , k − . In this notation e.g. ( d k Π( x )) | x =0 ( v l wu k − l − ) captures all terms of the form X l + ...l m = ll m +2 ··· + l L = k − − l C l ,...,l L ( ∂ l x i · · · ∂ l L x iL Π(0)) v l i · · · v l m i m w i m +1 u l m +2 i m +2 · · · u l L i L , i j ∈ { , . . . , L } . Then the argument above applies and we now want to construct a global solution of (1.8), whichreads as ∂ t u + ∆ u = dP u ( u t , u t ) + dP u (∆ u, ∆ u ) + 4 dP u ( ∇ u, ∇ ∆ u ) + 2 dP u ( ∇ u, ∇ u )+ 2 d P u ( ∇ u, ∇ u, ∆ u ) + 4 d P u ( ∇ u, ∇ u, ∇ u )+ d P u ( ∇ u, ∇ u, ∇ u, ∇ u ) , where d P u ( ∇ u, ∇ u, ∇ u ) = d P u ( ∂ i u, ∂ j u, ∂ i ∂ j u ) ,d P u ( ∇ u, ∇ u, ∇ u, ∇ u ) = d P u ( ∂ i u, ∂ i u, ∂ j u, ∂ j u ) , and dP u , d P u , d P u are derivatives of the orthogonal tangent projector P p : R L → T p N for p ∈ N . We extend this equation via Π ( d Π u = P u for u ∈ N ) to functions that only map tothe neighborhood V ε ( N ) . By direct calculation or comparison to (1.8), this can be verified for(1.9) and hence we solve Lv = L (Π( v + p )) − d Π v + p ( Lv ) , for v = u − p where p := lim | x |→∞ u ( x ) via the Z d ∩ Z s -limit of v k +1 = Sv [0] + V ( N ( v k )) , k ≥ , v ( t, x ) = 0 , N ( v ) = L (Π( v )) − d Π v ( Lv ) . (4.7)In particular, the smoothness of the solution follows from the persistence of higher regularityin the fixed point argument from above. Since for δ > small enough, we obtain (note that in ˙ B , d we have C data) sup t ∈ R dist ( u, N ) ≤ k u − p k L ∞ t,x . k v k L ∞ t B , d . k v k Z n . δ, (4.8)the map Π and thus (1.9) is welldefined in a B (0 , Cδ ) ball in Z d . The only thing left to showis that u ( t ) ∈ N for t ∈ R , such that in particular, (1.9) implies (1.8). LOBAL RESULTS FOR BIHARMONIC WAVE MAPS 23 If v = u − p ∈ B (0 , Cδ ) ⊂ Z d , then Π( u ) − p = Π( v + p ) − p ∈ Z d and Π( u ) − u =Π( v + p ) − p − v ∈ Z d with k Π( u ) − u k Z d + k Π( u ) − p k Z d . k v k Z d , provided δ > is small. We now have L ( u − Π( u )) = L ( v − Π( v + p )) = − d Π v + p ( Lv ) = − d Π v + p ( N ( v )) . (4.9)Since Π( u ) ∈ N , we have N (Π( u ) − p ) ⊥ T Π( u ) N and from Im ( d Π u ) ⊂ T Π( u ) N , u = v + p ,we obtain d Π v + p ( N (Π( u ) − p )) = 0 . At this point, however, we mention, that we cancel the linear part in series expansions for L (Π( v + p )) in N ( v ) and of L (Π( u ) − p ) = L (Π( v + p ) − p ) in N (Π( u ) − p ) at v = 0 (Thusthe constant part of d Π vansihes in N ). We then obtain L ( u − Π( u )) = − d Π v + p (cid:0) ( d Π (Π( u ) − p )+ p − d Π v + p ) L (Π( u ) − p ) (4.10) + d Π v + p ( L (Π( u ) − p − v ))+ L (Π( v + p ) − Π((Π( u ) − p ) + p )) (cid:1) . Note that we don’t want to use Π = Π , since technically we want the identity for the seriesexpressions for Π , d Π with missing linear parts. Especially, all terms appearing on the RHSare at least quadratic.This implies (note that u (0) = Π u (0) , u t (0) = ∂ t (Π u )(0) by assumption) k u − Π( u ) k Z d . (1 + k v k Z d ) (cid:13)(cid:13) ( d Π (Π( u ) − p )+ p − d Π v + p ) L (Π( u ) − p ) (cid:13)(cid:13) W d + (1 + k v k Z d ) k d Π v + p ( L (Π( u ) − p − v )) k W d + (1 + k v k Z d ) k L (Π( v + p ) − Π((Π( u ) − p ) + p )) k W d . (1 + k v k Z d ) k u − Π( u ) k Z d k L (Π( u ) − p ) k W d + (1 + k v k Z d ) k v k Z d k L (Π( u ) − p − v ) k W d + (1 + k v k Z d )( k v k Z d + k Π( u ) − p k Z d ) k u − Π( u ) k Z d . (1 + k v k Z d ) k v k Z d k u − Π( u ) k Z d . In particular, if k v k Z d ≤ δ is sufficiently small, we have u = Π( u ) ∈ N . Appendix A. Local Smoothing & lateral Strichartz inequalities
In this section, we recall the local smoothing effect (i.e. lateral Strichartz estimates with localizeddata) and a maximal function estimate for the linear Cauchy problem ( i∂ t u ( t, x ) ± ∆ u ( t, x ) = f ( t, x ) ( t, x ) ∈ R × R d u (0 , x ) = u ( x ) x ∈ R d (A.1)in the lateral space L pe L qt,e ⊥ for e ∈ S d − with norm k f k pL pe L qt,e ⊥ = Z ∞−∞ Z [ e ] ⊥ Z ∞−∞ | f ( t, re + x ) | q dt dx ! pq dr. (A.2)The norm (A.2) was used by Kenig, Ponce, Vega, see e.g. [11], in order to establish localsmoothing estimates for nonlinear Schrödinger equations.The estimates for L pe L qt,e ⊥ , L e L t,e ⊥ , L e L ∞ t,e ⊥ in Corollary A.3 and Lemma A.5 below aresubstantial in the wellposedness theory of Schrödinger maps and were proven by Ionescu, Kenigin [9], [10] (see also the work of Bejenaru in [1] and Bejenaru, Ionescu, Kenig in [2]).Similar ideas (however more involved due to the absence of the L e L ∞ t,e ⊥ estimate in d = 2 ) havebeen used by Bejenaru, Ionescu, Kenig and Tataru in [3] for global Schrödinger maps into S indimension d ≥ with small initial data in H d . Here we follow Bejenaru’s calculation in [1], which recovers the smoothing effect for (A.1)provided the data u , f is sufficiently localized in the sets A e = (cid:8) ξ | ξ · e ≥ | ξ |√ (cid:9) ,B ± e = (cid:26) ( τ, ξ ) | | ± τ − ξ | ≤ | τ | + ξ , ξ ∈ A e (cid:27) A λ = { ( τ, ξ ) | λ/ ≤ ( τ + | ξ | ) ≤ λ } , as defined in Section 2. Especially for ( τ, ξ ) ∈ B ± e ∩ A λ , there holds ± τ − ξ e ⊥ ≥ , ξ e ∼ λ, ξ e + q ± τ − ξ e ⊥ ∼ λ. (A.3) Remark A.1.
We note that our definition of B ± e slightly differs from [1].Taking the FT (in t, x ) of (A.1), with u being localized in B ± e , ˆ f ( τ, ξ ) = ( τ ∓ | ξ | )ˆ u ( τ, ξ ) = ± (cid:18)q ± τ − ξ e ⊥ − ξ e (cid:19) (cid:18)q ± τ − ξ e ⊥ + ξ e (cid:19) ˆ u ( τ, ξ ) . (A.4)Hence, considering (A.3), we proceed by taking the (inv.) FT in the coordinates t, x e ⊥ , ±F − ( ˆ f ( ξ e , τ,ξ e ⊥ ) (cid:0)q ± τ − ξ e ⊥ + ξ e (cid:1) − ) (A.5) = F − (cid:18)q ± τ − ξ e ⊥ ˆ u ( ξ e , τ, ξ e ⊥ ) (cid:19) − ξ e ˆ u ( ξ e , t, x e ⊥ ) . Thus, (A.1) is equivalent to an intial value problem of the following type ( i∂ r + D ± t,x ) v ( t, r, x ) = f,v ( t, , x ) = u ( t, x ) , (A.6)where \ D ± t,x v ( τ, ξ ) = p ± τ − | ξ | ˆ v ( τ, ξ ) . Thus (at least formally) the homogeneous solutionof (A.6) is represented as v ( t, r, x ) = e irD ± t,x u ( t, x ) . In the following, we only consider homogeneous estimates for (A.1), wich imply all linearestimates we need in Section2. Inhomogeneous bounds for the biharmonic problem (2.14) with F ∈ L e L t,e ⊥ can be proven similarly as for the Schrödinger equation using the calculation inSection 2.The equation (A.6) has the scaling v λ ( t, r, x ) = v ( λ t, λr, λx ) , λ > and we now prove thefollowing Strichartz estimate. Lemma A.2 (Strichartz estimate) . Let u ∈ S ′ ( R × R d − ) , f ∈ S ′ ( R × ( R × R d − )) haveFourier support in {± τ ≥ ξ } ∩ A λ for some dyadic λ ∈ Z . Then there holds (cid:13)(cid:13)(cid:13) e irD ± t,x u ( t, x ) (cid:13)(cid:13)(cid:13) L pr L qt,x . λ d +12 − p − d +1 q k u k L t,x , (A.7) (cid:13)(cid:13)(cid:13)(cid:13)Z r −∞ e i ( r − s ) D ± t,x f ( s, t, x ) ds (cid:13)(cid:13)(cid:13)(cid:13) L pr L qt,x . λ p ′ − p +( d +1)( q ′ − q ) − k f k L ˜ p ′ r L ˜ q ′ t,x , (A.8) where ( p, q ) , (˜ p, ˜ q ) are admissible, i.e. ≤ p, q ≤ ∞ , ( p, q ) = (2 , ∞ ) if d = 2 and p + dq ≤ d . (A.9) Proof.
We use a Littlewood-Paley decomposition \ P λ ( u )( τ, ξ ) = ϕ (( τ + | ξ | ) /λ )ˆ u ( τ, ξ ) , LOBAL RESULTS FOR BIHARMONIC WAVE MAPS 25 and by scaling of (A.6), we have P ( u λ − ) = ( P λ u ) λ − . Thus we reduce the estimate (A.7) to (cid:13)(cid:13)(cid:13) e irD ± t,x P u ( t, x ) (cid:13)(cid:13)(cid:13) L pr L qt,x ≤ C d,p,q (cid:13)(cid:13)(cid:13) ϕ (( τ + | ξ | ) )ˆ u ( τ, ξ ) (cid:13)(cid:13)(cid:13) L τ,ξ . (A.10)We further have e irD ± t,x P u ( t, x ) = K ∗ t,x P u, with kernel K ( t, r, x ) = Z Z e i ( x,t,r ) · ( ξ,τ, √ ± τ − ξ ) ψ ( τ, ξ ) dτ dξ, (A.11)where ψ ∈ C ∞ c ( R × R d − ) with ψ ( τ, ξ ) = 1 for ( τ, ξ ) ∈ supp (( τ, ξ ) ϕ (( τ + | ξ | ) ))) .which is the Fourier transform of a (compactly supported) surface carried measure on thehypersurface S = n(cid:16) ξ, τ, p ± τ − ξ (cid:17) | ξ ∈ R d − , τ ∈ R , ξ ≤ ± τ o , and S has d non-vanishing principal curvature functions in the relevant coordinate patch. Thuswe observe | K ( t, r, x ) | . (1 + | r | ) − d , t ∈ R , x ∈ R d − , (A.12)which gives (cid:13)(cid:13)(cid:13) e irD ± t,x P u ( t, x ) (cid:13)(cid:13)(cid:13) L ∞ t,x . (1 + | r | ) − d k P u k L t,x . (A.13)Now in the endpoint case ( p, q ) = (2 , dd − ) , we need to apply Keel-Tao’s argument and otherwisewe can use direct interpolation. More precisely, combining (A.13) and the fact that e irD ± t,x is agroup on L t,x with a classical T T ∗ argument and the Christ-Kiselev Lemma, we deduce (A.7)and (A.8) for P u . (cid:3) We remark that Lemma A.2 is valid if u, f have Fourier support in e.g. in A λ/ ∪ A λ ∪ A λ ,i.e. as long as frequency is controlled by λ . For a dyadic number λ ∈ Z , we recall the definitionof A dλ = { ξ | λ/ ≤ | ξ | ≤ λ } . An immediate consequence of the Strichartz estimate is thefollowing Corollary. Corollary A.3.
Let u ∈ L ( R d ) , e ∈ S d − , λ > dyadic with supp (ˆ u ) ⊂ A dλ ∩ A e . Thenthere holds (cid:13)(cid:13) e ± it ∆ u (cid:13)(cid:13) L pe L qt,e ⊥ ≤ Cλ d − p − ( d +1) q k u k L x , (A.14) where ( p, q ) is an admissible pair. Let u ∈ S ′ ( R × R d ) , e ∈ S d − , λ > dyadic such thatsupp (ˆ u ) ⊂ (cid:8) ( τ, ξ e ⊥ ) | ( τ, ξ ) ∈ B ± e ∩ A λ (cid:9) Then there holds (cid:13)(cid:13)(cid:13)(cid:13) e ix e D ± t,xe ⊥ u ( t, x e ⊥ ) (cid:13)(cid:13)(cid:13)(cid:13) L pt L qx ≤ C d,p,q λ d +12 − p − dq k u ( t, x e ⊥ ) k L t,xe ⊥ , (A.15) where ( p, q ) is an admissible pair.Proof. For the first statement, we identify ξ = ( ξ · e ) e + ξ e ⊥ ( ξ e , ξ e ⊥ ) and proceed asfollows. By the change of coordinates q ± τ − ξ e ⊥ = ξ e , we have Z R d e ix · ξ e ± it | ξ | ˆ u ( ξ ) dξ = Z [ e ] ⊥ Z {± τ ≥ ξ e ⊥ } e ± itτ e ix e ⊥ ξ e ⊥ e ix e q ± τ − ξ e ⊥ ˆ u ( q ± τ − ξ e ⊥ e + ξ e ⊥ ) dτ q ± τ − ξ e ⊥ dξ e ⊥ . Now we set ˆ u ( τ, ξ e ⊥ ) = ˆ u ( q ± τ − ξ e ⊥ e + ξ e ⊥ ) (cid:18) q ± τ − ξ e ⊥ (cid:19) − , if ± τ ≥ | ξ e ⊥ | , and ˆ u ( τ, ξ e ⊥ ) = 0 elsewhere. By assumption on ˆ u , we have u ∈ S ′ ( R × R d − ) (upon theidentification of [ e ] ⊥ = R d − ) and for ( τ, ξ e ⊥ ) ∈ supp (ˆ u ) there holds q ± τ − ξ e ⊥ = ξ e ∼ λ and λ/ ≤ q | τ | + ξ e ⊥ ≤ λ (since in particular u localizes in A e ∩ A dλ ). Thus, we applyLemma A.2 and conclude (cid:13)(cid:13) e ± it ∆ u ( x ) (cid:13)(cid:13) L pe L qt,e ⊥ = (cid:13)(cid:13)(cid:13)(cid:13)Z R d e ix · ξ e ± it | ξ | ˆ u ( ξ ) dξ (cid:13)(cid:13)(cid:13)(cid:13) L pe L qt,e ⊥ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z [ e ] ⊥ Z {± τ ≥ ξ e ⊥ } e ± itτ e ix e ⊥ ξ e ⊥ e ix e q ± τ − ξ e ⊥ ˆ u ( τ, ξ e ⊥ ) dτ dξ e ⊥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pe L qt,e ⊥ . λ d +12 − p − d +1 q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˆ u ( q τ − ξ e ⊥ e + ξ e ⊥ ) (cid:18) q ± τ − ξ e ⊥ (cid:19) − χ {± τ > ξ e ⊥ } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L τ,e ⊥ . λ d − p − d +1 q k u k L x , where, for the last inequality, we reverse the coordinate change and estimate the Jacobian. Forthe second statement, the estimate follows from Strichartz estimates for the Schrödinger groupand from the above coordinate transform in the backward direction. To be more precise, weestimate (cid:13)(cid:13)(cid:13)(cid:13) e − ix e D ± t,xe ⊥ u ( t, x e ⊥ ) (cid:13)(cid:13)(cid:13)(cid:13) L pt L qx = (cid:13)(cid:13)(cid:13)(cid:13)Z R d e ix · ξ e ± it | ξ | ˆ u ( ±| ξ | , ξ e ⊥ ) 2( ξ · e ) dξ (cid:13)(cid:13)(cid:13)(cid:13) L pt L qx . λ d +12 − p − dq (cid:13)(cid:13)(cid:13) ( ξ · e ) ˆ u ( ±| ξ | , ξ e ⊥ ) χ { ξ · e ≥ } (cid:13)(cid:13)(cid:13) L x . λ d +12 − p − dq k u ( t, x e ⊥ ) k L t,xe ⊥ . (cid:3) Remark A.4.
In the case supp (ˆ u ) ⊂ A dλ , we obtain from Corollary A.3 sup e ∈M (cid:13)(cid:13) e ± it ∆ P e ( ∇ ) u (cid:13)(cid:13) L pe L qt,e ⊥ ≤ Cλ d − p − ( d +1) q k u k L x , (A.16)and especially the L pe L ∞ t,e ⊥ estimate for q = ∞ , pd ≥ , d ≥ .The next Lemma (from [1]) shows that the P e ( ∇ ) localization on the LHS of (A.16) is notnecessary in the case q = ∞ if dp > . Lemma A.5.
Let u ∈ L ( R d ) such that supp (ˆ u ) ⊂ A dλ for some dyadic λ ∈ Z . Then thereholds sup e ∈M (cid:13)(cid:13) e ± it ∆ u (cid:13)(cid:13) L pe L ∞ t,e ⊥ ≤ C d,p λ d − p k u k L x , (A.17) where ≤ p ≤ ∞ and dp > . Let u ∈ S ′ ( R × R d ) , such thatsupp (ˆ u ) ⊂ { ( τ, ξ ˜ e ⊥ ) | ( τ, ξ ) ∈ B ± ˜ e ∩ A λ } for some λ ∈ Z and ˜ e ∈ M . Then there holds sup e ∈M (cid:13)(cid:13)(cid:13) e − irD ± t,xe u ( t, x e ⊥ ) (cid:13)(cid:13)(cid:13) L pr L ∞ t,xe ⊥ ≤ C d,p λ d +12 − p k u ( t, x ˜ e ⊥ ) k L t,x ˜ e ⊥ , (A.18) where ( d, p ) are as above.Proof. By scaling we reduce again to the unit frequency λ = 1 . Then estimate (A.17) is aconsequence of the T T ∗ argument for the Schrödinger group in the space L pe L ∞ t,e ⊥ and Young’sinequality. As mentioned before, we obtain the decay (cid:12)(cid:12)(cid:12)(cid:12)Z R d e ix · ξ e ± it | ξ | ϕ ( | ξ | ) dξ (cid:12)(cid:12)(cid:12)(cid:12) . (1 + | x · e | ) − d , which implies (for dp > ) (cid:13)(cid:13)(cid:13)(cid:13)Z R d e ix · ξ e ± it | ξ | ϕ ( | ξ | ) dξ (cid:13)(cid:13)(cid:13)(cid:13) L p e L ∞ t,e ⊥ . . LOBAL RESULTS FOR BIHARMONIC WAVE MAPS 27
Then by Young’s inequality (cid:13)(cid:13)(cid:13)(cid:13)Z e ± i ( t − s )∆ f ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L pe L ∞ t,e ⊥ . k f k L p ′ e L t,e ⊥ , which implies (A.17) by T T ∗ . For (A.18), we use again (note ˆ u is localized in B ± e , thus (A.3)holds) Z R d e ix · ξ e ± it | ξ | ϕ ( | ξ | ) dξ = Z [ e ] ⊥ Z ± τ ≥ ξ e ⊥ e ix e √ ± τ − ξ e ⊥ e i ( x e ⊥ ,t ) · ( ξ e ⊥ , ± τ ) ϕ ( √± τ ) dτ √± τ − ξ e ⊥ dξ e ⊥ and thus we obtain (A.18) also from T T ∗ and Young’s inequality for exp( − iD ± t,e ⊥ ) . (cid:3) Remark A.6.
The first estimate in Lemma A.5 holds more general by the same argument in thefollowing sense. Let u , u as above in Lemma A.5 and further ≤ p, q ≤ ∞ such that q > and ( qq − < dp, q < ∞ < dp, q = ∞ . (A.19)Then there holds sup e ∈M (cid:13)(cid:13) e ± it ∆ u (cid:13)(cid:13) L pe L qt,e ⊥ ≤ C d,p,q λ d − d +1 q − p k u k L x , (A.20)Provided (A.19) holds, it is verified that Z ∞∞ (cid:18)Z ∞ (1 + | x e | + r ) − dq r d − dr (cid:19) pq dx e < ∞ , which is required by the argument in the proof of Lemma A.5, if we use (cid:12)(cid:12)(cid:12)(cid:12)Z R d e ix · ξ e ± it | ξ | ϕ ( | ξ | ) dξ (cid:12)(cid:12)(cid:12)(cid:12) . (1 + | x e | + | ( t, x e ⊥ ) | ) − d . Under the assumption (A.19), we especially infer p + dq < d ( q −
4) + 4 d q = d ( q − q < d , so that ( p, q ) is admissible. This is a natural requirement, since typically Strichartz bounds withbounded frequency rely on estimating the truncated dispersion factor via Young’s inequality. Remark A.7.
We apply the estimates to Lemma 2.5 and Lemma 2.6 in Section 2. Also, inSection 2, we need to use Corollary A.3 and Lemma A.5 for functions on R d that have Fouriersupport in A dλ/ ∪ A dλ ∪ A d λ . This is observed (for all t ∈ R ) e.g. for functions on R d +1 localized (in ( τ, ξ ) ) in B e ∩ A λ , which have Fourier support in A dλ ∪ A dλ/ , and this poses noproblem to the proof. Acknowledgments
The author acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation) - Project-ID 258734477 - SFB 1173
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