Transport of Gaussian measures under the flow of one-dimensional fractional nonlinear Schrödinger equations
aa r X i v : . [ m a t h . A P ] F e b TRANSPORT OF GAUSSIAN MEASURES UNDER THE FLOW OFONE-DIMENSIONAL FRACTIONAL NONLINEAR SCHR ¨ODINGEREQUATIONS
JUSTIN FORLANO AND KIHOON SEONG
Abstract.
We study the transport property of Gaussian measures on Sobolev spaces ofperiodic functions under the dynamics of the one-dimensional cubic fractional nonlinearSchr¨odinger equation. For the case of second-order dispersion or greater, we establish anoptimal regularity result for the quasi-invariance of these Gaussian measures, followingthe approach by Debussche and Tsutsumi [15]. Moreover, we obtain an explicit formulafor the Radon-Nikodym derivative and, as a corollary, a formula for the two-point functionarising in wave turbulence theory. We also obtain improved regularity results in the weaklydispersive case, extending those in [20]. Our proof combines the approach introduced byPlanchon, Tzvetkov and Visciglia [47] and that of Debussche and Tsutsumi [15].
Contents
1. Introduction 21.1. Main results 21.2. The Radon-Nikodym derivative of the transported measure 61.3. Organization of the paper 102. Notations and preliminary estimates 102.1. Notations 102.2. Preliminary estimates 102.3. Function spaces 122.4. Linear and bilinear Strichartz estimates 123. The strongly dispersive case α ≥
1: Proof of Theorem 1.1 143.1. Proof of Proposition 1.8 153.2. Proof of Theorem 1.1 164. The strongly dispersive case α ≥
1: the energy estimate 175. Uniform L p -integrability of the Radon-Nikodym derivative 245.1. Variational formulation 245.2. Uniform exponential integrability 266. The weakly dispersive case < α < Mathematics Subject Classification.
Key words and phrases. fractional nonlinear Schr¨odinger equation; quasi-invariance; Gaussian measure. Introduction
Main results.
In this paper, we study the transport properties of Gaussian measureson periodic functions under the dynamics of the cubic fractional nonlinear Schr¨odingerequation (FNLS) on the one-dimensional torus T = R / (2 π Z ): ( i∂ t u + ( − ∂ x ) α u = ±| u | u,u | t =0 = φ, (1.1)where u : R × T C is the unknown function, and for α >
0, we denote by ( − ∂ x ) α the Fourier multiplier operator defined by (( − ∂ x ) α f ) b ( n ) := | n | α b f ( n ), n ∈ Z . Here, α measures the strength of the dispersion in FNLS (1.1). When α = 1, FNLS (1.1) is thecubic nonlinear Schr¨odinger equation (NLS) which arises in the study of nonlinear optics,fluids and plasma physics; for further discussion, see [50]. For α = 2, (1.1) correspondsto the cubic fourth order NLS (4NLS) and has applications in the study of solitons inmagnetic materials [26, 56]. In the weakly dispersive case < α <
1, FNLS (1.1) wasintroduced in the context of the fractional quantum mechanics [30] and in continuum limitsof long-range lattice interactions [27]. The equation FNLS (1.1) is no longer dispersivewhen α ≤ although the cubic nonlinear half-wave equation ( α = ) has many physicalapplications ranging from wave turbulence [31, 7] to gravitational collapse [19, 21]. See also[54] regarding FNLS (1.1) when < α < . In this paper, we focus on studying FNLS (1.1)for α > where dispersion is present.Our goal in this paper is to study the statistical properties of solutions to FNLS (1.1). Inparticular, we study the quasi-invariance of Gaussian measures µ s under the flow of (1.1)where, for s ∈ R , the measures µ s are formally written as dµ s = Z − s e − k φ k Hs dφ = Y n ∈ Z Z − s,n e − h n i s | b φ n | d b φ n , (1.2)where Z s and Z s,n are normalization constants. These measures (1.2) are the inducedprobability measure under the random Fourier series: ω ∈ Ω φ ω ( x ) = φ ( x ; ω ) = X n ∈ Z g n ( ω ) h n i s e inx , (1.3)where h · i = (1 + | · | ) and { g n } n ∈ Z is a sequence of independent standard complex-valued Gaussian random variables on a probability space (Ω , F , P ). From the randomFourier series representation, the random distribution φ ω in (1.3) lies in H σ ( T ) almostsurely if and only if σ < s − . (1.4)Consequently, µ s is supported in H σ ( T ) \ H s − ( T ) where σ satisfies (1.4). In particular, thetriplet ( H s , H σ , µ s ) forms an abstract Wiener space; see [23, 29]. In [20], with Trenberth,the first author studied the transport property of Gaussian measures µ s under the flow ofFNLS (1.1) for some range of s > , depending on α > . Our main goal is to improve theregularity restrictions in [20].In order to discuss the transport of Gaussian measures µ s under the flow of (1.1), weneed to understand when there is a well-defined flow for (1.1) in the support of µ s . The In the following, we often drop the harmless factor of 2 π . By convention, we set Var( g n ) = 1, n ∈ Z . UASI-INVARIANCE FOR FNLS 3 well-posedness theory of (1.1) is distinguished by the strength of the dispersion α : (i)strong dispersion α ≥ < α <
1. In the former case (i), forany α ≥
1, the cubic FNLS (1.1) is globally well-posed in H σ ( T ) for σ ≥ α = 1) and 4NLS( α = 2) as they are ill-posed in negative Sobolev spaces in the sense of non-existence ofsolutions [25]; see also [5, 32, 10, 45, 41, 34, 46, 28]. In the latter case (ii), the cubicFNLS (1.1) is locally well-posed in H σ ( T ) for σ ≥ − α [9, 51] and globally well-posed when σ > α +112 [17]. This latter result is not expected to be sharp. For future use, we defineΦ t ( · ) : φ ∈ H σ ( T ) u ( t, φ ) ∈ H σ ( T ) to be the flow map of FNLS (1.1) at time t .In view of this well-posedness theory, we consider Gaussian measures µ s for s > max (cid:18) , − α (cid:19) . (1.5)Our first main result is an optimal regularity result for the quasi-invariance of Gaussianmeasures under the flow of FNLS (1.1) in the strongly dispersive case α ≥
1. This extendsthe result in [20] to the optimal Sobolev regularity.
Theorem 1.1.
Let α ≥ and s > . Then, the Gaussian measure µ s in (1.2) is quasi-invariant under the dynamics of the cubic FNLS (1.1) . More precisely, for every t ∈ R ,the measures (Φ t ) ∗ µ s and µ s are mutually absolutely continuous. Our second main result is an improvement on the regularity restrictions in [20] for theweakly dispersive case < α < Theorem 1.2.
Let < α < and s > − α . Then, for every R > , there exists T > such that for every measurable set A ⊂ { u ∈ H s − − ε ( T ) : k u k H s − − ε ( T ) < R } satisfying µ s ( A ) = 0 , where ε > is sufficiently small, we have (Φ t ) ∗ µ s ( A ) = 0 for every t ∈ [ − T, T ] . Theorem 1.1 is an optimal regularity result for the quasi-invariance of µ s under thedynamics of (1.1) when α ≥
1. In particular, this includes the case of the cubic NLS ( α = 1);see Remark 1.5. Our result improves on the main result in [20], where the first author andTrenberth proved quasi-invariance of µ s under (1.1) for s > max( , − α ) when α ≥
1. Inaddition, we also have an explicit formula for the corresponding Radon-Nikodym derivativeand show that it is locally bounded with respect to µ s ; see Proposition 1.10. In the weaklydispersive setting, Theorem 1.2 improves the regularity restriction of s > min (cid:0) , − α (cid:1) in [20]. Whilst our regularity result does not cover the full range in (1.5), our problemforces us to go beyond current techniques. In particular, the key novelty in our proof is tocombine the approaches in [47] and [15]. Further improving the regularity restrictions inthe weakly dispersive case < α < Remark 1.3.
The local-in-time nature of Theorem 1.2 is an artefact of the lack of an almostsure global well-posedness result on the support of µ s for FNLS (1.1) when < α < s ≤ α +712 . Any improvement in this direction would lead to a corresponding improvementin Theorem 1.2; namely, we could ‘upgrade’ local-in-time quasi-invariance to (global-in-time) quasi-invariance.The transport properties of Gaussian measures have been well studied in probabilitytheory, beginning with the seminal work of Cameron-Martin [8]; see, [8, 48, 12, 13] forexample. In particular, Ramer [48] studied the quasi-invariance of Gaussian measures under J. FORLANO AND K. SEONG general nonlinear transformations. In [57], Tzvetkov initiated the study of the transportproperties of Gaussian measures on functions / distributions under nonlinear HamiltonianPDEs and there has been significant progress in this direction [57, 41, 43, 39, 40, 47, 24,20, 49, 15, 37].In [57, 41, 43, 39, 24, 20, 49, 37], an indirect method has been an effective tool in showingthe quasi-invariance property of the Gaussian measures under Hamiltonian flows. Namely,rather than directly proving the quasi-invariance of the Gaussian measure, one insteadproves the quasi-invariance of a weighted Gaussian measure which is absolutely continuouswith respect to the reference Gaussian measure. This indirect strategy is composed of (i)the construction of the weighted Gaussian measure, where the weight arises from correctionterms stemming from the nonlinearity, and (ii) an efficient energy estimate (with smoothing)on the time derivative of a corresponding modified energy.In contrast, the approach we use to prove Theorem 1.1 and Theorem 1.2 is based on ex-ploiting an explicit formula for the Radon-Nikodym derivative of the transported measurewith an L -cutoff. This approach was introduced by Debussche and Tsutsumi [15]. Moreprecisely, one obtains an explicit expression for the Radon-Nikodym derivative of this trans-ported measure under dynamics which have been truncated to finite dimensions (truncatedFNLS (1.8)) and prove the uniform L p -integrability, p >
1, of the Radon-Nikodym deriva-tive in this truncation. We then obtain an explicit representation for the Radon-Nikodymderivative of the transported measure under the flow of FNLS (1.1), from which Theorem1 . L t,x -Strichartz estimates on T . Namely, we establish an estimatefor the integral in time of the derivative of the H s -energy functional. As an example, forsolutions to FNLS (1.1), the energy estimate (with smoothing) is (cid:12)(cid:12)(cid:12)(cid:12) Re ˆ − T h i | Φ t ( v ) | Φ t ( v ) , D s Φ t ( v ) i L ( T ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( k Φ t ( v ) k X ,
12 + T ) k Φ t ( v ) k θX s − − ε,
12 + T , (1.6)where θ ≤ , ε > T > X s,bT are the local-in-time Fourier restriction norm spaces X s,b which are defined in Subsection 2.3. This is incontrast to the approaches in [57, 41, 43, 39, 24, 20, 49] where the main energy estimates areestablished only for each fixed time t . However, these approaches do have a complementingadvantage: by reducing the analysis to time t = 0, one can exploit random oscillations inthe random Fourier series (1.3) and obtain a probabilistic energy estimate. This approachseems to work well when the dispersion is weaker, such as for nonlinear wave equations; see[43, 24, 49].We now return to discussing our main results Theorem 1.1 and Theorem 1.2. Let usfirst consider the case of high dispersion α ≥ s > up to third-order dispersion( α ≥ ). Hence, below third-order dispersion (1 ≤ α < ), we need to refine the analysisin [15]. The essential difference between our analysis and that in [15] is that we employ asymmetrization argument, as in [20], which allows us to gain extra decay through the meanvalue theorem and the double mean value theorem (Lemma 2.1).In the weakly dispersive case < α <
1, the method in [15] is no longer appropriatesince we must work with the density of the transported measure with an L -cutoff. This L -cutoff is necessary in order to obtain the uniform L p -integrability of the corresponding One can actually allow for 4 ≤ θ < ε for some small ε >
0; for example see Lemma 5.1.
UASI-INVARIANCE FOR FNLS 5
Radon-Nikodym derivative. The disadvantage here is that we need deterministic control ofthe flow of individual solutions by the L -norm of the initial data. In view of the regularitygap between the known well-posedness of FNLS (1.1) and L x , we expect such control onlythrough the L -conservation of solutions to (1.1) at each fixed time. Hence, we must usethe weaker space-time norm L ∞ ([0 , T ]; L x ) instead of the X , + T norm in (1.6). Note thatthis gap is due to the weaker dispersion which causes a derivative loss in the L t,x -Strichartzestimate; see Lemma 2.9. Thus, following the approach in [15], we can only ever hope toobtain results for some α > α > and with a more restrictive range of s than stated inTheorem 1.2; see Remark 6.5 for more details.To go beyond this difficulty, we combine the local argument in [47] with the density basedapproach in [15]. The benefit of our approach is two-fold: (i) we are no longer constrainedby an L -cutoff in the measure and (ii) we weaken the space-time energy estimate (1.6) to (cid:12)(cid:12)(cid:12)(cid:12) Re ˆ − T h i | Φ t ( v ) | Φ t ( v ) , D s Φ t ( v ) i L ( T ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:0) k Φ t ( v ) k kX s − − ε,
12 + T (cid:1) , (1.7)for some k ∈ N . This allows us to obtain Theorem 1.2. In proving (1.7) for the weaklydispersive FNLS (1.1), we exploit both the linear and bilinear Strichartz estimates in [51]. Remark 1.4.
We point out that the same issue of dealing with L -cutoffs on the probabilitymeasures was faced in the analysis of FNLS (1.1) for < α < < α <
1, the first author and Trenberth appliedthe method due to Planchon, Tzvetkov and Visciglia [47]. The key point here is to arguelocally within H σ ( T ), which allows one to use deterministic growth bounds on solutions to(1.1) to weaken the necessary fixed-time energy estimate. In [24], the authors combinedthe approach in [47] with the energy method in [57, 41, 43] to produce a hybrid argumentwhich has a further weakened energy estimate. This hybrid argument was then applied in[20] to obtain an improved regularity restriction in the restricted range α > . Remark 1.5. (i) For the particular case of the cubic NLS ( α = 1), Theorem 1.1 impliesthat the Gaussian measures µ s are quasi-invariant under this NLS flow for any s > .This extends implicit results in the works of Bourgain [4] and Zhidkov [60]. Those authorsshowed that for each k ∈ N , NLS has an invariant weighted Gaussian measure ρ k whichis mutually absolutely continuous with the Gaussian measure µ k . The invariance of themeasures ρ k imply the quasi-invariance of the Gaussian measures µ k for each k ∈ N . Seealso [42] for further discussion.(ii) In this remark, we only consider the defocusing FNLS (1.1); that is, (1.1) with thenegative sign. Due to the Hamiltonian structure of FNLS (1.1), it is natural to study thetransport property of the following weighted Gaussian measure (Gibbs measure) dρ α = Z − e − ´ | u | dx dµ α . In [4], Bourgain proved that ρ is invariant under the dynamics of NLS. This extends to α > , in the sense that ρ α is invariant under the dynamics of FNLS (1.1); see [16]. In theremaining range < α ≤ , one no longer has deterministic well-posedness in the supportof µ α . In [52], Sun and Tzvetkov proved almost sure global well-posedness of FNLS (1.1)with respect to µ α for any α > −√ ≈ .
562 and hence the corresponding invarianceof ρ α ; see also [51]. These results also imply the quasi-invariance of µ α . Thus, the quasi-invariance of µ α under the dynamics of FNLS (1.1) with weak dispersion < α < J. FORLANO AND K. SEONG
Remark 1.6.
Whilst Theorem 1.1 is optimal with respect to (1.5), it is possible to inquireabout the quasi-invariance of Gaussian measures supported below L ( T ) under suitablyrenormalized FNLS dynamics. Recently, the second author with Oh [37] proved the quasi-invariance of Gaussian measures µ s in negative Sobolev spaces under the flow of the (renor-malized) 4NLS for any s > . In [44], Oh, Tzvetkov and Wang established the invarianceof the white-noise measure µ under the flow of the (renormalized) 4NLS. These resultsseem to be extendable to the (renormalized) FNLS for some α > Remark 1.7.
The transport property of Gaussian-measures under the flow of HamiltonianPDEs is intimately related to the strength of dispersive effects. Exploring this idea in thepositive sense motivates our study of FNLS (1.1). In the negative sense, Oh, Sosoe andTzvetkov [39] showed that Gaussian measures µ s are not quasi-invariant under the followingdispersionless ODE ( α = 0): i∂ t u = | u | u . See [49] for a similar result for a dispersionlesssystem. Due to the change of variables u ( t, x ) u ( t, x − t ), this implies that µ s is notquasi-invariant under the flow of the transport equation i∂ t u + i∂ x u = | u | u. We therefore expect that µ s is also not quasi-invariant under the flow of the half-waveequation ( α = ), and it would be of interest to prove or disprove this claim.1.2. The Radon-Nikodym derivative of the transported measure.
Before dis-cussing further consequences of our results, we state the explicit formula for the Radon-Nikodym derivative of the transported measure when α ≥
1; see Proposition 1.8 below. Tothis end, we first introduce some notations. Given
R >
0, we use B R to denote the ball ofradius R in L ( T ) centered at the origin. Define E N by E N = π ≤ N L ( T ) = span { e inx : | n | ≤ N } and let E ⊥ N be the orthogonal complement of E N in L ( T ), where π ≤ N is the (sharp)Dirichlet projection to frequencies {| n | ≤ N } . Given s ∈ R , let µ s be the Gaussian measureon H s − − ε ( T ) defined in (1.2). Then, we can write µ s as µ s = µ s,N ⊗ µ ⊥ s,N , where µ s,N and µ ⊥ s,N are the marginal distributions of µ s restricted to E N and E ⊥ N , respec-tively. In other words, µ s,N and µ ⊥ s,N are induced probability measures under the followingrandom Fourier series: π ≤ N φ : ω ∈ Ω π ≤ N φ ( x ; ω ) = X | n |≤ N g n ( ω ) h n i s e inx ,π >N φ : ω ∈ Ω π >N φ ( x ; ω ) = X | n | >N g n ( ω ) h n i s e inx , respectively. Formally, we can write µ s,N and µ ⊥ s,N as dµ s,N = Z − s,N e − k π ≤ N φ k Hs dφ N and dµ ⊥ s,N = b Z − s,N e − k π >N φ k Hs dφ ⊥ N , where dφ N and dφ ⊥ N are (formally) the products of the Lebesgue measures on the Fouriercoefficients: dφ N = Y | n |≤ N d b φ ( n ) and dφ ⊥ N = Y | n | >N d b φ ( n ) . UASI-INVARIANCE FOR FNLS 7
Given r >
0, we also define measures µ s,r and µ s,N,r by dµ s,r = {k φ k L ≤ r } dµ s and dµ s,N,r = {k φ N k L ≤ r } dµ s,N . Finally, we introduce the following truncated version of FNLS (1.1). Given N ∈ N , weconsider the Cauchy problem ( i∂ t u N + ( − ∂ x ) α u N = ± π ≤ N (cid:0) | u N | u N (cid:1) , ( t, x ) ∈ R × T ,u | t =0 = π ≤ N φ = φ N . (1.8)When α ≥
1, the global well-posedness of (1.8) (Lemma 4.1) implies we may define Φ
N,t ( · )to be the flow map of the truncated dynamics (1.8) at time t : φ N ∈ H σ ( T ) → Φ N,t ( φ ) =: u N ( t, φ N ) ∈ H σ ( T ).We now describe the approach of Debussche and Tsutsumi [15] for proving the quasi-invariance of Gaussian measures as applied to FNLS (1.1). The key result is the following: Proposition 1.8.
Let α ≥ , N ∈ N , t ∈ R and r > . Then, the Radon-Nikodymderivative of the transported measure (Φ N,t ) ∗ µ s,N,r with respect to µ s,N,r is given by d (Φ N,t ) ∗ µ s,N,r dµ s,N,r = f N ( t, · )= exp (cid:18) ∓ ˆ t Re h i ( | u N | u N )( − t ′ , · ) , D s u N ( − t ′ , · ) i L ( T ) dt ′ (cid:19) . (1.9) Suppose, in addition, that { f N ( t, · ) } N ∈ N is uniformly bounded in L p ( dµ s,r ) , for some
1. In Proposition 1.8, the crucial assumption is the uniform L p -integrability ofthe Radon-Nikodym derivative f N ( t, · ) in (1.9), corresponding to the truncated flow (1.8).This then implies the explicit formula for the Radon-Nikodym derivative of the transportedmeasure (Φ t ) ∗ µ s,r with respect to µ s,r and hence the mutual absolute continuity of (Φ t ) ∗ µ s,r and µ s,r . Theorem 1.1 follows from this mutual absolute continuity and a limiting argument;see Section 3.The proof of the uniform L p -integrability of the Radon-Nikodym derivative is split intoa deterministic step and a probabilistic step. In the deterministic step, one proves a deter-ministic energy estimate for log f N ( t, · ); see Lemma 4.2. The probabilistic step then verifiesthe uniform L p -integrability of this bound for f N ( t, · ). In [15], the proof of this latter stepuses a dyadic pigeon hole argument following [4], while we use the variational approach dueto Barashkov and Gubinelli [1] (the Bou´e-Dupuis variational formula).The approach outlined above is not the only way to verify the key assumption in Propo-sition 1.8. Indeed, higher integrability of the Radon-Nikodym derivative is equivalent to aquantitative quasi-invariance statement. Proposition 1.9. ([22, Proposition 3.4] , [6, Proposition 3.5]) Let s ∈ R , σ ∈ R satisfying (1.4) , ρ be a probability measure supported on H σ ( T ) and, for each t ∈ R , a measurable J. FORLANO AND K. SEONG map Ψ t : H σ ( T ) → H σ ( T ) . Suppose there exists ≤ δ < and C = C ( t, δ ) > such that (Ψ t ) ∗ ρ ( A ) ≤ C { ρ ( A ) } − δ , (1.10) for any measurable set A ⊆ H σ ( T ) . By the Radon-Nikodym theorem, there exists a non-negative f ( t, · ) ∈ L ( dρ ) such that d (Ψ t ) ∗ ρ = f ( t, · ) dρ . Then, (i) The property (1.10) holds for some ≤ δ < if and only if f ( t, · ) ∈ L ( dρ ) ∩ L pw ( dρ ) with p = δ . Namely, f ( t, · ) ∈ L ( dρ ) and there exists C ′ > such that we have ρ ( { φ : | f ( t, φ ) | > λ ) ≤ C ′ h λ i − p , for all λ > . (ii) The property (1.10) holds for δ = 0 if and only if f ( t, · ) ∈ L ∞ ( dρ ) . The above proposition holds in general whenever there are two finite measures µ, ν onsome measure space for which µ ≪ ν , so that the Radon-Nikodym derivative exists. Weconsidered particular measures in the above statement that are more relevant for our currentdiscussion. We remark that in [57, 41, 39, 43], the bounds (1.10) were established for aweighted Gaussian measure with an appropriate cutoff and thus one has L p -bounds for theassociated Radon-Nikodym derivatives.Applying our new approach in this paper to the strongly dispersive FNLS (1.1), weobtain the following result. Proposition 1.10.
Let α ≥ and s > . For every t ∈ R , the Radon-Nikodym derivative ofthe transported measure (Φ t ) ∗ µ s with respect to µ s , which exists from Theorem 1.1, belongsto L ( dµ s ) ∩ L ∞ loc ( dµ s ) . Moreover, we have the following explicit formula d (Φ t ) ∗ µ s dµ s ( φ ) = f ( t, φ ) = exp (cid:18) ∓ ˆ t Re h i ( | u | u )( − t ′ , φ ) , D s u ( − t ′ , φ ) i L ( T ) dt ′ (cid:19) (1.11) for all φ ∈ L ( T ) . The claim (1.11) follows from Theorem 1.1 and Proposition 1.8; see Section 6. The localboundedness follows from using Lemma 4.2 in the argument of Section 6, which impliesthat (1.10) holds for any measurable set A ⊆ B R ⊂ L ( T ), R >
0, and for δ = 0, where ρ = µ s and Ψ t is the flow of the strongly dispersive FNLS (1.1). This is a general featureof our argument and a similar partial quantitative estimate is obtained by the approach in[47] and the hybrid argument in [24] (albeit with some δ > µ s : Proposition 1.11.
Let α ≥ , s > , Φ t denote the flow of FNLS (1.1) and f ( t, · ) denotethe Radon-Nikodym derivative of (Φ t ) ∗ µ s with respect to µ s as in Proposition 1.10. Givenany ≤ p < ∞ , let g , g ∈ L ( dµ s ) ∩ L p loc ( dµ s ) . Then, for any t ∈ R , the transports of themeasures g ( φ ) dµ s ( φ ) , and g ( φ ) dµ s ( φ ) by Φ t are given by G ( t, φ ) dµ s ( φ ) and G ( t, φ ) dµ s ( φ ) , We thank Nikolay Tzvetkov for suggesting this application to us.
UASI-INVARIANCE FOR FNLS 9 resepectively, for suitable G ( t, · ) , G ( t, · ) ∈ L ( dµ s ) ∩ L p loc ( dµ s ) . Moreover, for any R > ,we have (cid:13)(cid:13) B R (cid:0) G ( t, · ) − G ( t, · ) (cid:1)(cid:13)(cid:13) L p ( dµ s ) ≤ (cid:13)(cid:13) B R f ( t, · ) (cid:13)(cid:13) − p L ∞ ( dµ s ) k B R ( g − g ) k L p ( dµ s ) , (1.12) when p > , and k G ( t, · ) − G ( t, · ) k L ( dµ s ) = k g − g k L ( dµ s ) . (1.13)We point out that if the Radon-Nikodym derivative f ( t, · ) was (globally) in L ∞ ( dµ s ),then (1.12) would become (cid:13)(cid:13) G ( t, · ) − G ( t, · ) (cid:13)(cid:13) L p ( dµ s ) ≤ (cid:13)(cid:13) f ( t, · ) (cid:13)(cid:13) − p L ∞ ( dµ s ) k g − g k L p ( dµ s ) . The application of quasi-invariance of Gaussian measures (more generally, of arbitraryprobability measures) to the L -stability statement, as in (1.13), was first noted by Sun andTzvetkov in [51]. The result of Proposition 1.11 shows that the higher (local) integrabilityof the Radon-Nikodym derivative with respect to the reference Gaussian measure µ s impliesstability locally in L p ( dµ s ), for p > µ s in (1.2),one then aims to study the evolution in time of certain averaged quantities of the solutions u ( t, x ). One such key quantity is the two-point function { N ( n, t ) } n ∈ Z , which represents theaverage energy stored at each frequency n ∈ Z , and are defined by N ( n, t ) = E [ | b u n ( t ) | ] . (1.14)The aim in wave turbulence theory is then to derive an effective equation, called the wavekinetic equation, for the evolution of the two-point function { N ( n, t ) } n ∈ Z and hence thedistribution of energy at each frequency; see for example [18].In [43, Corollary 1.4], Oh and Tzvetkov showed that there is a formula for the two-pointfunction (1.14), in terms of the Radon-Nikodym derivative of the transported Gaussianmeasure. This reduces the study of the two-point functions to studying the dynamicalproperties of the Radon-Nikodym derivative. Whilst the result there was stated under theassumption that the Radon-Nikodym derivative belongs to L ( dµ s ), it turns out that itstill holds under the weaker assumption L ( dµ s ) . In the following corollary, we furthershow that the two-point functions { N ( n, t ) } n ∈ Z can be expressed in terms of the explicitformula (1.11) for the Radon-Nikodym derivative. It is interesting then to understand ifthis explicit formula may be of further use in this setting. Corollary 1.12.
Let α ≥ , s > , and Φ t denote the flow of FNLS (1.1) . Then, we have N ( n, t ) = ˆ L ( T ) | b φ ( n ) | d (Φ t ) ∗ µ s dµ s ( φ ) dµ s ( φ )= ˆ L ( T ) | b φ ( n ) | exp (cid:18) ∓ ˆ t Re h i ( | u | u )( − t ′ , φ ) , D s u ( − t ′ , φ ) i L ( T ) dt ′ (cid:19) dµ s ( φ ) for any n ∈ Z and t ∈ R , where N ( n, t ) is the two-point function defined in (1.14) . We learnt from the authors in [43] that the hypothesis in [43, Corollary 1.4] may be weakened from L ( dµ s ) to L ( dµ s ). Organization of the paper.
In Section 2, we introduce some notations and pre-liminary estimates, such as the linear and bilinear Strichartz estimates. In Section 3, wepresent the proof of Proposition 1.8 and then prove Theorem 1.1 by assuming Lemma 4.2and the uniform L p -integrability of the Radon-Nikodym derivative. The energy estimate(Lemma 4.2) is proved in Section 4 while we show the uniform L p -integrability in Section5. In Section 6, first by assuming a weaker space-time energy estimate (Lemma 6.2) andusing a hybrid argument (combining the approaches in [47] and [15]), we present the proofof Theorem 1.2 and establish the weaker space-time energy estimate (Lemma 6.2).2. Notations and preliminary estimates
Notations.
In the following, we fix small ε > σ = s − − ε such that (1.4) is satisfied. Given N ∈ N ∪ {∞} , we use π ≤ N to denote the Dirichletprojection onto the frequencies {| n | ≤ N } and set π >N := Id − π ≤ N . When N = ∞ , it isunderstood that π ≤ N = Id. Given dyadic M >
0, we let P M denote the Littlewood-Payleyprojector onto frequencies {| n | ∼ M } , such that f = ∞ X M ≥ P M f. When M = 1, P is a smooth projector to frequencies {| n | . } .We use a + (and a − ) to denote a + ε (and a − ε , respectively) for arbitrarily small ε ≪ ε > ε → Preliminary estimates.
In this subsection, we record some elementary estimatesthat will be useful in our analysis.In our approach, the phase functionΦ α ( n ) := Φ( n , n , n , n ) := | n | α − | n | α + | n | α − | n | α (2.1)naturally arises as the source of dispersion. In order to exploit this for a smoothing benefit,we crucially rely on the following lower bound (see Lemma 2.4): for α > , we have | Φ α ( n ) | & | n − n || n − n | n α − when n = n − n + n and where n max := max( | n | , | n | , | n | , | n | ) + 1. This lower bound first appeared in thesetting < α ≤ α > ( n ) = n − n + n − n = − n − n )( n − n ) when n = n − n + n , and 4NLS (see [41, Lemma 3.1]). In particular, we rely on an understanding of the ratioΨ s ( n )Φ α ( n ) , (2.2)where Ψ s ( n ) := h n i s − h n i s + h n i s − h n i s . A key tool for this analysis is the doublemean value theorem; see [11, Lemma 2.3].
UASI-INVARIANCE FOR FNLS 11
Lemma 2.1.
Let ξ, η, λ ∈ R and f ∈ C ( R ) . Then, we have f ( ξ + η + λ ) − f ( ξ + η ) − f ( ξ + λ ) + f ( ξ ) = λη ˆ ˆ f ′′ ( ξ + t λ + t η ) dt dt . The following result is a direct consequence of Lemma 2.1, using that the map x
7→ h x i s − is non-decreasing when s ≥ Lemma 2.2.
Fix s ≥ and let n , n , n , n ∈ Z be such that n = n − n + n . Then,we have | Ψ s ( n ) | . | n − n || n − n | n s − , where n max = max( | n | , | n | , | n | , | n | ) + 1 and the implicit constant depends only on s . The use of Lemma 2.2 was sufficient to obtain the quasi-invariance results for (1.1) in[20] when < α < s >
1. In this paper, however, we need to refine the analysis inLemma 2.2 to consider s ≤ Lemma 2.3.
Let n , n , n , n ∈ Z such that n = n − n + n . Then: (i) If | n | ∼ | n | ∼ | n | ∼ | n | and max( | n − n | , | n − n | ) ≪ | n | , we have | Ψ s ( n ) | . | n − n || n − n | n s − , (2.3) for any s ∈ R . (ii) If | n | ∼ | n | ≫ min( | n | , | n | ) and s ≥ , we have | Ψ s ( n ) | . | n − n | n s − . (2.4) Proof.
The second estimate (2.4) follows from the mean value theorem. As for the firstestimate (2.3), Lemma 2.1 implies | Ψ s ( n ) | . | n − n || n − n | sup t ,t ∈ [0 , (cid:0) h n + t ( n − n ) + t ( n − n ) i s − (cid:1) . | n − n || n − n |h n i s − , thanks to the condition max( | n − n | , | n − n | ) ≪ | n | . Hence, we obtain the desiredresult. (cid:3) We now state the following estimate related to the phase function (2.1).
Lemma 2.4 ([17, 20]) . Fix α > and let n , n , n , n ∈ Z be such that n = n − n + n .Then, we have | Φ α ( n ) | & | n − n || n − n | ( | n − n | + | n − n | + | n | ) α − & | n − n || n − n | n α − where the implicit constant depends only on α . In particular, when { n , n } 6 = { n , n } , thephase function Φ α satisfies the following size estimates: (i) If | n | ∼ | n | ∼ | n | ∼ | n | , then | Φ α ( n ) | & | n − n || n − n | n α − . (ii) If | n | ∼ | n | ≫ min( | n | , | n | ) , then | Φ α ( n ) | & | n − n | n α − . (ii) If | n | ∼ | n | ≫ max( | n | , | n | ) or | n | ∼ | n | ≫ max( | n | , | n | ) , then | Φ α ( n ) | & n α max . Combining Lemma 2.3 and Lemma 2.4, we obtain an estimate on the ratio in (2.2). Thisresult is fundamental in our analysis; see the proof of Lemma 4.2 and Lemma 6.2.
Lemma 2.5.
Let α > and s ≥ . Then, for any n , n , n , n ∈ Z satisfying n = n − n + n with { n , n } 6 = { n , n } , we have | Ψ s ( n ) || Φ α ( n ) | . n s − α max . (2.5) Proof.
Apart from the subcase where | n | ∼ | n | ∼ | n | ∼ | n | and max( | n − n | , | n − n | ) & | n | , (2.5) follows from Lemma 2.3 and Lemma 2.4. As for this remaining subcase, we notethat the mean value theorem implies | Ψ s ( n ) | . n s − min( | n − n | , | n − n | ) , and thus | Ψ s ( n ) || Φ α ( n ) | . n s − α +1max | n − n | , | n − n | ) . n s − α max . This completes the proof of Lemma 2.5. (cid:3)
Function spaces.
We use the Fourier restriction norm spaces X s,b , which are adaptedto the linear flow of (1.1). More precisely, given s, b ∈ R , we define the space X s,b ( R × T )via the norm k v k X s,b ( R × T ) = kh n i s h τ − | n | α i b b v ( τ, n ) k L τ ℓ n ( R × Z ) , where b v ( τ, n ) denotes the space-time Fourier transform of v ( t, x ). Given T >
0, we alsodefine the local-in-time version X s,b ([0 , T ] × T ) of X s,b ( R × T ) as X s,b ([0 , T ] × T ) = inf {k v k X s,b ( R × T ) : v | [0 ,T ] = u } . We will denote by X s,b and X s,bT the spaces X s,b ( R × T ) and X s,b ([0 , T ] × T ), respectively.We have the following embedding: for any s ∈ R and b > , we have X s,bT ֒ → C ([0 , T ]; H s ( T )) . Given any function F on [0 , T ] × T , we denote by ˜ F any extension of F onto R × T . Lemma 2.6.
Let s ≥ and ≤ b < . Then, for any compact interval I , we have k I ( t ) f k X s,b . k f k X s,b , where the implicit constant depends only on b . For a proof of Lemma 2.6, see for example [14].2.4.
Linear and bilinear Strichartz estimates.
In this subsection, we present somelinear L t,x -Strichartz estimates on T as well as a bilinear estimate. We begin with the L t,x -Strichartz estimate. Lemma 2.7.
For α ≥ and b = (cid:0) α (cid:1) , we have k u k L t,x ( R × T ) . k u k X ,b ( R × T ) . (2.6) UASI-INVARIANCE FOR FNLS 13
We note that the value of b in Lemma 2.7 is sharp in the sense that the estimate (2.6)fails if b < (cid:0) α (cid:1) . To see this, one may easily adapt the counterexample in [41, Footnote9]. The estimate (2.6) is a generalization of the following L t,x -Strichartz estimate due toBourgain [3] for α = 1: k u k L t,x ( R × T ) . k u k X , ( R × T ) . See also [41] for a corresponding estimate when α = 2. The key difference between theproofs of these estimates and (2.6) is a lack of explicit factorisations due to the fractionalpowers | n | α . For this, we need the following counting estimate. Lemma 2.8.
Let α ≥ , n ∈ Z , τ ∈ R and M ≥ . Then, we have { n ∈ Z : || n | α + | n − n | α − τ | ≤ M } . α M α , (2.7) uniformly in n ∈ Z and τ ∈ R .Proof. Let Ψ n ( n ) := | n | α + | n − n | α and A ( n, τ, M ) be the set in (2.7). We claim that itsuffices to consider n >
0. First, if n = 0, we can argue directly as follows. We may assume τ ≥ M , since otherwise we have | n | . M α . Then, | n | ∈ [2 − α ( τ − M ) α , − α ( τ + M ) α ]and hence by the mean value theorem, A (0 , τ, M ) ≤ C ( α ) (cid:2) ( τ + M ) α − ( τ − M ) α (cid:3) . M α . In view of the property Ψ − n ( n ) = Ψ n ( − n ), we may assume n >
0. We claim that wemay also assume n ≥ n . Indeed, if n ∈ A ( n, τ, M ), then since Ψ n ( n − n ) = Ψ n ( n ), n − n ∈ A ( n, τ, M ) with n − n > n . Now suppose there are at least two elements in n , n ∈ A ( n, τ, M ) and without loss of generality, we assume n > n . Since α ≥
1, thefunction x ∈ R Ψ n ( x ) belongs to C ( R ) and we haveΨ ′ n ( n ) = 2 α | n | α − n − α | n − n | α − ( n − n ) , Ψ ′′ n ( n ) = 2 α (2 α − | n | α − + 2 α (2 α − | n − n | α − . Since n > n , Ψ ′ n ( n ) ≥ n ( n ) − Ψ n ( n ) = Ψ ′ n ( n )( n − n ) + 12 ˆ (1 − t )Ψ ′′ n ( tn + (1 − t ) n )( n − n ) dt & α ˆ (1 − t ) (cid:12)(cid:12) n + t ( n − n ) (cid:12)(cid:12) α − ( n − n ) dt & α | n − n | α . Now, for n , n ∈ A ( n, τ, M ), we have Ψ n ( n ) , Ψ n ( n ) ∈ [ τ − M, τ + M ] and hence | n − n | . M α . This completes the proof of (2.7). (cid:3)
Proof of Lemma 2.7.
We follow the standard argument as found in [53]. For each dyadicnumber M , we define b u M ( τ, n ) := M ≤h τ −| n | α i < M b u ( τ, n ) . Then, by symmetry, we have k u k L t,x = k uu k L t,x . X m ≥ X M ≥ k u m M u M k L t,x . Hence, if we have k u m M u M k L t,x . (2 m M ) α M k u m M k L t,x k u M k L t,x = 2 − θm (2 m M ) (1+ α ) M (1+ α ) k u m M k L t,x k u M k L t,x (2.8)with θ := ( − α ) >
0, then, from the Cauchy-Schwarz inequality, we obtain (2.6). There-fore, it suffices to show (2.8). From Plancherel’s theorem and the Cauchy-Schwarz inequal-ity, we have k u m M u M k L t,x . M sup ( τ,n ) ∈ R × Z A ( τ, n, m M ) k u m M k L t,x k u M k L t,x . (2.9)It follows from (2.7) that we have A ( τ, n, m M ) . (2 m M ) α . (2.10)Thus, from (2.9) and (2.10), we obtain (2.8), which completes the proof of Lemma 2.7. (cid:3) Note that by interpolation between X , + ֒ → L ∞ t L x , (2.11)and X , = L t,x , we have X , (cid:0) − p (cid:1) + ֒ → L pt L x , (2.12)for any 2 < p < ∞ . Similarly, by interpolation between (2.6) and (2.11), we have X , − s (cid:0) α (cid:1) +2 s + ֒ → L s − s t L − s s x , for any 0 < s < . Then, the Sobolev embedding W s, − s s ֒ → L implies X s, − s (cid:0) α (cid:1) +2 s + ֒ → L s − s t L x . (2.13)By Bernstein’s inequality, we have k P N f k L ∞ t,x . N k P N f k X ,
12 + . (2.14)For the weakly dispersive case of (1.1) ( < α < L t,x -Strichartz estimate loosesderivatives due to the weaker curvature of the phase function (2.1). Lemma 2.9. [51, Corollary 2.11]
Given < α ≤ and N ≫ M dyadic, we have k P N f k L t,x . N − α k P N f k X , , (2.15) k P N f · P M g k L t,x . M − α k P N f k X , k P M g k X , , (2.16)3. The strongly dispersive case α ≥ : Proof of Theorem 1.1 In this section, we prove one of our main theorems (Theorem 1.1) by assuming Lemma4.2 and the uniform L p -integrability of the Radon-Nikodym derivative f N ( t, · ) (1.9) of thetransported measure (Φ N,t ) ∗ µ s,N,r . We present the proof of Lemma 4.2 and the uniform L p -integrability in Section 4 and 5, respectively. UASI-INVARIANCE FOR FNLS 15
Proof of Proposition 1.8.
Whilst the proof of Proposition 1.8 closely follows [15,Section 4], we include details in order to make this paper self-contained. We recall the basicinvariance property of the Lebesgue measures dφ N = Q | n |≤ N d b φ ( n ). Lemma 3.1 (Liouville’s theorem) . Let N ∈ N . Then, the Lebesgue measure dφ N = Q | n |≤ N d b φ ( n ) is invariant under the flow Φ N,t .Proof of Proposition 1.8.
It follows from a change of variables, Lemma 3.1 and the L -conservation that for any measurable set A in E N , we have(Φ N,t ) ∗ µ s,N,r ( A ) = Z − s,N ˆ φ N ∈ Φ − N,t ( A ) {k φ N k L ≤ r } e − k φ N k Hs dφ N = Z − s,N ˆ φ N ∈ A {k Φ − N,t ( φ N ) k L ≤ r } e − k Φ − N,t ( φ N ) k Hs dφ N = ˆ φ N ∈ A {k φ N k L ≤ r } e − k Φ − N,t ( φ N ) k Hs e k φ N k Hs µ s,N ( dφ N )= ˆ φ N ∈ A e − k Φ − N,t ( φ N ) k Hs e k φ N k Hs µ s,N,r ( dφ N ) . Therefore, we have f N ( t, φ N ) := d (Φ N,t ) ∗ µ s,N,r dµ s,N,r ( φ N ) = e − k Φ − N,t ( φ N ) k Hs e k φ N k Hs . (3.1)By differentiating f N ( t, · ) in time, we obtain the following differential equation: ddt f N ( t, φ N ) = − Re h i ( − ∂ x ) α u N ( − t, φ N ) ± i ( | u N | u N )( − t, φ N ) , D s u N ( − t, φ N ) i f N ( t, φ N )= ∓ Re h i ( | u N | u N )( − t, φ N ) , D s u N ( − t, φ N ) i f N ( t, φ N ) . By solving this differential equation with f N (0 , · ) = 1, we have the explicit expression forthe Radon-Nikodym derivative: f N ( t, φ N ) = d (Φ N,t ) ∗ µ s,N,r dµ s,N,r ( φ N )= exp (cid:18) ∓ ˆ t Re h i ( | u N | u N )( − t ′ , φ N ) , D s u N ( − t ′ , φ N ) i L ( T ) dt ′ (cid:19) . This proves (1.9).We now define the natural extension of f N ( t, · ) on L ( T ) by f N ( t, φ ) = f N ( t, φ N ). Itfollows from the uniform L p ( dµ s,r )-integrability of the Radon-Nikodym derivative f N ( t, · )that by passing to a subsequence, f N ( t, · ) converges weakly in L p ( dµ s,r ). Moreover, fromLemma 4.2, f N ( t, φ ) converges pointwise to f ( t, φ ) = exp (cid:18) ∓ ˆ t Re h i ( | u | u )( − t ′ , φ ) , D s u ( − t ′ , φ ) i L ( T ) dt ′ (cid:19) for each φ ∈ L ( T ). Hence, we obtain f N ( t, · ) ⇀ f ( t, · ) in L p ( dµ s,r ) (3.2)as N → ∞ (i.e. f ( t, · ) is the weak limit of f N ( t, · ) in L p ( dµ s,r )).It remains to show that f ( t, · ) is the Radon-Nikodym derivative of the transported mea-sure (Φ t ) ∗ µ s,r with respect to the Gaussian measure µ s,r with L -cutoff. It follows from Fubini’s theorem and (3.1) that for any bounded and continuous function ψ on L ( T ), wehave ˆ ψ ( φ N ) f N ( t, φ ) {k φ N k L ≤ r } µ s ( dφ )= ˆ ψ ( φ N ) f N ( t, φ ) {k φ N k L ≤ r } µ s,N ( dφ N ) ⊗ µ ⊥ s,N ( dφ ⊥ N )= ˆ φ ⊥ N ∈ E ⊥ N (cid:26) ˆ φ N ∈ E N ψ ( φ N )(Φ N,t ) ∗ µ s,N,r ( dφ N ) (cid:27) µ ⊥ s,N ( dφ ⊥ N )= ˆ φ ⊥ N ∈ E ⊥ N (cid:26) ˆ φ N ∈ E N ψ ( u N ( t, φ N )) µ s,N,r ( dφ N ) (cid:27) µ ⊥ s,N ( dφ ⊥ N )= ˆ ψ ( u N ( t, φ N )) {k φ N k L ≤ r } µ s ( dφ ) . (3.3)From the approximation property of the truncated dynamics (4.3) and the Lebesgue dom-inated convergence theorem, we have ˆ ψ ( u N ( t, φ N )) {k φ N k L ≤ r } µ s ( dφ ) −→ ˆ ψ ( u ( t, φ )) {k φ k L ≤ r } µ s ( dφ ) . (3.4)By (3.2), we have ˆ ψ ( φ N ) f N ( t, φ ) {k φ N k L ≤ r } µ s ( dφ ) −→ ˆ ψ ( φ ) f ( t, φ ) {k φ k L ≤ r } µ s ( dφ ) . (3.5)Hence, it follows from (3.3), (3.4) and (3.5) that we have ˆ ψ ( φ ) f ( t, φ ) {k φ k L ≤ r } µ s ( dφ ) = ˆ ψ ( u ( t, φ )) {k φ k L ≤ r } µ s ( dφ ) = ˆ ψ ( φ )(Φ t ) ∗ µ s,r ( dφ ) . This shows that we obtain d (Φ t ) ∗ µ s,r dµ s,r = f ( t, · ) = exp (cid:18) ∓ ˆ t Re h i ( | u | u )( − t ′ , · ) , D s u ( − t ′ , · ) i L ( T ) dt ′ (cid:19) . This completes the proof of Proposition 1.8. (cid:3)
Proof of Theorem 1.1.
We are now ready to present the proof of Theorem 1.1. Here,we suppose the uniform L p ( dµ s,r )-integrability of the Radon-Nikodym derivative f N ( t, · )(1.9) whose proof is presented in Section 5.Fix t ∈ R . Let A ⊂ L ( T ) be a measurable set such that µ s ( A ) = 0. Then, for any r > µ s,r ( A ) = 0 . (3.6)Assume that f N ( t, · ) ∈ L p ( dµ s,r ) , uniformly in N ∈ N (i.e. f N ( t, · ) is L p ( dµ s,r )-integrablewith a uniform (in N ) bound). Then, it follows from Proposition 1.8 and (3.6) that weobtain (Φ t ) ∗ µ s,r ( A ) = ˆ A f ( t, φ ) µ s,r ( dφ ) = 0 . (3.7) UASI-INVARIANCE FOR FNLS 17
From the Lebesgue dominated convergence theorem and (3.7), we have(Φ t ) ∗ µ s ( A ) = lim r →∞ ˆ Φ − t ( A ) {k φ k L ≤ r } µ s ( dφ )= lim r →∞ ˆ Φ − t ( A ) µ s,r ( dφ )= lim r →∞ (Φ t ) ∗ µ s,r ( A ) = 0 . This completes the proof of Theorem 1.1.4.
The strongly dispersive case α ≥ : the energy estimate In this section, we estimate the Radon-Nikodym derivative (1.9) of the transported mea-sure (Φ
N,t ) ∗ µ s,r,N . More precisely, we obtain Lemma 4.2 which will be used to show theuniform L p ( dµ s,r )-integrability of the Radon-Nikodym derivative (1.9) in Section 5.As a consequence of [3], [41, Appendix 1] and [20, Appendix 2], we have the followingglobal well-posedness and approximation property of solutions to FNLS (1.1) when α ≥ L t,x -Strichartz estimate in Lemma 2.7, one can prove properties(4.1), (4.2) and (4.3). Lemma 4.1 (GWP of FNLS (1.1) for α ≥ . Let α ≥ , σ ≥ and u ∈ H σ ( T ) . Given N ∈ N , there exists a unique global solutions u N ∈ C ( R ; H σ ( T )) to thetruncated FNLS (1.8) with u N | t =0 = π ≤ N u and a unique global solution u ∈ C ( R ; H σ ( T )) to FNLS (1.1) with u | t =0 = u . Moreover, these solutions satisfy: sup t ∈ [ − T,T ] k u ( t ) k H σ + k u k X s,bT ≤ C k u k H σ , T > , (4.1) k u ( t ) k L = k u k L , t ∈ R , sup t ∈ [ − T,T ] k u N ( t ) k H σ + k u N k X s,bT ≤ C k π ≤ N u k H σ , T > , (4.2) k u N ( t ) k L = k π ≤ N u k L , t ∈ R , where C > depends only on T and k u k L and b = + δ for fixed < δ ≪ dependingonly on α . Moreover, for any T > , we have sup t ∈ [ − T,T ] k u ( t ) − u N ( t ) k H σ + k u − u N k X σ,bT → , (4.3) as N → ∞ . We now prove the crucial lemma (Lemma 4.2) to control the Radon-Nikodym derivative(1.9) of the transported measure (Φ
N,t ) ∗ µ s,N . Lemma 4.2.
Let α ≥ , s ∈ ( , and put σ = s − − ε for sufficiently small ε > .Given N ∈ N , let u N ∈ C ( R ; H σ ( T )) be the solution to (1.8) satisfying u N | t =0 = π ≤ N u ,as assured by Lemma 4.1. Then, for any T > , we have (cid:12)(cid:12)(cid:12)(cid:12) ˆ T Re h i | u N | u N ( − t, · ) , D s u N ( − t, · ) i L ( T ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k u N k X ,bT k u N k X σ,bT ≤ C ( k u k L , T ) k u k H σ . (4.4) for some b = + δ , < δ ≪ . Furthermore, if we define F ( u ) as the functional on the lefthand side of (4.4) for u ∈ C ( R ; H σ ( T )) the solution to FNLS (1.1) with u | t =0 = u , then F ( u ) satisfies the same bound in (4.4) and F ( u N ) → F ( u ) as N → ∞ . Proof.
In the following, we assume (1.8) is focusing; that is, with the positive sign on thenonlinearity. Note that ˆ T Re h i | u | u ( − t, · ) , D s u ( − t, · ) i dt = Im ˆ T ˆ T | u | u ( − t, x ) D s u ( − t, x ) dxdt = Im ˆ T ˆ T (cid:0) | u ( − t, x ) | − π k u ( − t, · ) k L (cid:1) × u ( − t, x ) D s u ( − t, x ) dxdt = Im ˆ T ˆ T (cid:0) N ( u )( − t, x ) + R ( u )( − t, x ) (cid:1) × D s u ( − t, x ) dxdt, where N ( u , u , u )( t, x ) := X n ∈ Z X n = n − n + n n ,n = n b u ( t, n ) b u ( t, n ) b u ( t, n ) e inx , R ( u , u , u )( t, x ) := X n ∈ Z b u ( t, n ) b u ( t, n ) b u ( t, n ) e inx , and N ( u, u, u ) =: N ( u ) and R ( u, u, u ) =: R ( u ). By Parseval’s theorem, we see thatIm ˆ T R ( u )( − t, x ) D s u ( − t, x ) dx = 0 . Therefore, we have ˆ T Re h i | u | u ( − t, · ) , D s u ( − t, · ) i dt = Im ˆ T ˆ T N ( u )( − t, x ) D s u ( − t, x ) dxdt. Let N ∈ N ∪ {∞} . With b v N ( t, n ) = e it | n | α b u N ( t, n ) and a symmetrisation argument , wehaveLHS of (4.4)= Im ˆ T X n ∈ Z X n = n − n + n n ,n = n e − it Φ α ( n ) b v N ( − t, n ) b v N ( − t, n ) b v N ( − t, n ) h n i s b v N ( − t, n ) dt = 14 Im ˆ − T X n ∈ Z X n = n − n + n n ,n = n Ψ s ( n ) e it Φ α ( n ) b v N ( t, n ) b v N ( t, n ) b v N ( t, n ) b v N ( t, n ) dt =: 14 F ( v N ) . (4.5) Using Im a = − Im a . UASI-INVARIANCE FOR FNLS 19
By symmetry, we may also assume | n | ≥ | n | and | n | ≥ | n | . We perform an integrationby parts in time and use (1.8) to obtain F ( v N )= Im X n ∈ Z X n = n − n + n n ,n = n Ψ s ( n ) e it Φ α ( n ) i Φ α ( n ) b v N ( t, n ) b v N ( t, n ) b v N ( t, n ) b v N ( t, n ) dt (cid:12)(cid:12)(cid:12)(cid:12) t = − T − Im ˆ − T X n ∈ Z X n = n − n + n n ,n = n Ψ s ( n ) e it Φ α ( n ) i Φ α ( n ) ∂ t (cid:0)b v N ( t, n ) b v N ( t, n ) b v N ( t, n ) b v N ( t, n ) (cid:1) dt = Im X n ∈ Z X n = n − n + n n ,n = n Ψ s ( n ) i Φ α ( n ) b u N ( t, n ) b u N ( t, n ) b u N ( t, n ) b u N ( t, n ) dt (cid:12)(cid:12)(cid:12)(cid:12) t = − T + 2Im ˆ − T X n ∈ Z X n = n − n + n n ,n = n Ψ s ( n ) i Φ α ( n ) b u N ( t, n ) b u N ( t, n ) b u N ( t, n ) × (cid:18) X n = n − n + n b u N ( t, n ) b u N ( t, n ) b u N ( t, n ) (cid:19) dt + 2Im ˆ − T X n ∈ Z X n = n − n + n n ,n = n Ψ s ( n ) i Φ α ( n ) b u N ( t, n ) b u N ( t, n ) b u N ( t, n ) × (cid:18) X n = n − n + n b u N ( t, n ) b u N ( t, n ) b u N ( t, n ) (cid:19) dt =: N ( u N )( t ) | t = − T + N ( u N ) + N ( u N ) . (4.6)In the following, we will simply write u for u N . First, we will show |N ( u ) | = (cid:12)(cid:12)(cid:12)(cid:12) ˆ − T X n ∈ Z X n = n − n + n n ,n = n Ψ s ( n )Φ α ( n ) (cid:18) X n = n − n + n b u ( t, n ) b u ( t, n ) b u ( t, n ) (cid:19) × b u ( t, n ) b u ( t, n ) b u ( t, n ) dt (cid:12)(cid:12)(cid:12)(cid:12) . k u k X ,bT k u k X σ,bT , (4.7)for some b = + δ , 0 < δ ≪
1. The same argument can be applied to prove (4.7) for N ( u N ). Thus, we neglect to show an estimate for N ( u N ). In the following, we let w beany extension of u on [ − T, T ]. Namely, w ( t, x ) | [ − T,T ] = u ( t, x ). We split the frequencyregion into a few cases. We suppose that | n | ≥ | n | and let n (1)max = max( | n | , | n | ) + 1. • Case 1: n (1)max ≫ n max In this case, n (1)max ≫ | n | so | n | ∼ | n | or | n | ∼ | n | . Without loss of generality, weassume | n | ∼ | n | . • Case 1.1: | n | ∼ | n | We may assume n max = | n | . We write N ( u ) = X n ∈ Z X n = n − n + n n = n − n + n Ψ s ( n )Φ α ( n ) F t (cid:8) ( [ − T, ( t ) b w ( t, n )) b w ( t, n ) b w ( t, n ) × b w ( t, n ) b w ( t, n ) b w ( t, n ) (cid:9) (0)= ˆ τ − τ + τ − τ + τ − τ =0 X n ∈ Z X n = n − n + n n = n − n + n Ψ s ( n )Φ α ( n ) b w ( τ , n ) b w ( τ , n ) b w ( τ , n ) × b w ( τ , n ) b w ( τ , n ) F { [ − T, b w ( · , n ) } ( τ ) . Given ε >
0, we choose δ > δ δ = 4 ε . We define f := F − t,x ( h n i σ | b w ( τ, n ) | ) ,g := F − t,x ( | b w ( τ, n ) | ) ,g T := F − t,x |F t,x { [ − T, w } ( τ, n ) | . By Lemma 2.5, we have | Ψ s ( n ) || Φ α ( n ) | h n i σ h n i σ h n i σ . h n max i s − α h n (1)max i σ . h n (1)max i ε h n max i ν +6 ε , where ν := 3 σ + 2 α − s − ε and the second inequality follows as s + 2 α ≥ + 12 ε . Then,Parseval’s theorem implies |N ( u ) | . ˆ τ − τ + τ − τ + τ − τ =0 X n ∈ Z X n = n − n + n n = n − n + n h n (1)max i ε h n max i ν +6 ε F t,x f ( τ , n ) F t,x f ( τ , n ) × F t,x f ( τ , n ) F t,x g ( τ , n ) F t,x g ( τ , n ) F t,x g T ( τ , n ) . ˆ R ˆ T | D − ν g T || D − ν g || D − ε g || D − ε f | dxdt. By using H¨older’s inequality, Sobolev’s inequality, the L t,x -Strichartz inequality (2.13),(2.12), and Lemma 2.6, we have ˆ R ˆ T | D − ν g T || D − ν g || D − ε g || D − ε f | dxdt . ˆ R k D − ν g k L
2+ 2 δx k D − ν g T k L
2+ 2 δx k ( D − ε g )( D − ε f ) k L δx dt . ˆ R k D − ν + δ ) g k L x k D − ν + δ ) g T k L x k D − ε g k L δ )1 − δx k D − ε f k L x dt . k g k L εt L x k g T k L εt L x k D − ε + δ δ g k L
4+ 16 ε − εt L x k D − ε f k L
4+ 16 ε − εt L x . k w k X σ, − k D − ε + δ δ g k X , − k w k X , − . k w k X σ, − k w k X , − . UASI-INVARIANCE FOR FNLS 21
Here, we supposed that − ν δ ) ≤ ⇐⇒ s + 2 α ≥
52 + 8 ε, which is assured by assuming that ε > s ≥ + 8 ε . By takingan infimum over all such extensions w of u , we have shown |N ( u ) | . k u k X ,
12 + T k u k X σ,
12 + T . • Case 1.2: | n | ≫ | n | ≫ n max In this case, we have | Ψ s ( n ) || Φ α ( n ) | h n i σ h n i σ h n i σ . h n (1)max i ε h n i ε h n max i ν +6 ε , where ν = 3 σ + 2 α − s − ε . Therefore, we have |N ( u ) | . ˆ R ˆ T | D − ν g T || D − ν g || D − ε g || D − ε f | dxdt and hence one can proceed as in Case 1.1 above. • Case 1.3: | n | ≪ n max We may assume | n max | ∼ | n | . Then, we have | Ψ s ( n ) || Φ α ( n ) | h n i σ h n i σ h n i σ . h n (1)max i ε h n i ε h n max i ν +6 ε where ν = 3 σ + 2 α − s − ε , and hence one can proceed as in Case 1.1.In the following, unless explicitly stated otherwise (for example, as in Subcase 2.2.2), weset ν := 3 σ + 2 α − s − ε . • Case 2: | n (1)max | ∼ | n max |• Case 2.1: | n max | ≫ | n | In this case, | n max | ∼ | n | ∼ | n | . Then, | Ψ s ( n ) || Φ α ( n ) | h n i σ h n i σ h n i σ . h n max i ε h n max i ν +6 ε and hence one can proceed as in Case 1.1. • Case 2.2: | n max | ∼ | n | We have | n | ∼ | n | or | n | ∼ | n | . We assume | n | ∼ | n | , which leads to two subcases. • Subcase 2.2.1: | n | ≫ | n | In this case, we have | n | ∼ | n | . Without loss of generality, we assume | n | =max( | n | , | n | , | n | , | n | ). If | n | ∼ n max , then we have | Ψ s ( n ) || Φ α ( n ) | h n i σ h n i σ h n i σ . h n max i ε h max( | n | , | n | , | n | ) i ν +6 ε and we can proceed as in Case 1.1. If | n | ≪ n max , we only have two frequencies similar to n max , so we need to proceed slightly differently. We note that | Ψ s ( n ) || Φ α ( n ) | h n i σ h n i σ . h n max i σ +2 α − s = 1 h n max i ν , where ν := 2 σ + 2 α − s . We define f : = F − t,x ( h n i σ | b w ( τ, n ) | ) f T : = F − t,x ( h n i σ |F { [ − T, w } ( τ, n ) | ) g : = F − t,x ( h n i − ε | b w ( τ, n ) | ) . Given ε >
0, we define δ > δ δ = 4 ε . By using H¨older’s inequality, Sobolev’sinequality, the L t,x -Strichartz inequality (2.13), (2.12), and Lemma 2.6, we have ˆ τ − τ + τ + ···− τ =0 X n ∈ Z X n = n − n + n n = n − n + n h F t,x g ( τ , n ) F t,x g ( τ , n ) F t,x g ( τ , n ) × (cid:16) h n i − ν F t,x f ( τ , n ) (cid:17)(cid:16) h n i − σ +3 ε F t,x f ( τ , n ) (cid:17)(cid:16) h n i − ν F t,x f T ( τ , n ) (cid:17)i . ˆ R k D − ν f k L
2+ 2 δx k D − ν f T k L
2+ 2 δx k ( D − σ +3 ε f ) g k L δ dt . ˆ R k D − ν + δ ) f k L x k D − ν + δ ) f T k L x k D − σ +3 ε f k L δ )1 − δx k g k L x dt . k f k L εt L x k f T k L εt L x k D − σ +3 ε + δ δ f k L
4+ 16 ε − εt L x k g k L
4+ 16 ε − εt L x . k w k X σ, − k D − σ +4 ε + δ δ f k X , − k w k X , − . k w k X σ, − k w k X , − . Here, we use the following conditions − σ + 4 ε + δ δ ≤ − ν δ ) ≤ . The first is justified by choosing ε > s ≥ + 9 ε and the second followsfrom α ≥ − ε . • Subcase 2.2.2: | n | ∼ | n | ∼ | n | With | n | = max( | n | , | n | , | n | ), we have | Ψ s ( n ) || Φ α ( n ) | h n i σ h n i σ h n i σ . h n max i ε h n i ν +6 ε and hence we proceed as in Case 1.1. • Case 3: | n (1)max | ≪ | n max | Since | n | . n (1)max , we have n max ∼ | n | ∼ | n | . Suppose | n | ∼ n max . Then, we have | Ψ s ( n ) || Φ α ( n ) | h n i σ h n i σ h n i σ . h n max i ε h n i ν +6 ε , and hence we proceed as in Case 1.1. Otherwise, we have | n | ≪ n max . In this case, withoutloss of generality, we assume | n | = max( | n | , | n | , | n | , | n | ) . We note that | Ψ s ( n ) || Φ α ( n ) | h n i σ h n i σ . h n max i σ +2 α − s = 1 h n max i ν , where ν := 2 σ + 2 α − s . Hence, we can proceed as in Subcase 2.2.1. UASI-INVARIANCE FOR FNLS 23
To finish the proof of (4.4), it remains to estimate the boundary term N in (4.6). Fix t ∈ R . We denote by n ( j ) the j -th largest frequency among ( n , n , n , n ). • Case 1: | n (1) | ∼ | n (2) | ∼ | n (3) | & | n (4) | Without loss of generality, we may assume | n | ∼ | n | ∼ | n | . We define f ( n ) = h n i σ | b u ( t, n ) | and using Lemma 2.5, we have |N ( u )( t ) | . X n ∈ Z X n = n − n + n n ,n = n h n max i ν f ( n ) f ( n ) f ( n ) | b u ( t, n ) | . (cid:18) X n ∈ Z X n = n − n + n f ( n ) h n i ν | b u ( t, n ) | (cid:19) (cid:18) X n ∈ Z X n = n − n + n f ( n ) h n i ν f ( n ) (cid:19) . k u ( t ) k L k u ( t ) k H σ , where ν := 3 σ + 2 α − s and the sums above converge, since if s > − α + 3 ε , we ensurethat ν >
1, for σ = s − − ε . • Case 2: | n (1) | ∼ | n (2) | ≫ | n (3) | & | n (4) | Without loss of generality, we may assume | n | ∼ | n | ≫ | n | ≥ | n | . From Lemma 2.5,we have |N ( u )( t ) | . X n ∈ Z X n = n − n + n n ,n = n h n max i ν h n i σ f ( n ) | b u ( t, n ) | f ( n ) f ( n ) , where ν := 2 σ + 2 α − s which is positive as α > . By Cauchy-Schwarz inequality, webound the above by (cid:18) X n ∈ Z X n = n − n + n f ( n ) f ( n ) f ( n ) (cid:19) (cid:18) X n ∈ Z X n = n − n + n | b u ( t, n ) | h n max i ν h n i σ (cid:19) . k u ( t ) k L k u ( t ) k H σ (cid:18) X n ,n h n i ε h n i ε h n i ν +2 σ − − ε (cid:19) and the above sums converge since s > − α + 3 ε implies ν + σ − − ε ≥
0. We also usedthat s ≤ ε so the condition − σ + 1 + ε ≥ F ( u N ) → F ( u ) as N → ∞ . We set w N = v − v N . From (4.5), we have (cid:12)(cid:12) F ( u ) − F ( u N ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)
14 Im ˆ − T X n ∈ Z X n = n − n + n n ,n = n Ψ s ( n ) e it Φ α ( n ) (cid:18) b w N ( t, n ) b v ( t, n ) b v ( t, n ) b v ( t, n )+ b v N ( t, n ) b w N ( t, n ) b v ( t, n ) b v ( t, n ) + b v N ( t, n ) b v N ( t, n ) b w N ( t, n ) b v ( t, n )+ b v N ( t, n ) b v N ( t, n ) b v N ( t, n ) b w N ( t, n ) (cid:19) dt (cid:12)(cid:12)(cid:12)(cid:12) . Note that e it | n | α ∂ t b w N ( t, n )= X n = n − n + n (cid:18)(cid:16)b u ( t, n ) − b u N ( t, n ) (cid:17)b u ( t, n ) b u ( t, n )+ b u N ( t, n ) (cid:16)b u ( t, n ) − b u N ( t, n ) (cid:17)b u ( t, n ) + b u N ( t, n ) b u N ( t, n ) (cid:16)b u ( t, n ) − b u N ( t, n ) (cid:17)(cid:19) . Therefore, by proceeding as in the proof of (4.4) and from Lemma 4.1, we have (cid:12)(cid:12) F ( u ) − F ( u N ) (cid:12)(cid:12) . sup t ∈ [0 ,T ] (cid:0) k u ( t ) k H σ + k u N ( t ) k H σ (cid:1) k u ( t ) − u N ( t ) k H σ + (cid:0) k u k X σ,
12 + T + k u N k X σ,
12 + T (cid:1) k u − u N k X σ,
12 + T −→ N → ∞ . This completes the proof of Lemma 4.2. (cid:3) Uniform L p -integrability of the Radon-Nikodym derivative In this section, we prove the uniform L p -integrability of the Radon-Nikodym derivative f N ( t, · ) in (1.9). It follows from (1.9) and Lemma 4.2 that we have ˆ | f N ( t, φ ) | p µ s,r ( dφ ) = ˆ {k φ k L ≤ r } | f N ( t, φ ) | p µ s ( dφ ) ≤ ˆ {k φ k L ≤ r } exp( C k φ N k H σ ) µ s ( dφ ) , (5.1)where C = C ( t, p, r ) depends only on t, p , and r . Therefore, to show the uniform L p -integrability, it suffices to prove the following lemma. Lemma 5.1.
Let < s ≤ , σ = s − − ε for small ε > and q ≥ such that σq < s .Then, for any r > , we have the following exponential integrability sup N ∈ N E µ s h { ´ T | φ N | dx ≤ r } e ( ´ T | D σ φ N | dx ) q i < ∞ . (5.2)Note that given < s ≤
1, (5.2) holds for q = . This implies the uniform boundednessof (5.1) and hence completes the proof of Proposition 1.8. A similar, slightly more generalstatement (replacing the H σ -norm in (5.2) with a Besov norm) was proved in [15, Lemma2.2 and Corollary 2.3], adapting arguments in [4]. Our proof of Lemma 5.1 is based on thevariational approach due to Barashkov and Gubinelli [1]. More precisely, we will rely onthe Bou´e-Dupuis variational formula [2, 58]; see Lemma 5.2.5.1. Variational formulation.
In order to prove (5.2), we use a variational formula forthe partition function as in [55, 35, 38]. Let us first introduce some notations. Let W ( t )be a cylindrical Brownian motion in L ( T ). Namely, we have W ( t ) = X n ∈ Z B n ( t ) e inx , where { B n } n ∈ Z is a sequence of mutually independent complex-valued Brownian motions.Then, define a centered Gaussian process Y ( t ) by Y ( t ) = h∇i − s W ( t ) . (5.3) By convention, we normalize B n such that Var( B n ( t )) = t . In particular, B is a standard real-valuedBrownian motion. UASI-INVARIANCE FOR FNLS 25
Note that we have Law P ( Y (1)) = µ s , where µ s is the Gaussian measure in (1.2). By setting Y N = π N Y , we have Law P ( Y N (1)) = ( π N ) ∗ µ s , i.e. the push-forward of µ s under π N . Inparticular, a typical function u in the support of µ s belongs to L ∞ ( T ) when s > .Next, let H a denote the space of drifts, which are progressively measurable processesbelonging to L ([0 , L ( T )), P -almost surely. We now state the Bou´e-Dupuis variationalformula [2, 58]; in particular, see Theorem 7 in [58]. Lemma 5.2.
Let Y be as in (5.3) . Fix N ∈ N . Suppose that F : C ∞ ( T d ) → R is measurablesuch that E (cid:2) | F ( π N Y (1)) | p (cid:3) < ∞ and E (cid:2) | e − F ( π N Y (1)) | q (cid:3) < ∞ for some < p, q < ∞ with p + q = 1 . Then, we have − log E h e − F ( π N Y (1)) i = inf θ ∈ H a E (cid:20) F ( π N Y (1) + π N I ( θ )(1)) + 12 ˆ k θ ( t ) k L x dt (cid:21) , where I ( θ ) is defined by I ( θ )( t ) = ˆ t h∇i − s θ ( t ′ ) dt ′ and the expectation E = E P is an expectation with respect to the underlying probabilitymeasure P . Before proceeding to the proof of Lemma 5.1, we state a lemma on the pathwise regularitybounds of Y (1) and I ( θ )(1). Lemma 5.3. (i)
For any finite p ≥ , we have E h k Y N (1) k pW σ, ∞ i ≤ C p < ∞ , (5.4) uniformly in N ∈ N . (ii) For any θ ∈ H a , we have k I ( θ )(1) k H s ≤ ˆ k θ ( t ) k L dt. (5.5) Proof.
It follows from the Sobolev embedding theorem that for any sufficiently small 0 <η ≪
1, there exists finite r = r ( η ) > k u k W s, ∞ ( T ) . k u k W s + η,r ( T ) . Here, we reduce the r = ∞ case to the case of large but finite r by paying the expense ofa slight loss of spatial derivative. Hence, we assume r < ∞ in the following.Let p ≥ r . Note that h∇i σ + ε Y N (1) = X | n |≤ N B n (1) h n i + ε e inx . Then, one can see that h∇i σ + ε Y N (1) is a mean zero Gaussian random variable with variance kh n i − − ε | n |≤ N k ℓ n . Hence, there exists a constant C > kh∇i σ + ε Y N (1) k L pω ≤ Cp kh n i − − ε | n |≤ N k ℓ n . (5.6)Then, from Minkowski’s inequality and (5.6), we have (cid:0) E k Y N (1) k pW σ + ε ,rx (cid:1) p ≤ (cid:13)(cid:13) kh∇i σ + ε Y N (1) k L pω (cid:13)(cid:13) L rx ≤ Cp (cid:0) X | n |≤ N h n i ε (cid:1) ≤ e Cp . This proves (5.4).As for (ii), the estimate (5.5) follows from Minkowski’s and Cauchy-Schwarz’ inequalities.See the proof of Lemma 4.7 in [24]. (cid:3)
Uniform exponential integrability.
In this section, we present the proof of Lemma5.1. Since the argument is identical for any finite p ≥
1, we only present details for the case p = 1. Note that {| · |≤ K } ( x ) ≤ exp (cid:0) − A | x | γ (cid:1) exp( AK γ ) (5.7)for any K, A, γ >
0. Given N ∈ N and A ≫ R N ( u ) = (cid:18) ˆ T | D σ u N | dx (cid:19) q − A (cid:18) ˆ T | u N | dx (cid:19) γ (5.8)for some γ > q which will be determined later (see (5.16)). Then, the following uniformexponential bound (5.9) with (5.7)sup N ∈ N (cid:13)(cid:13)(cid:13) e R N ( u ) (cid:13)(cid:13)(cid:13) L p ( µ s ) ≤ C p,A,γ < ∞ (5.9)implies the uniform exponential integrability (5.2). Hence, it remains to prove the uniformexponential integrability (5.9). In view of the Bou´e-Dupuis formula (Lemma 5.2), it sufficesto establish a lower bound on W N ( θ ) = E (cid:20) − R N ( Y (1) + I ( θ )(1)) + 12 ˆ k θ ( t ) k L x dt (cid:21) , (5.10)uniformly in N ∈ N and θ ∈ H a . We set Y N = π N Y = π N Y (1) and Θ N = π N Θ = π N I ( θ )(1).From (5.8), we have R N ( Y + Θ) = (cid:18) ˆ T | D σ Y N | dx + 2 ˆ T Re( D σ Y N D σ Θ N ) dx + ˆ T | D σ Θ N | dx (cid:19) q − A (cid:26) ˆ T (cid:16) | Y N | + 2 Re( Y N Θ N ) + | Θ N | (cid:17) dx (cid:27) γ . (5.11)Hence, from (5.10) and (5.11), we have W N ( θ ) = E (cid:20) − (cid:18) ˆ T | D σ Y N | dx + 2 ˆ T Re( D σ Y N D σ Θ N ) dx + ˆ T | D σ Θ N | dx (cid:19) q + A (cid:26) ˆ T (cid:16) | Y N | + 2 Re( Y N Θ N ) + | Θ N | (cid:17) dx (cid:27) γ + 12 ˆ k θ ( t ) k L x dt (cid:21) . (5.12)We first state a lemma, controlling the terms appearing in (5.12). We present the proofof this lemma at the end of this section. Lemma 5.4.
Let s , σ and q be as given in Lemma 5.1. Then, we have the following: (i) There exists c > such that k Θ N k qH σ ≤ k Θ N k H s + A k Θ N k q (1+2 ε ) q (1+2 ε ) − s ( q − L , (5.13) (cid:18) ˆ T (cid:12)(cid:12) D σ Y N D σ Θ N (cid:12)(cid:12) dx (cid:19) q ≤ c k Y N k qW σ, ∞ + 1200 k Θ N k H s + A k Θ N k q (1+2 ε ) q (1+2 ε ) − s ( q − L (5.14) for any sufficiently large A > , uniformly in N ∈ N . UASI-INVARIANCE FOR FNLS 27 (ii)
Let
A, γ > . Then, there exists c = c ( A, γ ) > such that A (cid:26) ˆ T (cid:16) | Y N | + 2 Re( Y N Θ N ) + | Θ N | (cid:17) dx (cid:27) γ ≥ A k Θ N k γL − c k Y N k γL ∞ , (5.15) uniformly in N ∈ N . Set γ = q (1 + 2 ε ) q (1 + 2 ε ) − s ( q − . (5.16)Note that to ensure the denominator is non-vanishing, we impose σq < s . Then, as in[1, 24, 36, 35], the main strategy is to establish a pathwise lower bound on W N ( θ ) in (5.12),uniformly in N ∈ N and θ ∈ H a , by making use of the positive terms: U N ( θ ) = E (cid:20) A k Θ N k γL + 12 ˆ k θ ( t ) k L x dt (cid:21) . (5.17)coming from (5.12) and (5.15). From (5.12) and (5.17) together with Lemmas 5.4 and 5.3,we obtain inf N ∈ N inf θ ∈ H a W N ( θ ) ≥ inf N ∈ N inf θ ∈ H a n − C + 110 U N ( θ ) o ≥ − C > −∞ . (5.18)Then, the uniform exponential integrability (5.9) follows from (5.18) and Lemma 5.2. Thiscompletes the proof of Lemma 5.1.We conclude this section by presenting the proof of Lemma 5.4. Proof of Lemma 5.4. (i) It follows from interpolation and Young’s inequality that k Θ N k qH σ . k Θ N k qσs H s k Θ N k q (1 − σs ) L ≤ k Θ N k H s + A k Θ N k γL , where A > qσs + qγ (1 − σs ) = 1, where γ is given in (5.16). This yields the first estimate (5.13).As for the second estimate (5.14), the Cauchy-Schwarz and Cauchy inequalities imply (cid:18) ˆ T (cid:12)(cid:12) D σ Y N D σ Θ N (cid:12)(cid:12) dx (cid:19) q ≤ k Y N k qH σ k Θ N k qH σ ≤ k Y N k qW σ, ∞ + 12 k Θ N k qH σ . Now we apply (5.13).(ii) Note that there exists a constant C γ > | a + b + c | γ ≥ | c | γ − C γ ( | a | γ + | b | γ ) (5.19)for any a, b, c ∈ R (see Lemma 5.8 in [35]). Then, from (5.19), we have A (cid:26) ˆ T (cid:16) | Y N | + 2 Re( Y N Θ N ) + | Θ N | (cid:17) dx (cid:27) γ ≥ A (cid:18) ˆ T | Θ N | dx (cid:19) γ − AC γ (cid:26)(cid:18) ˆ T | Y N | dx (cid:19) γ + (cid:18) ˆ T | Y N Θ N | dx (cid:19) γ (cid:27) . (5.20)From Young’s inequality, we have (cid:18) ˆ T | Y N Θ N | dx (cid:19) γ ≤ k Y N k γL ∞ k Θ N k γL ≤ c k Y N k γL ∞ + 1100 C γ k Θ N k γL . (5.21)Hence, (5.15) follows from (5.20) and (5.21). (cid:3) The weakly dispersive case < α < < α < Truncated dynamics in the weakly dispersive case.
We consider two kinds ofapproximating flows: ( i∂ t u N + ( − ∂ x ) α u N = ± π ≤ N (cid:0) | u N | u N (cid:1) , ( t, x ) ∈ R × T ,u | t =0 = π ≤ N φ, (6.1)and ( i∂ t u N + ( − ∂ x ) α u N = ± π ≤ N (cid:0) | π ≤ N u N | π ≤ N u N (cid:1) , ( t, x ) ∈ R × T ,u | t =0 = φ. (6.2)The difference between these is that (6.2) is the linear evolution on high frequencies {| n | >N } , whilst (6.1) vanishes on high frequencies. We denote the solution maps of (6.1) and(6.2) at time t > e Φ N,t and Φ
N,t respectively. We have the identitiesΦ
N,t = e Φ N,t ◦ π ≤ N + S ( t ) ◦ π >N and π ≤ N ◦ Φ N,t = e Φ N,t ◦ π ≤ N . (6.3)In this section, we define E N by E N = π ≤ N H σ ( T ) = span { e inx : | n | ≤ N } and let E ⊥ N be the orthogonal complement of E N in H σ ( T ). Given R >
0, we use B R todenote the ball of radius R in H σ ( T ) centered at the origin. We need some boundednessand approximation properties for these flows. Lemma 6.1.
Given < α < , let s > − α be such that the flow Φ of FNLS (1.1) is(locally) well-defined in H σ ( T ) , σ = s − − ε . Then, the following statements hold: (i) Then, for every
R > , there exist T ( R ) > and C ( R ) > such that Φ N,t ( B R ) ⊂ B C ( R ) for all t ∈ [ − T ( R ) , T ( R )] and for all N ∈ N ∪ {∞} . (ii) For every
R > , there exists T ( R ) > and C ( R ) > such that sup u ∈ B R k Φ N,t ( u ) k X σ,
12 + T ( R ) ≤ C ( R ) , uniformly in N ∈ N ∪ {∞} (iii) Let A ⊂ B R ⊂ H σ ( T ) be a compact set and denote by T ( R ) > the local existencetime of the solution map Φ defined on B R . Then, for every δ > , there exists N ∈ N ,such that k Φ t ( u ) − Φ N,t ( u ) k H σ < δ, for any u ∈ A , N ≥ N and t ∈ [ − T ( R ) , T ( R )] . Furthermore, we have Φ t ( A ) ⊂ Φ N,t ( A + B δ ) for all t ∈ [ − T ( R ) , T ( R )] and for all N ≥ N . UASI-INVARIANCE FOR FNLS 29
Proof.
The uniform growth bounds (i) and (ii) follow from the local well-posedness argu-ment in [9]. The approximation property (iii) follows from a modification of the contractionargument as in [41, Lemma 6.20] by using the nonlinear estimate (the trilinear estimate)in [9, Proposition 4.1] and the following uniform estimate which plays the same role as in[41, Lemma 6.19]: there exists N = N ( ε, R ) ∈ N such that we have k P >N Φ t ( u ) k X σ,
12 + T ( R ) < ε (6.4)for all u ∈ A and N ≥ N . This estimate (6.4) can be proved by following the proof of[41, Lemma 6.19] with the continuity of the flow map Φ t from H σ to X σ, + T ( R ) . (cid:3) Proof of Theorem 1.2.
In this subsection, we present the proof of our second mainresult (Theorem 1.2). First, suppose that we have the following energy estimate withsmoothing which will be proved in the next subsection.
Lemma 6.2.
Let < α < and s ∈ (cid:0) − α , (cid:3) . Given N ∈ N , R > and u ∈ B R , let u N ∈ C ([0 , T ( R )); H σ ( T )) be the local-in-time solution to (6.1) satisfying u N | t =0 = π ≤ N u ,as assured by Lemma 6.1. Then there exists k ∈ N such that, we have (cid:12)(cid:12)(cid:12)(cid:12) ˆ T Re h i | u N | u N ( − t, · ) , D s u N ( − t, · ) i L ( T ) dt (cid:12)(cid:12)(cid:12)(cid:12) . k u N k kX σ,bT ≤ C ( R ) . (6.5) for some b = + δ , < δ ≪ . We are now ready to present the proof of Theorem 1.2.
Proof of Theorem 1.2.
Following the same argument as in the proof of Proposition 1.8, weobtain an explicit expression for the Radon-Nikodym derivative: f N ( t, φ N ) = d ( e Φ N,t ) ∗ µ s,N dµ s,N ( φ N )= exp (cid:18) ∓ ˆ t Re h i ( | u N | u N )( − s, φ N ) , D s u N ( − s, φ N ) i L x ds (cid:19) . Now we want to show (local-in-time and local-in-phase space) quasi-invariance. Given
R >
0, let A ⊂ B R ⊂ H σ ( T ) be a measurable set. We want to show that given any t ∈ [0 , T ( R )], we have µ s ( A ) = 0 = ⇒ µ s (Φ t ( A )) = 0 . By the inner regularity of the measure µ s , it is enough to show that A ⊂ B R compact and µ s ( A ) = 0 = ⇒ µ s (Φ t ( A )) = 0 . Now, from Lemma 6.1 (ii), we have µ s (Φ t ( A )) ≤ µ s (Φ N,t ( A + B δ )) (6.6) for any fixed δ >
0, provided that N is large enough. Let D ⊂ B R be an arbitrarymeasurable set. By Fubini’s theorem and (6.3), we have µ s (Φ N,t ( D )) = ˆ D (Φ N, − t ( φ )) dµ s ( φ )= ˆ E ⊥ N (cid:26) ˆ E N D (Φ N, − t ( φ )) dµ s,N (cid:27) dµ ⊥ s,N = ˆ E ⊥ N (cid:26) ˆ E N D ( e Φ N, − t ( π ≤ N φ ) + S ( − t ) π >N φ ) dµ s,N (cid:27) dµ ⊥ s,N = ˆ E ⊥ N (cid:26) ˆ E N D ( φ N + S ( − t ) π >N φ ) f N ( − t, φ N ) dµ s,N (cid:27) dµ ⊥ s,N Since D ⊂ B R , we can use Lemma 6.1 and Lemma 6.2 to findsup N ∈ N sup φ N ∈ B R sup t ∈ [ − T ( R ) ,T ( R )] f N ( − t, φ N ) ≤ e C ( T ( R ) ,R ) =: C ( R ) . By the invariance of µ ⊥ s,N under S ( t ), which follows from [41, Lemma 4.1], we then have µ s (Φ N,t ( D )) ≤ C ( R ) ˆ E ⊥ N ˆ E N D ( φ N + S ( − t ) π >N φ ) dµ s,N dµ ⊥ s,N = C ( R ) ˆ E N ˆ E ⊥ N D ( φ N + S ( − t ) π >N φ ) dµ ⊥ s,N dµ s,N = C ( R ) ˆ E N ˆ E ⊥ N D ( φ N + π >N φ ) dµ ⊥ s,N dµ s,N = C ( R ) ˆ D ( φ ) dµ s = C ( R ) µ s ( D ) . We now go back and apply this inequality with D = A + B δ to (6.6) (supposing δ < R ),and we get µ s (Φ t ( A )) ≤ C ( R ) µ s ( A + B δ )Since A is compact, by the continuity of probability measures from above, we havelim δ → µ s ( A + B δ ) = µ s ( A )and hence µ s (Φ t ( A )) ≤ C ( R ) µ s ( A ) (6.7)for any compact set A ⊂ B R . Now since µ s ( A ) = 0, we have µ s (Φ t ( A )) = 0 . This completes the proof of Theorem 1.2. (cid:3)
We note that we can remove the compactness assumption in (6.7). This general obser-vation is crucial for the proof of Proposition 1.10 below.
Corollary 6.3.
Let < α < and s ∈ (cid:0) − α , (cid:3) . Given any R > , let A ⊆ B R be ameasurable set. Then, µ s (Φ t ( A )) ≤ C ( R ) µ s ( A ) , for any t ∈ [0 , T ( R )] . UASI-INVARIANCE FOR FNLS 31
Proof.
We follow the argument in [41, Lemma 6.10]. Let A be a measurable set in B R ⊂ H σ .Then, from the inner regularity of µ s , there exists a sequence { K j } of compact sets suchthat K j ⊂ Φ t ( A ) and lim j →∞ µ s ( K j ) = µ s (Φ t ( A )) . (6.8)From the bijectivity of Φ t , we have K j = Φ t (Φ − t ( K j )) . Since Φ( − t ) is the continuous map, one can observe that Φ( − t )( K j ) is compact. Also, wehave Φ( − t )( K j ) ⊂ Φ( − t )Φ t ( A ) = A . Hence, by applying (6.7) to Φ( − t )( K j ), we have µ s ( K j ) = µ s (Φ t (Φ( − t )( K j ))) ≤ C ( R ) µ s (Φ( − t )( K j )) ≤ C ( R ) µ s ( A ) . (6.9)Then, after taking a limit as j → ∞ , it follows from (6.9) and (6.8) that we have the desiredresult. (cid:3) Proof of Proposition 1.10.
In the following, we fix t ∈ R . We first show (1.11). By Propo-sition 1.8, we know that the density of (Φ t ) ∗ µ s,r with respect to µ s,r is given by d (Φ t ) ∗ µ s,r dµ s,r = f ( t, · ) = exp (cid:18) ∓ ˆ t Re h i ( | u | u )( − t ′ , · ) , D s u ( − t ′ , · ) i L ( T ) dt ′ (cid:19) (6.10)for every t ∈ R and r >
0. Now, fix t ∈ R , r > B r := { φ ∈ L ( T ) : k φ k L ≤ r } . Let A ⊆ L ( T ) be a measurable set. From the L -conservation of the flow of (1.1) and (6.10),we have (Φ t ) ∗ µ s ( A ∩ B r ) = ˆ L ( T ) A ∩ B r (Φ t ( φ )) dµ s ( φ )= ˆ L ( T ) A (Φ t ( φ )) B r ( φ ) dµ s ( φ )= ˆ L ( T ) A ( φ ) d (Φ t ) ∗ µ s,r ( φ )= ˆ L ( T ) A ( φ ) B r ( φ ) f ( t, φ ) dµ s ( φ ) . (6.11)It follows from the continuity from below of a measure thatlim r →∞ (Φ t ) ∗ µ s ( A ∩ B r ) = (Φ t ) ∗ µ s ( A ) . (6.12)From the Lebesgue monotone convergence theorem, we havelim r →∞ ˆ A B r ( φ ) f ( t, φ ) dµ s ( φ ) = ˆ A f ( t, φ ) dµ s ( φ ) . (6.13)Hence, by combining (6.11), (6.12), and (6.13), we have(Φ t ) ∗ µ s ( A ) = ˆ A f ( t, φ ) dµ s ( φ ) (6.14)for any measurable set A ⊆ L ( T ). It follows from the definition of the Radon–Nikodymderivative d (Φ t ) ∗ µ s dµ s (that is, d (Φ t ) ∗ µ s dµ s is a function (up to a µ s -null set) which satisfies (6.14))that we have d (Φ t ) ∗ µ s dµ s = f ( t, · ) = exp (cid:18) ∓ ˆ t Re h i ( | u | u )( − t ′ , · ) , D s u ( − t ′ , · ) i L ( T ) dt ′ (cid:19) and hence we have verified (1.11).Now, we proceed to show that f ( t, · ) ∈ L ∞ loc ( dµ s ). By following the proof of Theorem 1.2but replacing each instance of Lemma 6.1 and Lemma 6.2 by Lemma 4.1 and Lemma 4.2,we obtain: µ s (Φ t ( A )) ≤ C ( R ) µ s ( A ) , (6.15)for any compact A ⊆ B R ⊂ L ( T ) and for some C ( R ) > A . Following theargument in the proof of Corollary 6.3, we obtain (6.15) for any measurable set A ⊆ B R ⊂ L ( T ). Fix t ∈ R and R >
0. We claim that B R f ( t, · ) ∈ L ∞ ( dµ s ). We argue as in [6,Proposition 3.5]. Suppose, in order to obtain a contradiction, that B R f ( t, · ) / ∈ L ∞ ( dµ s ).Then, choosing M = 2 C ( R ), there exists a measurable set A R such that B R ( φ ) f ( t, φ ) > M for all φ ∈ A R and µ s ( A R ) >
0. We necessarily have that A R ⊆ B R . Now µ s (Φ t ( A R )) = ˆ A R f ( t, φ ) dµ s ( φ ) = ˆ A R B R ( φ ) f ( t, φ ) dµ s ( φ ) > M µ s ( A R ) . Now our choice of M yields a contradiction with (6.15). Hence, B R f ( t, · ) ∈ L ∞ ( dµ s ) andsince R > f ( t, · ) ∈ L ∞ loc ( dµ s ). (cid:3) Proof of Lemma 6.2.
In this subsection, we present the proof of Lemma 6.2. Wefirst proceed exactly as in the proof of Lemma 4.2 by observing a cancellation of resonantinteractions, symmetrizing, and integrating by parts in time. This reduces the proof of(6.5) to establishing the following two estimates:sup t ∈ [0 ,T ( R )] (cid:12)(cid:12) N ( u N )( t ) (cid:12)(cid:12) . sup t ∈ [0 ,T ( R )] k u N ( t ) k H σ , (6.16) X j =1 (cid:12)(cid:12) N j ( u N ) (cid:12)(cid:12) . k u N k X σ,bT ( R ) , (6.17)for any N ∈ N ∪ {∞} and some b = + δ , 0 < δ ≪
1, where the multilinear operators N , N and N are defined in (4.6), with | n | ≥ | n | and | n | ≥ | n | . As our estimates willbe uniform in the parameter N , in the following, we will simply write u for u N . First, wewill show |N ( u N ) | = (cid:12)(cid:12)(cid:12)(cid:12) ˆ − T X n ∈ Z X n = n − n + n n ,n = n Ψ s ( n )Φ( n ) (cid:18) X n = n − n + n b u ( t, n ) b u ( t, n ) b u ( t, n ) (cid:19) × b u ( t, n ) b u ( t, n ) b u ( t, n ) dt (cid:12)(cid:12)(cid:12)(cid:12) . k u k X σ,bT , (6.18)for some b = + δ , 0 < δ ≪
1. The same argument can be applied to prove (6.18) for N ( u N ). Thus, we neglect to show an estimate for N ( u N ). In the following, we let w be anyextension of u on [ − T, T ]. We suppose that | n | ≥ | n | . Let n (1)max = max( | n | , | n | ) + 1.In this setting, it will be convenient to dyadically decompose the frequencies of all the UASI-INVARIANCE FOR FNLS 33 functions. In view of our symmetry assumptions, we therefore have N ≥ N , N ≥ N and N ≥ N . We let f N : = F − t,x ( |h n i σ F { P N w } ( τ, n ) | ) ,f N,T : = F − t,x ( |h n i σ F { [ − T, P N w } ( τ, n ) | ) . We split the frequency region into a few cases in a similar way as we did in the proof ofLemma 4.2. • Case 1: n (1)max ≫ n max In this case, we have n (1)max ≫ | n | which implies | n | ∼ | n | or | n | ∼ | n | . We assume | n | ∼ | n | . Then, we have |N ( u N ) | ≤ X Nj, j =1 ,..., N k,k =1 , , N ≥ N ,N ≥ N N ≥ N ,N ∼ N ( N N N N N N ) − σ ˆ τ − τ τ − τ τ − τ X n − n n − n n n − n n | Ψ s ( n ) || Φ α ( n ) |× Y j =2 F t,x f N j ( τ j , n j ) F t,x f N ,T ( τ , n ) F t,x f N ( τ , n ) F t,x f N ( τ , n ) dτ . . . dτ =: X ∗ I ( N ) , where ∗ represents the conditions on the summations in the first line above. We split intoa few subcases. • Case 1.1: | n | ∼ | n | Without loss of generality, we assume | n | ∼ n max . From Lemma 2.5, we have | Ψ s ( n ) || Φ α ( n ) | . N s − α . Hence, by H¨older’s inequality, (2.16), (2.15), (2.14) and Lemma 2.6, we have I ( N ) . N s − α − σ N − σ ( N N ) − σ (cid:13)(cid:13) f N ,T f N k L t,x k f N k L t,x k f N k L t,x k f N k L ∞ t,x k f N k L ∞ t,x . N s − α − σ + − α N − σ + − α ( N N ) − σ + Y j =2 k f N j k X ,
12 + k f N k X , k f N k X , × k f N k X , k f N ,T k X , . N s − α − σ + − α +14 N − σ + − α
11 4 Y j =2 k P N j w k X σ,
12 + Y ℓ =1 k P N ℓ w k X σ,
12 + . To sum over the dyadic scales, we enforce − σ + − α ≤ − ε <
0, which requires s ≥ − α +2 ε .Then, we can write N s − α − σ + − α +14 N − σ + − α . N − ε N s − α − σ +2+3 ε . N − ε , provided that 2 s − α − σ + 2 + 3 ε ≤
0. This conditions requires s > − α + ε . Hence,in this case, we need s > max (cid:18) − α ε, − α ε (cid:19) = 5 − α ε, (6.19) for 0 < ε ≪
1. Thus, we may sum over the dyadic scales and show that this contributioncan be bounded by the right hand side of (6.18). • Case 1.2: | n | ≫ | n | ≫ n max We proceed similar to Case 1.1, with I ( N ) . N s − α − σ N − σ N − σ ( N N ) − σ (cid:13)(cid:13) f N ,T f N k L t,x k f N k L t,x k f N k L t,x k f N k L ∞ t,x k f N k L ∞ t,x . N s − α − σ + − α N − σ + − α N − σ + − α ( N N ) − σ + Y j =2 k P N j w k X σ,
12 + Y ℓ =1 k P N ℓ w k X σ,
12 + . N s − α − σ +2+4 N − ε
11 4 Y j =2 k P N j w k X σ,
12 + Y ℓ =1 k P N ℓ w k X σ,
12 + . N − ε
11 4 Y j =2 k P N j w k X σ,
12 + Y ℓ =1 k P N ℓ w k X σ,
12 + provided s , α and ε satisfy (6.19). • Case 1.3: | n | ≪ n max We may assume | n max | ∼ | n | . Then, in this case, we proceed in a similar way as in Case1.2 above. • Case 2: n (1)max ∼ n max • Case 2.1: n max ≫ | n | In this case, | n max | ∼ | n | ∼ | n | . Hence, we can proceed as in Case 1.1. • Case 2.2: | n max | ∼ | n | We can have | n | ∼ | n | or | n | ∼ | n | . We assume | n | ∼ | n | . This leads to two naturalsubcases. • Subcase 2.2.1: | n | ≫ | n | In this case, we have | n | ∼ | n | . We assume that | n | = max( | n | , | n | , | n | , | n | ). If | n | ∼ n max , we may proceed as in Case 1.1. Otherwise, if | n | ≪ n max , we need to applythe bilinear estimate (2.16) twice. Indeed, suppose that | n | = max( | n | , | n | , | n | ). Then,by H¨older’s inequality, (2.16), (2.14) and Lemma 2.6, we have I ( N ) . N s − α − σ N − σ ( N N N ) − σ k f N ,T f N k L t,x k f N f N k L t,x k f N k L ∞ t,x k f N k L ∞ t,x . N s − α − σ ( N N ) − σ + − α ( N N ) − σ + Y j =2 k P N j w k X σ,
12 + Y ℓ =1 k P N ℓ w k X σ,
12 + . N − ε N s − α − σ + − α +14 4 Y j =2 k P N j w k X σ,
12 + Y ℓ =1 k P N ℓ w k X σ,
12 + . N − ε Y j =2 k P N j w k X σ,
12 + Y ℓ =1 k P N ℓ w k X σ,
12 + , UASI-INVARIANCE FOR FNLS 35 where in the third inequality we need α ≥ + 2 ε and in the final inequality, we need s > − α ε. • Subcase 2.2.2: | n | ∼ | n | ∼ | n | In this case, we have | n | ∼ | n | ∼ | n | , so we can proceed as in Case 1.1. • Case 3: n (1)max ≪ n max Since | n | . n (1)max , we have n max ∼ | n | ∼ | n | . If | n | ∼ n max , we can proceed as in Case1.1. Otherwise, if | n | ≪ n max , we may proceed as in Subcase 2.2.1.Compiling these cases, overall we require < α < s > − α + ε . This completesthe proof of (6.18).We now estimate the boundary term N and thus establish (6.16). We fix t ∈ [0 , T ( R )].This leads to the following cases. • Case 1: | n (1) | ∼ | n (4) | From Lemma 2.5, we have | Ψ s ( n ) || Φ α ( n ) | . | n (1) | s − α . We define f ( t, n ) = h n i σ | b u ( t, n ) | . Then, by Cauchy-Schwarz, we have |N ( u )( t ) | . X n ∈ Z X n = n − n + n n ,n = n h n (1) i ν f ( n ) f ( n ) f ( n ) f ( n ) . (cid:18) X n ∈ Z X n = n − n + n f ( n ) f ( n ) h n i ν (cid:19) (cid:18) X n ∈ Z X n = n − n + n f ( n ) f ( n ) h n i ν (cid:19) . k u ( t ) k H σ , where ν := 4 σ + 2 α − s and we can perform the summations provided s > − α + ε . • Case 2: | n (1) | ∼ | n (3) | ≫ | n (4) | We assume, without loss of generality, that | n | ∼ | n | ∼ | n | ≫ | n | . Then, we have |N ( u )( t ) | . (cid:18) X n ∈ Z X n = n − n + n f ( n ) f ( n ) f ( n ) (cid:19) (cid:18) X n ∈ Z X n = n − n + n | n | & | n | f ( n ) h n i ν h n i σ (cid:19) . (cid:18) X n ,n | n | & | n | h n i ε h n i ε h n i ν +2 σ − − ε (cid:19) k u ( t ) k H σ . k u ( t ) k H σ , where ν := 3 σ + 2 α − s and we can sum provided 2 ν + 2 σ − − ε ≥
0, which requires s ≥ − α + 5 ε . We have also used here the condition − σ + 1 + ε ≥
0, which requires s ≤ ε but this condition is satisfied since we have only considered the case s ≤ • Case 3: | n (1) | ∼ | n (2) | ≫ | n (3) | ≥ | n (4) | With ν := 2 σ + 2 α − s >
0, we have1 h n (1) i ν h n (3) i σ h n (4) i σ . h n (3) i σ + ν h n (4) i σ + ν . Hence, by Cauchy-Schwarz, |N ( u )( t ) | . (cid:18) X n (3) ,n (4) h n (3) i σ + ν h n (4) i σ + ν (cid:19) k u ( t ) k H σ , where we can sum provided that 2 σ + ν >
1. This is satisfied if s > − α + 2 ε . Remark 6.4.
Combining the regularity restrictions required in estimating both the bound-ary and remainder piece above, we see that we need s > max (cid:18) − α , − α (cid:19) = 3 − α , since < α <
1, which gives rise to the restriction in Lemma 6.2. We point out that,contrary to the higher dispersion case (Lemma 4.2), the worst regularity restriction comesfrom the boundary estimate (6.16). This is an artefact of the weaker dispersion since thelower bound on the phase function in Lemma 2.4 is much less effective and we can also nolonger use space-time estimates like the L t,x -Strichartz estimate (2.15). Remark 6.5.
Our reduction to the energy estimate in Lemma 6.2 is essential for studyingthe weakly dispersive FNLS (1.1). If instead we tried to obtain an energy estimate as inLemma 4.2, following [15], we would need to place two functions into the L ∞ t L x -norm inorder to be controlled using the L x -conservation. To illustrate why this does not yieldresults for all < α <
1, we consider the Subcase 2.2.1 in the above proof of Lemma 6.2.Using the assumptions and notations from there along with Sobolev embedding and the L t,x -Strichartz estimate (2.15), we have I ( N ) . N s − α N − σ N − σ N − σ k f N k L t L ∞ x k f N k L t L ∞ x k f N k L t L ∞ x k f N k L t L ∞ x × k u N k L ∞ t L x k u N k L ∞ t L x . N s − α − σ + + − α +3 N − σ + + − α +4 k f N k X , k f N k X , k f N k X , k f N k X , × k u N k L ∞ t L x k u N k L ∞ t L x . In order to sum over the dyadic scales, we need2 s − α − σ + 12 + 1 − α < s − σ − α + 2 < . The first condition requires α > and the second imposes s > − α . Thus, such anapproach does not seem to cover the full range < α < Proof of Proposition 1.11.
In this subsection, we prove Proposition 1.11. Theargument is the same as that of Corollary 1.3 in [51] and diverges only when we prove thestability in L p loc ( dµ s ). We will include details for the benefit of the reader. We define themeasures dν j ( φ ) = g j ( φ ) dµ s ( φ ) , j = 1 , . UASI-INVARIANCE FOR FNLS 37
Then, for any test function ϕ , we have ˆ L ϕ ( φ ) d (Φ t ) ∗ ν j ( φ ) = ˆ L ϕ (Φ t ( φ )) dν j ( φ )= ˆ L ϕ (Φ t ( φ )) g j ( φ ) dµ s ( φ )= ˆ L ϕ ( φ ) g j (Φ − t ( φ )) f ( t, φ ) dµ s ( φ ) . Therefore, d (Φ t ) ∗ ν j ( φ ) = G j ( t, φ ) dµ s ( φ ), j = 1 ,
2, where G j ( t, φ ) = g j (Φ − t ( φ )) f ( t, φ ).Now, we have ˆ L | G ( t, φ ) − G ( t, φ ) | dµ s ( φ ) = ˆ L | g (Φ − t ( φ )) − g (Φ − t ( φ )) | f ( t, φ ) dµ s ( φ )= ˆ L | g ( φ ) − g ( φ ) | dµ s ( φ ) . Now, suppose p > g , g ∈ L ( dµ s ) ∩ L p loc ( dµ s ) and fix R >
0. Then, by L -conservation, we have Φ − t ( B R ) = B R , and hence ˆ B R | G ( t, φ ) − G ( t, φ ) | p dµ s ( φ )= ˆ B R | g (Φ − t ( φ )) − g (Φ − t ( φ )) | p f ( t, φ ) p dµ s ( φ ) ≤ k B R f ( t, · ) k p − L ∞ ( dµ s ) ˆ B R | g (Φ − t ( φ )) − g (Φ − t ( φ )) | p f ( t, φ ) dµ s ( φ ) ≤ k B R f ( t, · ) k p − L ∞ ( dµ s ) ˆ Φ − t ( B R ) | g ( φ ) − g ( φ ) | p dµ s ( φ )= k B R f ( t, · ) k p − L ∞ ( dµ s ) ˆ B R | g ( φ ) − g ( φ ) | p dµ s ( φ ) . This shows (1.12).
Acknowledgments.
The authors would like to kindly thank Tadahiro Oh and NikolayTzvetkov for suggesting the problem, for their continued support and for informing us thatthe L ( dµ s )-integrability assumption in [43, Corollary 1.4] can be weakened to L ( dµ s )-integrability. The authors are also grateful to Nikolay Tzvetkov for suggesting the applica-tion of Proposition 1.10 to the L p -stability result in Proposition 1.11.J. F. was supported by The Maxwell Institute Graduate School in Analysis and its Ap-plications, a Centre for Doctoral Training funded by the UK Engineering and PhysicalSciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh and Tadahiro Oh’s ERC starting grantno. 637995 “ProbDynDispEq”. K.S. was partially supported by National Research Foun-dation of Korea (grant NRF-2019R1A5A1028324). K.S. would like to express his gratitudeto the School of Mathematics at the University of Edinburgh for its hospitality during hisvisit, where this manuscript was prepared. References [1] N. Barashkov, M. Gubinelli
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Justin Forlano, Maxwell Institute for Mathematical Sciences, Department of Mathe-matics, Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom and Department ofMathematics, University of California, Los Angeles, CA 90095, USA
Email address : [email protected] Kihoon Seong, Department of Mathematical Sciences, Korea Advanced Institute of Scienceand Technology, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
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