aa r X i v : . [ m a t h . A P ] M a y A NOTE ON GLOBAL EXISTENCE FOR THE ZAKHAROV SYSTEM ON T E. COMPAAN
Abstract.
We show that the one-dimensional periodic Zakharov system is globally well-posed in a class of low-regularity Fourier-Lebesgue spaces. The result is obtained by combining the I-method with Bourgain’s high-lowdecomposition method. As a corollary, we obtain probabilistic global existence results in L -based Sobolev spaces.We also obtain global well-posedness in H ` ˆ L , which is sharp (up to endpoints) in the class of L -basedSobolev spaces. Introduction & Statement of Results
In this note, we consider the periodic Zakharov system $’’’’’’’’&’’’’’’’’% iu t ` ∆ u “ nu, x P T , t P R ,n tt ´ ∆ n “ ∆ | u | ,u p x, q “ u p x q ,n p x, q “ n p x q , n t p x, q “ n p x q . This system was introduced in the 1970s as a model of Langmuir turbulence in ionized plasma [14]. The function u , the Schr¨odinger part, represents the envelope of a oscillating electric field, while the wave part n represents thedeviation from the mean of the ion density. The purpose of this note is to derive a global well-posedness result thatholds for initial data at a low regularity level.Recasting the Zakharov model as a first-order system by setting n ˘ “ n ˘ iD ´ n t , where D “ p´ ∆ q , we obtain(1) $’’’’’’’’&’’’’’’’’% iu t ` ∆ u “ p n ` ` n ´ q u, x P T , t P R ,in ˘ t ¯ Dn ˘ “ ˘ D | u | ,u p x, q “ u p x q ,n ˘ p x, q “ n ˘ p x q . The Zakharov system (1) conserves the Schr¨odinger part mass and a Hamiltonian energy: M p u q “ } u } L E p u, n ˘ q “ } u x } L ` ´ } n ` } L ` } n ´ } L ¯ ` ż p n ` ` n ´ q| u | d x. Note also that the mean values of n and n t are preserved under the flow. Thus we may without loss of generalityassume n and n are mean-zero. Sharp local well-posedness on T holds for initial data p u , n ˘ q P H p T q ˆ L p T q . This result is due to Takaoka [13], building on the result of Ginibre-Tsutsumi-Velo [9] for the Euclidean space system.Well-posedness was proved used the Fourier restriction norm technique introduced in [2].
The local result in sharp in the class of L -based Sobolev spaces. An earlier work of Bourgain [3] gives well-posedness in certain Fourier-Lebesgue type spaces. This result is neither stronger nor weaker than the Sobolev spaceresult [13], in the sense than neither result implies the other. However, Bourgain’s Fourier-Lebesgue space result alsoenables him to obtain a probabilistic global well-posedness result for (1). This will be discussed further below.The Hamiltonian conservation together with the Galgiardo-Nirenburg-Sobolev inequality yield global existence inthe energy space H p T q ˆ L p T q .In the two-dimensional case, the sharp Sobolev space local theory remains at the same level – it holds for initial data p u , n ˘ q P H p T q ˆ L p T q . However, to prove this, Kishimoto employed modified Besov-type Fourier restrictionnorms [11]. Similar arguments had previously been used by Bejenaru-Herr-Holmer-Tataru [1] to prove the parallelresult for the Zakharov system on R . We will use these norms as well in our work on T .In the T setting, Kishimoto has also obtained global existence below the energy space, in H ` p T q ˆ L p T q for Schr¨odinger data with sufficiently small L p T q norm (in relation to that of the ground-state solution) [11]. Theresult is established using the I-method of Colliander-Keel-Staffilani-Takaoka-Tao [7]. This result also applies to T ,since a solution to (1) on T corrresponds to a solution on T which is constant along one spatial dimension. Theconstraint on norm of the Schro¨odinger part in L is necessary on T to ensure that the Hamiltonian constrols therelevant Sobolev norm.In the one-dimensional case, Bourgain gave the following probabilistic global well-posedness result for (1): Theorem 1.1 ([3]) . There are Sobolev exponents ă σ ă s ă ă µ ă such that the Zakharov system (1) iswell-posed for data p u , n , n q satisfying u P H s p T q , sup k | k | µ | x u p k q| ă 8 , sup k | k | ´ σ | x n p k q| ă 8 , sup k | k | ´ σ ´ | x n p k q| ă 8 . Moreover, the system is almost surely globally well-posed with respect to the normalized Gibbs measure associated withthe Zakharov equation, which is supported on Ş s ă ´ H s ˆ H s ´ ˆ H s ´ ¯ . An examination of Bourgain’s proof reveals that one should take σ very close to zero, and s and µ very closeto one half. The almost-sure global existence was established by recasting the equation as an infinite dimensionalHamiltonian system for the spatial Fourier coefficients. The global well-posedness holds almost surely with respectto the Gibbs measure, a Sobolev space probability measure based on the Hamiltonian of the equation. This approachwas inspired by Lebowitz-Rose-Speer [12]. See also [4] for a similar argument for the nonlinear Schr¨odinger equation.Thus, global existence is known deterministically in H ` p T qˆ L p T q and probabilistically in H ´ p T qˆ H ´ ´ p T q .The object of this note is to prove a global existence result which, at least partially, fills this gap. Note that onecannot simply use a preservation of regularity argument to conclude some sort of global existence between thesespaces. This is because the support of the Gibbs measure is in H ´ z H ˆ H ´ ´ z H ´ ; that is, the probabilisticresult does not give any clear insight into the dynamics in smoother spaces.The idea of our proof is to combine the high-low decomposition method of Bourgain [5] with the I-method ofColliander-Keel-Staffilani-Takaoka-Tao [7]. This combination has appeared in the work of Bourgain for the quinticSchr¨odinger equation on T [6]. We note that we were motivated in part by the probabilistic global existence result ofColliander-Oh [8], which employed the high-low method. We now give an outline of the method. AKHAROV SYSTEM ON T The high-low method works by breaking the initial data into high-frequency and low-frequency parts, at somecut-off value N . One then solves the system with the low-frequency (and hence smooth) data in the energy space.To obtain a solution to the system with the full initial data, one solves a difference equation for the remainder, withthe high-frequency initial data. To iterate, it must be proved that the nonlinear part of the solution to the differenceequation is in the energy space (i.e. smoother than the initial data) with small norm. One can then add this smootherpart to the low-frequency solution and repeat the process. If one can obtain strong enough bounds, any time interval r , T s can be covered via iteration, by choosing N large enough.The I-method works by applying a smoothing operator, again with some cut-off parameter N , to the system. Alocal theory is then obtained for this smoothed version of the system in the energy space. The next step is to showthat a modified energy corresponding to the smoothed system is “almost conserved”, i.e. it doesn’t not grow toorapidly. If the growth is slow enough, one can again iterate to cover any interval r , T s by choosing a sufficiently largecut-off value N .In general, the I-method will yield well-posedness at lower Sobolev regularities than the high-low method. However,in our case the I-method cannot be directly applied. It requires a local theory in L -based Sobolev spaces whosenorms can be controlled by the modified energy. However, there is no such local theory for wave data below L forthe Zakharov system. We therefore must subtract out the linear flow in order to solve in Sobolev spaces. This leavesa difference equation, which is no longer expected to satisfy an almost-conservation law.On the other hand, if we attempt to apply the high-low method alone, we are required to obtain estimates of theform } uW p t qp n ˘ q} X , ´ , À } n ˘ } Y r, , } u } X , , . Here W p t qp n ˘ q is the linear wave flow. The norms which appear here are defined below. The relevant point for themoment is that the left-hand side is a norm with Sobolev regularity one, while the wave flow is rougher than L . Forsuch rough wave data, it seems impossible to obtain the necessary estimate. Randomization of the wave data doesnot appear to improve matters. In the H s ˆ H r local theory, this case corresponds to the (unattainable) s ą r ` u P H s p T q , for s P p , q , and wave initial data such that: n ˘ p x q “ ÿ k P Z {t u h ˘ k x k y β e ikx with sup k | h ˘ k | ă 8 . (2)That is, we take n ˘ in the Fourier-Lebesgue space FL β, : “ ! f P D p T q | sup k x k y β | p f p k q| ă 8 ) . We fix β P p , s , so that n ˘ P H r p T q for any r ă β ´ . We require β ą s ´ . This corresponds to requiring that n ˘ is in a Sobolev space H r with s ´ ă r . Note that the case s ´ “ r is critical in some sense. When s ´ ą r ,it seems impossible to obtain any well-posedness. Our main result is as follows. E. COMPAAN
Theorem 1.2.
Suppose u P H s p T q and n ˘ P FL β, , with s ą and β ą p ´ s q . Then for any T ą , the Zakharov system (1) with initial data p u , n ˘ q has a solution on the time interval r , T s .Furthermore, the norm of the nonlinear part of the solution grows at most polynomially in time. In particular, max ď t ď T ´ } u ´ e itδ u } H s ` } n ˘ ´ e ˘ it B x n ˘ } L ¯ À C p} u } H s , } n ˘ } FL β, , s, β qx T y max t α p ´ s q γ, p ´ β q γ u , where α P ˜ ´ β, min ! β ` ´ s p ´ s q , β ´ s ´ s )¸ and γ ą max ! p s ´ q α , β ` ´ s ´ α p ´ s q , α ` β ´ , β ´ s ´ α p ´ s q ) . Remark 1.3.
It is probable that our methods would also apply to rougher Schr¨odinger data (that is, to u P FL α, ,with α ď ). However, we do not pursue this here. Remark 1.4.
We note that Bourgain gives a local theory result for wave data in a subset of H ´ σ , with ă σ ! . Weprovide an alternate proof of this, which allows ă σ ă and is couched in terms suited to our proof of deterministicglobal existence. The theorem above implies a probabilistic well-posedness result as well. To state it, we first recall the definitionof the Gaussian measure associated to H s p T q . The measure is given byd µ s “ Z ´ s e ´ } n } Hs ź x P T d n p x q . A typical element in the support of the measure is of the form n “ n ω “ ÿ k P Z g k p ω qx k y s e ikx , where t g k p ω qu k are independent standard Gaussian random variables. We see that the n is ω -almost-surely in H s ´ ´ z H s ´ , so the measure is supported on Ş r ă s ´ H r p T q . This leads to the following corollary. Corollary 1.5.
The Zakharov system (1) is almost-surely globally well-posed for initial data in the space H s ˆ H r endowed with the Gaussian probability measure if s ą and r ą ´ ´ s p ´ s q . In particular, we have global existence almost-surely in H ` ˆ H ´ ` . The corollary follows by noting that typical data in H r is of the form n “ n ω “ ÿ k P Z g k p ω qx k y r ` ` ǫ e ikx , and thus its Fourier coefficients satisfy sup k x k y r ` | x n p k q| ă 8 almost surely. Hence the conditions of Theorem 1.2 are met for r “ β ´ . AKHAROV SYSTEM ON T As a corollary of the proof of the main theorem, we also have the following Sobolev space global well-posednessresult, which matches the sharp local theory up to the endpoint.
Corollary 1.6.
The one-dimensional Zakharov system (1) is globally well-posed for initial p u , n ˘ q P H s p T q ˆ L p T q for any s ą . We will work with the Besov-type Fourier restriction norms defined by } u } X s,b, “ ››› N s L b } P N,L u } L x L t ››› ℓ N ℓ L , } n } Y s,b, ˘ “ ››› N s L b } Q ˘ N,L n } L x L t ››› ℓ N ℓ L , where N Á L Á { P N,L u p k, τ q : “ χ | k |« N χ | τ ´ k |« L p u p k, τ q { Q ˘ N,L n p k, τ q : “ χ | k |« N χ | τ ˘| k ||« L p n p k, τ q . We also have time-localized versions of these spaces, denoted X s,b, δ and Y s,b, ˘ δ , which are defined in the usual fashion.These norms were used for the Zakharov system in [1]; their properties may be found there. In the following, weoften drop the ˘ from the wave part notation for simplicity. Acknowledgments
The author is grateful to Professor G. Staffilani for many helpful discussions. This work was supported by NSFMSPRF
Proof of Theorem 1.2
Proof.
The idea of the proof is to combine the high-low method of Bourgain with an I-method argument. In thefollowing, N HL " N I " n ˘ into high and low parts: n ˘ p x q “ P ď N HL n ˘ ` P n ą N HL n ˘ “ : n ˘ L0 p x q ` n ˘ H0 p x q . Here P ď N HL denotes the spatial frequency projection onto frequencies of magnitude at most N HL , i.e. { P ď N HL f : “ χ t| k |ď N HL u p f and P ą N HL : “ Id ´ P ď N HL . Let p u , n ˘ q be the solution to (1) with initial data p u , n ˘ L0 q . To obtain this solution, we use the local theoryadapted to the I-method from [11, Prop. 4.5]. This says that we can obtain a solution p u , n ˘ q P X s, , δ ˆ Y , , ˘ ,δ on r , δ s , for δ « ´ } I u } H ` } n ˘ L0 } L ¯ ´ ´ . Here the smoothing operator I : H s Ñ H is defined by p I f p ξ q “ m p ξ q p f p ξ q , where m p ξ q “ $’&’% | ξ | ă N I , p N I {| ξ |q ´ s for | ξ | ą N I . E. COMPAAN
The multiplier m is taken to be smooth and non-increasing in | ξ | . The multiplier, and hence I itself, is dependant onthe choice of N I , but we elect to suppress this in the notation for simplicity.For initial data satisfying } u } H s “ K and sup k | y n ˘ L0 p k q| ď K x k y ´ β , we may choose δ « ” KN ´ s I ` KN ´ β HL ı ´ ´ “ : N ´ ´ . and obtain the bound(3) δ ´ ˜ } I u } X , , δ ` } n ˘ } Y , , ˘ ,δ ¸ À . To continue the argument, we need a local theory for the difference equation which results when we subtract thissolution p u , n ˘ q from the solution for the Zakharov system (1) with the full initial data p u , n q . This local theoryis given by Proposition 3.1 below. Essentially, it says that as long as | z n ˘ H0 p k q| ď C x k y ´ β , the difference equationcan be solved on r , δ s , and the H s ˆ L norm of its nonlinear part is at most order N s ´ β ´ HL . That is, the nonlinearpart is small and smooth. We add this nonlinear part p v , r m ˘ q to p u p δ q , n ˘ p δ qq and evolve according to (1) again to obtain functions p u , n ˘ q which solve (1) on the interval r δ, δ s . At time 2 δ , we again add in the smooth part p v , r m ˘ q of thesolution to the difference equation, and again evolve according (1) to obtain p u , n ˘ q . This can continue as long asthe norm of the solution does not grow too much. To control the growth, we carefully choose the frequency thresholds N HL and N I .To understand the growth of the norm, we study the Hamiltonian energy. The growth of the Hamiltonian isbounded as follows. Recall that p u j , n ˘ j q is defined on the interval rp j ´ q δ, jδ s with p u j , n ˘ j qp δ p j ´ qq “ ´ u j ´ ` v j ´ , n ˘ j ´ ` r m ˘ j ´ ¯ p δ p j ´ qq . Then the growth of the Hamiltonian is bounded by ˇˇˇ H p I u J , n ˘ J qp δJ q ´ H p I u , n ˘ qp q ˇˇˇ ď ˇˇˇ H p I u J , n ˘ J qp δJ q ´ r H p u J , n ˘ J qp δJ q ˇˇˇ ` J ÿ j “ ˇˇˇ r H p u j , n ˘ j qp δj q ´ r H p u j , n ˘ j qp δ p j ´ qq ˇˇˇ ` J ´ ÿ j “ ˇˇˇ r H p u j ` v j , n ˘ j ` r m ˘ j qp δj q ´ r H p u j , n ˘ j qp δj q ˇˇˇ ` ˇˇˇ r H p u , n ˘ qp q ´ H p I u , n ˘ qp q ˇˇˇ , where r H is the modified Hamiltonian given in [11, Section 3]. It is defined by r H p u, n ˘ q “ } I u } H ` } n ˘ } L ` ÿ ř k j “ p u p k q p u p k q ” σ ` p k , k q p n ` p k q ` σ ´ p k , k q p n ´ p k q ı . If β “ , we must include an additional factor of log N HL . This is harmless, so the case is ommitted for simplicity. This holds as long as δ is sufficiently small. Specifically, it is necessary that δ ǫ ´ C À
1, where ǫ ! δ ´ ǫ ´ } I u } X , , δ ` } n ˘ } Y , , ˘ ,δ ¯ À
1. This is possible to achieve for arbitrarily large K and C andarbitrarily small ǫ by taking N I and N HL sufficiently large. AKHAROV SYSTEM ON T Here σ ˘ is the bounded nonsingular Fourier multiplier operator defined in [11, (3.2)-(3.4)]. The effect of the multiplier σ ˘ is to eliminate certain low-order terms from dd t p H p u, n ˘ q , resulting in a more favorable growth bound for themodified Hamiltonian. The precise definition of σ ˘ is rather lengthy (and not used here) so we omit it.We proceed by bounding the growth of each term in the sum above. By [11, Prop. 3.2], for s ą , we can boundthe difference between the Hamiltonian energy and the modified Hamiltonian by ˇˇˇ H p I u J , n ˘ J qp δJ q ´ r H p u J , n ˘ J qp δJ q ˇˇˇ ` ˇˇˇ r H p u , n ˘ qp q ´ H p I u , n ˘ qp q ˇˇˇ À N ´ ` I ´ } I u J p δJ q} H } n ˘ J p δJ q} L ` } I u } H } n ˘ } L ¯ À N ´ ` I N . Next we bound the growth of the modified energy under the Zakharov flow. We use the following estimate. It isalmost identical to the T bound [11, Prop. 4.1]. However, it is slightly stronger because of more favorable estimatesavailable in the one-dimensional case. The proof is in Section 4. Proposition 2.1.
Fix ă s ă and δ P p , q . Suppose p u, n ˘ q is a smooth solution to (1) on the time interval r , δ s . Then | r H p u, n ˘ qp δ q ´ r H p u, n ˘ qp q| À N ´ ` I δ ´ } I u } X , , δ } n ˘ } Y , , ˘ δ ` ” N ´ ` I ` N ´ ` I δ ´ ` N ´ I δ ´ ı ˜ } I u } X , , δ ` } I u } X , , δ } n ˘ } Y , , ˘ δ ¸ . This gives ˇˇˇ r H p u j , n ˘ j qp δj q ´ r H p u j , n ˘ j qp δ p j ´ qq ˇˇˇ À N ´ ` I δ ´ N ` ´ N ´ ` I ` N ´ ` I δ ´ ` N ´ ` I δ ´ ¯ N « N ´ ` I N ` N ´ ` I N ` N ´ ` I N . Finally, we use the definition of the modified energy to note that ˇˇˇ r H p u j ` v j , n ˘ j ` r m ˘ j qp δj q ´ r H p u j , n ˘ j qp δj q ˇˇˇ À } I v j } H } I p u j ` v j q} H ` } r m ˘ j } L } n ˘ j ` r m ˘ j } L ` } u j ` v j } L } u j ` v j } H ` } r m ˘ j } L ` ´ } u j ` v j } L } v } H ` ` } u } L } v } H ` ¯ } n ˘ j } L À N ´ s I N s ´ ´ β HL N. Thus the total growth of the Hamiltonian over r , δJ s is bounded by N ´ ` I N ` p J ´ q ´ N ´ ` I N ` N ´ ` I N ` N ´ ` I N ` N ´ s I N s ´ ´ β HL N ¯ “ N ” N ´ ` I N ` p J ´ q ´ N ´ ` I ` N ´ ` I N ` N ´ ` I N ` N ´ s I N s ´ ´ β HL N ´ ¯ı . This is acceptable as long as N ´ ` I N À J À min " N ´ I , N ´ I N ´ , N ´ I N ´ , N ´p ´ s q I N ´p s ´ ´ β q HL N * . For such J , we can iterate to cover an interval of length δJ « min " N ´ I N ´ ´ , N ´ I N ´ ´ , N ´ I N ´ ´ , N ´p ´ s q I N ´p s ´ ´ β q HL N ´ ´ * . E. COMPAAN
Recalling that N « max t KN ´ s I , C N ´ β HL u , we obtain the following bounds. To simplify the expressions, here the implicit constants here depend on K , C , s ,and β (but not T ): N ´ I N ´ ´ Á T ô N I Á T s ´ ` & N I Á T ` N ´ β ` HL ,N ´ I N ´ ´ Á T ô N I Á T s ´ ` & N I Á T ` N ´ β ` HL ,N ´ I N ´ ´ Á T ô N I Á T s ´ ` & N I Á T ` N ´ β ` HL . The final term in the minimum yields the most complicated constraint: N ´p ´ s q I N ´p s ´ ´ β q HL N ´ ´ Á T ô N I À T ´ ` p ´ s q N β ` ´ s p ´ s q ´ HL & N I À T ´ ´ s N β ´ s ´ s ´ HL . To satisfy all these constraints simultaneously, we require s ą N I “ N α HL for some α P ˜ ´ β, min ! β ` ´ s p ´ s q , β ´ s ´ s )¸ and some N HL very large, dependent on K , T , s , and r . This is possible as long as1 ´ β ă min ! β ` ´ s p ´ s q , β ´ s ´ s ) . Solving this with s ą , we find the constraint β ą ´ min ! ´ s p ´ s q ` , ´ s p ´ s q ` ) “ ´ ´ s p ´ s q ` “ p ´ s q . To obtain a polynomial bound, note that we may take N HL “ C p K, C , s, β q T γ , where γ “ max ! p s ´ q α , β ` ´ s ´ α p ´ s q , α ` β ´ , β ´ s ´ α p ´ s q ) ` . Therefore we can take N « ˆ T α p ´ s q γ ` T ` ´ β ˘ γ ˙ . This allows us to conclude that the nonlinear part of the Zakharov flow grows at most polynomially in the H s ˆ L norm. (cid:3) Local Theory Result
The section contains the proof of the required local theory for the difference equation. The statement is as follows.
AKHAROV SYSTEM ON T Proposition 3.1.
Fix δ ! , ǫ ! , C ą , and input functions p u, n q such that $’’&’’% δ ´ ǫ } u } X s, , δ ď C δ ´ ǫ } n } Y , , δ ď C . Consider the difference equation on r , δ s given by (4) $’&’% iv t ` ∆ v “ Re p n ` m qp u ` v q ´ Re p n q uim t ´ Dm “ D r| u ` v | ´ | u | s with initial data v p x, q “ m p x, q “ m : “ W p t q ˜ ÿ | k |ě N HL h k x k y β e ikx ¸ . We assume that the coefficients h k satisfy sup k | h k | ď C for some C ą . We further assume that s ´ ´ β ă . Then for N HL " sufficiently large and δ sufficiently small, the differenceequation (4) has a solution in H s ˆ H β ´ ´ . Furthermore, if we write m p x, t q “ W p t q m p x q ` r m p x, t q , then we have for t P r , δ s } v } H sx ` } r m } L x À N s ´ β ´ HL ! . The proof of this amounts to proving a contraction for the difference equation(5) $’&’% iv ` ∆ v “ Re p n ` r m ` W p t q m qp u ` v q ´ Re p n q u,i r m t ´ D r m “ D r| u ` v | ´ | u | s with zero initial data on the interval r , δ s in a ball of radius « N s ´ ´ β HL in the space X s, , δ ˆ Y , , δ .The only problematic term to estimate is Re p W p t q m qp u ` v q . All others are covered by existing bilinear estimates due to Kishimoto [10] as long as we take δ sufficiently small.Furthermore, since u typically has much larger norm that v , it suffices to considerRe p W p t q m q u. To close the contraction, it suffices to obtain a bound of the form ›››› η δ p t q ż t e i p t ´ t q ∆ ´ Re p W p t q m q u ¯ d t ›››› X s, , δ À C δ ´ N s ´ ´ β HL } u } X s, , δ À C C δ ǫ ´ N s ´ ´ β HL . The remainder of the paper is devoted to obtaining this bound, relying heavily on estimates previously establishedby Kishimoto.The requirement that δ be small exists because we will require C δ ǫ ´ À C C δ ǫ ´ À P N ,L Re p W ˘ p t q m ˘ q u , P N ,L u , and Q ˘ N ,L W ˘ m ˘ ;i.e. the product Re p W ˘ p t q m ˘ q u is associated with the dyadic variables N and L , the linear wave flow is associatedthe dyadic variables N and L , etc.We also use bars to denote maxima and minima of the dyadic variables, i.e. L jk : “ max t L j , L k u , L : “ L “ max t L , L , L u ,L jk “ min t L j , L k u , L “ L “ min t L , L , L u . We also define L m : “ median t L , L , L u . Resonant case.
Here we confine our attention to the resonant frequencies. These occur when0 “ | L ´ L ´ L | À L. We have | L ´ L ´ L | “ | τ ` τ ´ p k ` k q ´ τ ` k ´ τ ¯ | k || “ | k || k ` k ˘ λ p k q| , where λ p k q is the sign of k . Since k ‰
0, solving this yields k “ k ˘ λ p k q k “ ¯ λ p k q ´ k , for k ‰ . On these frequencies, W ˘ p t q m ˘ u can be estimated in X s, ´ , δ as follows. We use the fact that | h ˘ k | ď C , andcompute } W ˘ p t q m ˘ u } X s, ´ , δ À ››››››› N s L ´ ›››››ż h ˘ k ´ λ p k q x k y β p u p¯ λ p k q ´ k , τ q δ p η p δ p τ ´ τ ˘ | k ´ λ p k q|qq d τ ››››› ℓ k p| k |« N q L τ p| τ ´ k |« L q ››››››› ℓ N ℓ L À C δ ´ ››››› N ´ β L ´ ››››› ż x τ ´ k y ´ ` x τ ´ p¯ λ p k q ´ k q y ´ x τ ´ τ ˘ | k ˘ λ p k q|y ` ˆ ˜ÿ L N s L { P N ,L u p¯ λ p k q ´ k , τ q ¸ d τ ››››› ℓ k p| k |« N q L τ p| τ ´ k |« L q ››››› ℓ N ℓ L AKHAROV SYSTEM ON T À C δ ´ ›››››››› N ´ β L ´ ››››››› x τ ´ k y ´ ` ›››››ÿ L N s L { P N ,L u p¯ λ p k q ´ k , τ q ››››› L τ ››››››› ℓ k p| k |« N q L τ p| τ ´ k |« L q ›››››››› ℓ N ℓ L À C δ ´ ›››››››› N ´ β L ´ ››››››››››››ÿ L N s L { P N ,L u p¯ λ p k q ´ k , τ q ››››› L τ ››››››› ℓ k p| k |« N ›››››››› ℓ N ℓ L À C δ ´ ››››› N ´ β L ´ ÿ L N s L } P N ,L u } L L ››››› ℓ N ℓ L À C δ ´ ››››› N ´ β ÿ L N s L } P N ,L u } L L ››››› ℓ N À C δ ´ N ´ β HL } u } X s, , I À C C δ ǫ ´ N ´ β HL . High Schr¨odinger frequencies.
We decompose dyadically in frequency space as follows. For general m , } η δ ż t S p t ´ t qp mu q dt } X s, , I « «ÿ N } P N η δ ż t S p t ´ t qp mu q dt } X s, , I ff À «ÿ N ˜ ÿ N ,N ÿ L ,L ,L ›››› η δ ż t S p t ´ t q P N ,L r P N ,L p η δ u q Q N ,L p η δ m qs dt ›››› X s, , I ¸ ff . Furthermore, we calculate that } Q N ,L p η δ W p t q m q} L L “ ›››› h k x k y β δ p η p δ p τ ˘ | k |qq ›››› L τ ℓ k p| k |« N , | τ ˘| k |« L q “ δ ›››› h k x k y β ›››› ℓ p| k |« N q } p η p δτ q} L p| τ |« L q À C δ ˆ N ´ β ˙ L x δL y ´ a “ C δN ´ β L x δL y ´ a . N « N " N and L Á N . Using Kishimoto’s Lemma 4.1 and Corollary 3.3, «ÿ N ˜ ÿ N ,N ÿ L ,L ,L ›››› η δ ż t S p t ´ t q P N ,L r P N ,L p η δ u q Q N ,L p η δ w qs dt ›››› X s, , I ¸ ff À C «ÿ N ˜ ÿ N ,N ÿ L ,L ,L δ ´ b N s L ´ b L L ` m L ` N ´ N ´ N ´ β L x δL y ´ a } P N ,L p η δ u q} L L ¸ ff À C «ÿ N ˜ÿ N ÿ L ,L ,L δ ´ a ´ b N s ` ´ L ´ b L L ` m L ` N ´ ´ β L ´ a } P N ,L p η δ u q} L L ¸ ff . If L ě L , then L À L , so we have the bound C «ÿ N ˜ÿ N ÿ L ,L ,L δ ´ a ´ b L ´ a ` L ´ b ` N s ` ´ N ´ s ´ ´ β ˆ N s L } P N ,L p η δ u q} L L ˙¸ ff . Take a “ ` and b “ ` to obtain C «ÿ N ˜ÿ N ÿ L δ ´ N s ` ´ N ´ s ´ ´ β ˆ N s L } P N ,L p η δ u q} L L ˙¸ ff À C δ ´ } u } X s, , « ÿ N " N N s ` ´ N ´ s ´ ´ β ff À C δ ´ N ´ β HL } u } X s, , À C C δ ǫ ´ N ´ β HL . If L À L , then L À L and we have the bound C «ÿ N ˜ÿ N ÿ L ,L ,L δ ´ a ´ b L ´ a L ´ b ` N s ` ´ N ´ s ´ ´ β ˆ N s L ` } P N ,L p η δ u q} L L ˙¸ ff . Take a “ ` and b “ ` to obtain C δ ´ N ´ β HL } u } X s, ` , À δ ´ N ´ β HL } u } X s, , À C C δ ǫ ´ N ´ β HL . N « N " N and L ! N . Using the same results as in the previous case, and noting that L “ L « N ,we have the bound C «ÿ N ˜ÿ N ÿ L ,L ,L δ ´ a ´ b L ´ a ` L ´ b N s ` ´ N ´ s ´ ´ β ˆ N s L ` } P N ,L p η δ u q} L L ˙¸ ff . Take b “ and a “ ` and proceed as above to obtain the bound C δ ´ N ´ β HL } u } X s, ` , À C δ ´ N ´ β HL } u } X s, , À C C δ ǫ ´ N ´ β HL . N « N , L ! N , and L Á N . In this case, we in fact have L “ L « N . Also note that N À N , N .We use Kishimoto’s Proposition 3.1. This gives the bound C ÿ N ,N ÿ L ,L ,L δ ´ a ´ b L ´ a L ´ b N ´ β N ´ ˆ N s L } P N ,L p η δ u q} L L ˙ . Here we take a “ ` and b “ and again obtain C δ ´ N ´ β HL } u } X s, , À C C δ ǫ ´ N ´ β HL . N « N and L Á N . Here N À N and L À L m . We use Kishimoto’s Prop. 3.1 again.If L ě L , then we arrive at C ÿ N ,N ÿ L ,L ,L δ ´ a ´ b L ´ a L ´ b N ´ β N ´ ˆ N s L } P N ,L p η δ u q} L L ˙ . This closes by taking a “ ` and b “ ` . Otherwise L ě L and we arrive at C ÿ N ,N ÿ L ,L ,L δ ´ a ´ b L ´ a L ´ b N ´ β N ´ ˆ N s L } P N ,L p η δ u q} L L ˙ . This closes by taking a “ ` and b “ ` . AKHAROV SYSTEM ON T N « N and L ! N . Here, since the resonant case has already been addressed, we also have L Á N , andhence we can apply the proof of Kishimoto’s Prop. 3.5. If L “ L with L Á N , or if L “ L or L , with L m Á N or L m ! N À L { N , Kishimoto’s proof translates directly to the one-dimensional case, and we get C ÿ N ,N ÿ L ,L ,L δ ´ a ´ b L ´ a L ´ b L ´ c L L m L N ´ β p N s L c } P N ,L p η δ u q} L L q . To close this, we need a ` b ` c “ ` , and each dyadic sum in L i to converge. Note that N À L , so the sum in N contributes L ` .If L “ L , then take a “ ` , and b “ c “ ` .If L “ L , we can take a “ ` , b “ ` , and c “ ` .If L “ L , then take a “ ` , b “ ` , and c “ ` .These lead to the bound C C δ ǫ ´ N ´ β HL as desired.Otherwise, if L “ L and L m À N , we note from Kishimoto’s proof that for fixed k , the frquency k is confinedto an interval of length À L { N . This leads to the bound ÿ N ,N ÿ L ,L ,L δ ´ a ´ b L ´ a L ´ b L ´ c L L N ´ β N ´ p N s L c } P N ,L p η δ u q} L L q . This can be bounded by δ ´ N ´ β HL } u } X s, , δ by taking a “ ` , b “ ` , and c “ ` .The final possibility is L “ L or L and L m ! N with L ! N N . Exactly the same argument can be used totreat this case.3.3. Low Schr¨odinger Frequencies.
Here we assume that N ! N « N . Noting that L Á N , we can useKishimoto’s Prop. 3.1, which carries through for dimension one, to obtain ÿ N ,N ÿ L ,L ,L δ ´ a ´ b L ´ a L ´ b L ´ c L L m L N s ´ ´ β N ´ s p N s L c } P N ,L p η δ u q} L L q . This closes by taking a ` b ` c “ ` as in the previous case and yields the desired bound δ ´ N s ´ ´ β HL } u } X s, , δ À C C δ ǫ ´ N s ´ ´ β HL . Proof of Proposition 2.1
The proof of this proposition is almost identical to that of [11, Prop. 4.1]. Note that the only difference betweenour result and that of Kishimoto is that we replace an factor of N ´ ` I δ ´ in the estimate by a factor of N ´ ` I δ ´ .We can thus repeat the proof of [11, Prop. 4.1] verbatim, except for the estimate of term (4.2), Cases 1(ii) and2(ii). These are the cases which lead to the factor N ´ ` I δ ´ . In those cases, an L x,t Strichartz estimate [11, Lemma2.11] was used to obtain the bounds. To prove our result, it suffices to obtain a one-dimensional analogue of thisStrichartz estimate. Specifically, we wish to show the following.
Proposition 4.1.
For u , v P L p T ˆ R q , we have } nm } L x,t À L N } n } L x,t } m } L x,t , where for some fixed N and L dyadic, supp r n p k, τ q , supp r m p k, τ q Ď t| k | « N u X t| τ ˘ | k || « L u . Proof.
We have, using Cauchy-Schwartz and Young’s inequalities, } nm } L x,t “ } y nm } L τ ℓ k “ } p n ‹ p m } L τ ℓ k ď sup | k |« N,τ | B p k, τ q| } p n } L τ ℓ k } p m } L τ ℓ k , where B p k, τ q “ ! p k , τ q ˇˇˇ | k | , | k ´ k | « N, | τ ˘ | k || , | τ ´ τ ˘ | k ´ k || « L ) . We can bound the size of the set by noting that τ can range in an interval of length at most L and k can rangeover an interval of size N . This gives | B p k, τ q| ď LN.
Inserting this bound on | B | into the estimate above gives the desired result. (cid:3) Applying this Strichartz estimate to a frequency-constrained function n , we obtain } n } L x,t À N L } n } L x,t « N } n } Y , , ˘ . Note that in the T case, Kishimoto instead obtained } n } L x,t À N } n } Y , , ˘ . Lowering the power to saves us a factor of N and gives us an additional power of δ . Since the L x,t estimateis used on two terms in a product, the net result is gain of N ´ δ in the estimate, as desired. References [1] I. Bejenaru, S. Herr, J. Holmer, D. Tataru,
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