On the one-dimensional cubic nonlinear Schrodinger equation below L^2
aa r X i v : . [ m a t h . A P ] J u l ON THE ONE-DIMENSIONAL CUBIC NONLINEAR SCHR ¨ODINGEREQUATION BELOW L TADAHIRO OH AND CATHERINE SULEM
Abstract.
In this paper, we review several recent results concerning well-posedness ofthe one-dimensional, cubic Nonlinear Schr¨odinger equation (NLS) on the real line R andon the circle T for solutions below the L -threshold. We point out common results forNLS on R and the so-called Wick ordered NLS (WNLS) on T , suggesting that WNLS maybe an appropriate model for the study of solutions below L ( T ). In particular, in contrastwith a recent result of Molinet [34] who proved that the solution map for the periodiccubic NLS equation is not weakly continuous from L ( T ) to the space of distributions, weshow that this is not the case for WNLS. Introduction
In this paper, we consider the one-dimensional cubic nonlinear Schr¨odinger equation(NLS):(1.1) ( iu t − u xx ± | u | u = 0 u (cid:12)(cid:12) t =0 = u , ( x, t ) ∈ T × R or R × R , where u is a complex-valued function and T = R / π Z . (1.1) arises in various physicalsettings for the description of wave propagation in nonlinear optics, fluids and plasmas (see[36] for a general review.) It also arises in quantum field theory as a mean field equationfor many body boson systems. It is known to be one of the simplest partial differentialequations (PDEs) with complete integrability [1, 2, 22].As a complete integrable PDE, (1.1) enjoys infinitely many conservation laws, startingwith conservation of mass, momentum, and Hamiltonian:(1.2) N ( u ) = ˆ | u | dx, P ( u ) = Im ˆ uu x dx, H ( u ) = 12 ˆ | u x | dx ± ˆ | u | dx. In the focusing case (with the − sign), (1.1) admits soliton and multi-soliton solutions.Moreover, (1.1) is globally well-posed in L thanks to the conservation of the L -norm(Tsutsumi [37] on R and Bourgain [3] on T .)It is also well-known that (1.1) is invariant under several symmetries. In the following, weconcentrate on the dilation symmetry and the Galilean symmetry. The dilation symmetrystates that if u ( x, t ) is a solution to (1.1) on R with initial condition u , then u λ ( x, t ) = λ − u ( λ − x, λ − t ) is also a solution to (1.1) with the λ -scaled initial condition u λ ( x ) = λ − u ( λ − x ). Associated to the dilation symmetry, there is a scaling-critical Sobolev index s c such that the homogeneous ˙ H s c -norm is invariant under the dilation symmetry. In thecase of the one-dimensional cubic NLS, the scaling-critical Sobolev index is s c = − . It iscommonly conjectured that a PDE is ill-posed in H s for s < s c . Indeed, on the real line, Mathematics Subject Classification.
Key words and phrases.
Schr¨odinger equation; Wick ordering; well-posedness.C.S. is partially supported by N.S.E.R.C. Grant 0046179-05.
Christ-Colliander-Tao [12] showed that the data-to-solution map is unbounded from H s ( R )to H s ( R ) for s < − . The Galilean invariance states that if u ( x, t ) is a solution to (1.1) on R with initial condition u , then u β ( x, t ) = e i β x e i β t u ( x + βt, t ) is also a solution to (1.1)with the initial condition u β ( x ) = e i β x u ( x ). Note that the L -norm is invariant under theGalilean symmetry. It turned out that this symmetry also leads to a kind of ill-posednessin the sense that the solution map cannot be smooth in H s for s < s ∞ c = 0. Indeed, asimple application of Bourgain’s idea in [4] shows that the solution map of (1.1) cannot be C in H s for s < s ∞ c = 0. See Section 2 for more results in this direction.Recently, Molinet [34] showed that the solution map for (1.1) on T cannot be continuousin H s ( T ) for s <
0. (See Christ-Colliander-Tao [13] and Carles-Dumas-Sparber [9] forrelated results.) His argument is based on showing that the solution map is not continuousfrom L ( T ) endowed with weak topology to the space of distributions ( C ∞ ( T )) ∗ . Severalremarks are in order. First, on the real line, there is no corresponding result (i.e. failure ofcontinuity of the solution map for s < . ) Also, the discontinuity in [34] is precisely causedby 2 µ ( u ) u , where µ ( u ) := ffl | u | dx = π ´ π | u | dx .Our main goal in this paper is to propose an alternative formulation of the periodic cubicNLS below L ( T ) to avoid this non-desirable behavior. In particular, we show that thismodel has properties similar to those of (1.1) on the real line even below L . We considerthe Wick ordered cubic NLS (WNLS):(1.3) ( iu t − u xx ± ( | u | − ffl | u | ) u = 0 u (cid:12)(cid:12) t =0 = u for ( x, t ) ∈ T × R . Clearly, (1.1) and (1.3) are equivalent for u ∈ L ( T ). If u satisfies (1.1)with u ∈ L ( T ), then v ( t ) = e ∓ iµ ( u ) t u ( t ) satisfies (1.3). However, for u / ∈ L ( T ), wecannot freely convert solutions of (1.3) into solutions of (1.1). The effect of this modificationcan be seen more clearly on the Fourier side. By writing the cubic nonlinearity as [ | u | u ( n ) = P n = n − n + n b u ( n ) b u ( n ) b u ( n ), we see that the additional term in (1.3) precisely removesresonant interactions caused by n = n or n . See Section 4. Such a modification doesnot seem to have a significant effect on R , since ξ = ξ or ξ is a set of measure zero inthe hyperplane ξ = ξ − ξ + ξ (for fixed ξ .)It turns out that (1.3) on T shares many common features with (1.1) on R (see Section2.) Equation (1.3) (in the defocusing case on T ) first appeared in the work of Bourgain[6, 7], in the study of the invariance of the Gibbs measure, as an equivalent formulationof the Wick ordered Hamiltonian equation, related to renormalization in the Euclidean ϕ quantum field theory (see Section 3.)There are several results on (1.3). Using a power series method, Christ [10] provedthe local-in-time existence of solutions in F L p ( T ) for p < ∞ , where the Fourier-Lebesguespace F L p ( T ) is defined by the norm k f k F L p ( T ) = k b f ( n ) k l pn ( Z ) . In the periodic case, wehave F L p ( T ) ) L ( T ) for p >
2. Gr¨unrock-Herr [20] established the same result (withuniqueness) via the fixed point argument.On the one hand, Molinet’s ill-posedness result does not apply to (1.3) since we removedthe part responsible for the discontinuity. On the other hand, by a slight modification ofthe argument in Burq-G´erard-Tzvetkov [8], we see that the solution map for (1.3) is not The Galilean symmetry does not preserve the momentum. Indeed, P ( u β ) = β N ( u ) + P ( u ). Strictly speaking, Molinet’s result applies to the flow map, i.e. for each nonzero u ∈ L ( T ), the map: u → u ( t ) is not continuous. UBIC NLS BELOW L uniformly continuous below L ( T ), see [16]. This, in particular, implies that one cannotexpect well-posedness of (1.3) in H s ( T ) for s < H s ( T ) for s <
0. Christ-Holmer-Tataru[15] established an a priori bound on the growth of (smooth) solutions in the H s -topologyfor s ≥ − . In Section 4, we show that the solution map for (1.3) is continuous in L ( T )endowed with weak topology. These results have counterparts for (1.1) on R .In [16], Colliander-Oh considered the well-posedness question of (1.3) below L ( T ) withrandomized initial data of the form(1.4) u ( x ; ω ) = X n ∈ Z g n ( ω ) p | n | α e inx , where { g n } n ∈ Z is a family of independent standard complex-valued Gaussian random vari-ables. It is known [40] that u ( ω ) ∈ H α − − ε \ H α − almost surely in ω for any ε > u of the form (1.4) is a typical element in the support of the Gaussian measure(1.5) dρ α = Z − α exp (cid:16) − ˆ | u | − | D α u | dx (cid:17) Y x ∈ T du ( x ) , where D = p − ∂ x . In [16], local-in-time solutions were constructed for (1.3) almost surely(with respect to ρ α ) in H s ( T ) for each s > − ( s = α − − ε for small ε > H s ( T ) for all s > − . The argument is based onthe fixed point argument around the linear solution, exploiting nonlinear smoothing underrandomization on initial data.The same technique can be applied to study the well-posedness issue of (1.3) with initialdata of the form(1.6) u ( x ; ω ) = v ( x ) + X n ∈ Z g n ( ω ) p | n | α e inx , where v is in L ( T ). The initial data of the form (1.6) may be of physical importancesince smooth data may be perturbed by a rough random noise. i.e. initial data, which aresmooth in an ideal situation, may be of low regularity in practice due to a noise. This isone of the reasons that we are interested in having a formulation of NLS below L .Another physically relevant issue is the study of (1.3) with initial data of the form (1.4)when α = 0. The Gaussian measure ρ α then corresponds to the white noise on T (up to amultiplicative constant.) It is conjectured [40] that the white noise is invariant under theflow of the cubic NLS (1.1). In [35], Oh-Quastel-Valk´o proved that the white noise is a weaklimit of probability measures that are invariant under the flow of (1.1) and (1.3). Note thatthe white noise ρ is supported on H − − ε ( T ) \ H − ( T ) for ε > B − , ∞ .)Such a low regularity seems to be out of reach at this point. Hence, the result in [35] impliesonly a version of “formal” invariance of the white noise due to lack of well-defined flow ofNLS on the support of the white noise. In view of Molinet’s ill-posedness below L ( T ), weneed to pursue this issue with (1.3) in place of (1.1). In this respect, the result in [16] canbe regarded as a partial progress toward this goal.Note that the white noise (i.e. u in (1.4) with α = 0 up to multiplicative constant)can be regarded as a Gaussian randomization (on the Fourier coefficients) of the deltafunction δ ( x ) = P n e inx . It is known [28] that in considering the Cauchy problem (1.1)on R with the delta function as initial condition, we have either non-existence or non-uniqueness in C ([ − T, T ]; S ′ ( R )). Moreover, on T , Christ [11] proved a non-uniqueness TADAHIRO OH AND CATHERINE SULEM result of (1.3) in the class C ([ − T, T ]; H s ( T )) for s <
0. Christ’s result states that one cannot have unconditional uniqueness in H s ( T ), s <
0. However, this is not an issue since, indiscussing well-posedness, we usually construct a unique solution in C ([ − T, T ]; H s ) ∩ X T ,where X T is an auxiliary function space (such as Strichartz spaces or X s,b spaces.)Lastly, another physical motivation for the study of NLS in the low regularity setting isthe localized induction approximation model for the flow of a vortex filament. The filamentat time t is given by a curve X ( x, t ) in R , satisfying(1.7) X t = X x × X xx , where x is the arclength. Then, under the Hasimoto transform [24]:(1.8) u ( x, t ) = c ( x, t ) exp (cid:18) i ˆ x τ ( y, t ) dy (cid:19) , where c ( x, t ) and τ ( x, t ) are the curvature and the torsion of X ( x, t ), the transformedfunction u satisfies the focusing cubic (1.1) on R . Guti´errez-Rivas-Vega [23] showed that asmooth filament can develop a sharp corner in finite time, which corresponds, under (1.8),to a Dirac delta singularity for u in (1.1). This necessitates the study of NLS in the lowregularity setting.This paper is organized as follows. In Section 2, we compare the results for NLS (1.1)on R and Wick ordered NLS (1.3) on T . In Section 3, we recall basic aspects of the Wickordering and the derivation of (1.3) on T following [7]. In Section 4, we present the proofof the weak continuity of the solution map for (1.3) in L ( T ).2. NLS on R and Wick ordered NLS on T In this section, we present several results that are common to (1.1) on R and (1.3) on T .We show a summary of these results in Table 1 below. This analogy suggests that Wickordered NLS (1.3) on T is an appropriate model to study when interested in solutions below L ( T ).2.1. Well-posedness in L . On the real line, Tsutsumi [37] proved global well-posednessof (1.1) in L ( R ). His argument is based on the smoothing properties of the linearSchr¨odinger operator expressed by the Strichartz estimates and the conservation of the L -norm. For the problem on the circle, Bourgain [3] introduced the X s,b space and provedglobal well-posedness of (1.1) in L ( T ). His argument is based on the periodic L Strichartzand the conservation of the L -norm. The same argument can be applied to establish globalwell-posedness of (1.3) in L ( T ).2.2. Ill-posedness in H s for s < : An application of Bourgain’s argument in [4] showsthat the solution maps for (1.1) on R and (1.3) on T are not C in H s for s <
0. Themethod consists of examining the differentiability at δ = 0 of the solution map with initialcondition u = δφ for some suitable φ i.e. differentiability at the zero solution in a certaindirection.On R , Kenig-Ponce-Vega [28] proved the failure of uniform continuity of the solution mapfor (1.1) in H s ( R ) for s < T , Burq-G´erard-Tzvetkov [8] We say that a solution u is unconditionally unique if it is unique in C ([0 , T ]; H s ) without intersectingwith any auxiliary function space. Unconditional uniqueness is a concept of uniqueness which does notdepend on how solutions are constructed. See Kato [26]. UBIC NLS BELOW L NLS on R WNLS on T NLS on T GWP in L [37] [3] [3]Ill-posedness below L [28], [12] [8] [8], [34]Well-posedness in F L p , p < ∞ [19] (GWP for p ∈ (2 , )) [10], [20] False [10]A priori bound for s ≥ − [29], ([14] for s > − ) [15] Not knownWeak continuity in L [21] Theorem 2.1 False [34] Table 1.
Corresponding results for NLS on R and WNLS on T (and NLSon T .)(also see [12]) constructed a family of explicit solutions supported on a single mode andshowed the corresponding result for (1.1). By a slight modification of their argument, wecan also establish the same result for (1.3). It is worthwhile to note that the momentumdiverges to ∞ in these examples.The above ill-posedness results state that the solution map is not smooth or uniformlycontinuous in H s below s < s ∞ c = 0. This does not say that (1.1) on R and (1.3) on T are ill-posed below L , i.e. it is still possible to construct continuous flow below L . Theseresults instead state that the fixed point argument cannot be used to show well-posedness of(1.1) on R and (1.3) on T below L , since such a method would make solution maps smooth.Compare the above results with the ill-posedness result by Molinet [34] - the discontinuityof the solution map below L ( T ) for the periodic NLS (1.1).2.3. Well-posedness in F L p : Define the Fourier Lebesgue space F L s,p ( R ) equipped withthe norm k f k F L s,p ( R ) = kh ξ i s b f ( ξ ) k L p ( R ) with h·i = 1+ |·| . When s = 0, we set F L p = F L ,p .The homogeneous F ˙ L s,p norm is invariant under the dilation scaling when sp = − L -norm under certain conditions. This class of initial data, in particular,contains those satisfying(2.1) (cid:12)(cid:12)(cid:12)(cid:12) d j dξ j b u ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) . h ξ i − α − j , j = 0 , , for some α > . We point out that u satisfying (2.1) is in F L p ( R ) with p > α . Gr¨unrock [19] considered(1.1) on R with initial data in F L p ( R ) and proved local well-posedness for p < ∞ and globalwell-posedness for 2 < p < . The method relies on the Fourier restriction method. For theglobal-in-time argument, he adapted Bourgain’s high-low method [5], where he separated afunction in terms of the size of its Fourier coefficient instead of its frequency size as in [5].On T , Christ [10] applied the power series method to construct local-in-time solutions(without uniqueness) for (1.3) in F L p ( T ) for p < ∞ . Gr¨unrock-Herr [20] proved the sameresult (with uniqueness in a suitable X s,b space) via the fixed point argument. A subtractionof 2 ffl | u | dx u in the nonlinearity in (1.3) is essential for continuous dependence. In [10], itis also stated (without proof) that (1.3) is global well-posed in F L p for sufficiently small(smooth) initial data. TADAHIRO OH AND CATHERINE SULEM
A priori bound:
Koch-Tataru [29] established an a priori bound on (smooth) solu-tions for (1.1) in H s ( R ) for s ≥ − in the form: given any M >
0, there exist
T, C > u ∈ L with k u k H s ≤ M , we have sup t ∈ [0 ,T ] k u ( t ) k H s ≤ C k u k H s ,where u is a solution of (1.1) with initial condition u . See Christ-Colliander-Tao [14] fora related result. This result yields the existence on weak solutions (without uniqueness).In the periodic setting, Christ-Holmer-Tataru [15] proved the same result for (1.3) when s ≥ − . In [29], relating mKdV and NLS through modulated wave train solutions, Koch-Tataru indicate how the regularity s = − arises by associating mKdV with initial data in L to (1.1) with initial data in H − .2.5. Weak continuity in L : The Galilean invariance for (1.1) yields the critical regularity s ∞ c = 0. i.e. the solution map is not uniformly continuous in H s for s < s ∞ c = 0. However,it does not imply that the solution map is not continuous in H s for s < R .) Heuristically speaking, given s ∈ R , one can consider the weak continuity of thesolution map in H s as an intermediate step between establishing the continuity in (thestrong topology of) H s and proving the continuity in H s for s < s . For example, recallthat if f n converges weakly in H s , then it converges strongly in H s for s < s (at least inbounded domains.) Indeed if there is sufficient regularity for the solution map in H s forsome s < s , then its weak continuity in H s can be treated by the approach used in theworks of Martel-Merle [32, 33] and Kenig-Martel [27] related to the asymptotic stabilityof solitary waves. In these works, weak continuity of the flow map plays a central role inthe study of the linearized operator around the solitary wave and in rigidity theorems. SeeCui-Kenig [17] for a nice discussion on this issue.There are several recent results in this direction. On R , Goubet-Molinet [21] proved theweak continuity of the solution map for (1.1) in L ( R ). Cui-Kenig [17, 18] proved the weakcontinuity in the s ∞ c -critical Sobolev spaces for other dispersive PDEs. However, on T ,Molinet [34] showed that the solution map for (1.1) is not continuous from L ( T ) endowedwith weak topology to the space of distributions ( C ∞ ( T )) ∗ . This, in particular, impliesthat the solution map for (1.1) is not weakly continuous in L ( T ).When considering the Wick ordered cubic NLS (1.3), we remove one of the resonantinteractions. Indeed, we have the following result on the weak continuity of the solutionmap for (1.3). Theorem 2.1 (Weak continuity of WNLS on L ( T )) . Suppose that u ,n converges weaklyto u in L ( T ) . Let u n and u denote the unique global solutions of (1.3) with initial data u ,n and u , respectively. Then, given T > , we have the following. (a) u n converges weakly to u in L T,x := L ([ − T, T ]; L ( T )) . (b) For any | t | ≤ T , u n ( t ) converges weakly to u ( t ) in L ( T ) . Moreover, this weakconvergence is uniform for | t | ≤ T . i.e. for any φ ∈ L ( T ) , lim n →∞ sup | t |≤ T |h u n ( t ) − u ( t ) , φ i L | = 0 . We do not expect the weak continuity in the Strichartz space, i.e. in L T,x (with | t | ≤ T .)This is due to the failure of the L x,t Strichartz estimate in the periodic setting [3]. Althoughthe proof of Theorem 2.1 is essentially contained in [34], we present it in Section 4 for thecompleteness of our presentation.
UBIC NLS BELOW L Wick ordering
Gaussian measures and Hermite polynomials.
In this subsection, we briefly goover the basic relation between Gaussian measures and Hermite polynomials. For thefollowing discussion, we refer to the works of Kuo [30], Ledoux-Talagrand [31], and Janson[25]. A nice summary is given by Tzvetkov in [38, Section 3] for the hypercontractivity ofthe Ornstein-Uhlenbeck semigroup related to products of Gaussian random variables.Let ν be the Gaussian measure with mean 0 and variance σ , and H n ( x ; σ ) be the Hermitepolynomial of degree n with parameter σ . They are defined by e tx − σt = ∞ X n =0 H n ( x ; σ ) n ! t n . The first three Hermite polynomials are: H ( x ; σ ) = 1, H ( x ; σ ) = x , and H ( x ; σ ) = x − σ .It is well known that every function f ∈ L ( ν ) has a unique series expansion f ( x ) = ∞ X n =0 a n H n ( x ; σ ) √ n ! σ n , where a n = ( n ! σ n ) − ´ ∞−∞ f ( x ) H n ( x ; σ ) dν ( x ), n ≥
0. Moreover, we have k f k L ( ν ) = P ∞ n =0 a n . In the following, we set H n ( x ) := H n ( x ; 1).Now, consider the Hilbert space L ( R d , µ d ) with dµ d = (2 π ) − d exp( −| x | / dx , x =( x , . . . , x d ) ∈ R d . We define a homogeneous Wiener chaos of order n to be an element ofthe form Q dj =1 H n j ( x j ), n = n + · · · + n d . Denote by K n the collection of the homogeneouschaoses of order n . Given a homogeneous polynomial P n ( x ) = P n ( x , . . . , x d ) of degree n ,we define the Wick ordered monomial : P n ( x ): to be its projection onto K n . In particular,we have : x nj := H n ( x j ) and : Q dj =1 x n j j := Q dj =1 H n j ( x j ) with n = n + · · · + n d .In the following, we discuss the key estimate for the well-posedness results of the Wickordered cubic NLS of [7, 16]. Consider the Hartree-Fock operator L = ∆ − x · ∇ , which isthe generator for the Ornstein-Uhlenbeck semigroup. Then, by the hypercontractivity ofthe Ornstein-Uhlenbeck semigroup U ( t ) = e Lt , we have the following proposition. Proposition 3.1.
Fix q ≥ . For every f ∈ L ( R d , µ d ) and t ≥ log( q − , we have (3.1) k U ( t ) f k L q ( R d ,µ d ) ≤ k f k L ( R d ,µ d ) . It is known that the eigenfunction of L with eigenvalue − n is precisely the homogeneousWiener chaos of order n . Thus, we have the following dimension-independent estimate. Proposition 3.2.
Let F ( x ) be a linear combination of homogeneous chaoses of order n .Then, for q ≥ , we have (3.2) k F ( x ) k L q ( R d ,µ d ) ≤ ( q − n k F ( x ) k L ( R d ,µ d ) . The proof is basically the same as in [38, Propositions 3.3–3.5]. We only have to notethat F ( x ) is an eigenfunction of U ( t ) with eigenvalue e − nt . The estimate (3.2) followsfrom (3.1) by evaluating (3.1) at time t = log( q − | u | :, the Wick ordered | u | , for a complex-valued function u , weconsider the Wick ordering on complex Gaussian random variables. Let g denote a standardcomplex-valued Gaussian random variable. Then, g can be written as g = x + iy , where TADAHIRO OH AND CATHERINE SULEM x and y are independent standard real-valued Gaussian random variables. Note that thevariance of g is Var( g ) = 2.Next, we investigate the Wick ordering on | g | n for n ∈ N , that is, the projection of | g | n onto K n . When n = 1, | g | = x + y is Wick-ordered into: | g | := ( x −
1) + ( y −
1) = | g | − Var( g ) . When n = 2, | g | = ( x + y ) = x + 2 x y + y is Wick-ordered into: | g | : = ( x − x + 3) + 2( x − y −
1) + ( y − y + 3)= x + 2 x y + y − x + y ) + 8(3.3) = | g | − g ) | g | + 2Var( g ) , where we used H ( x ) = x − x + 3. In general, we have : | g | n : ∈ K n . Moreover, we have(3.4) : | g | n : = | g | n + n − X j =0 a j | g | j = | g | n + n − X j =0 b j : | g | j : . This follows from the fact that | g | n , as a polynomial in x and y only with even powers, isorthogonal to any homogeneous chaos of odd order, and it is radial, i.e., it depends onlyon | g | = x + y . Note that : | g | n : can also be obtained from the Gram-Schmidt processapplied to | g | k , k = 0 , . . . , n with µ = (2 π ) − exp( − ( x + y ) / dxdy .3.2. Wick ordered cubic NLS.
In [7], Bourgain considered the issue of the invariantGibbs measure for (1.1) on T in the defocusing case. In this subsection, we present hisargument to derive (1.3) on T . First, consider the finite dimensional approximation to(1.1):(3.5) ( iu Nt − ∆ u N + P N ( | u N | u N ) = 0 u (cid:12)(cid:12) t =0 = P N u , ( x, t ) ∈ T × R , where u N = P N u and P N is the Dirichlet projection onto the frequencies | n | ≤ N . Thisis a Hamiltonian equation with Hamiltonian H ( u N ), where H is as in (1.2) with the +sign. On T , the Gaussian part dρ = Z − exp (cid:0) − ´ |∇ u | dx (cid:1) Q x ∈ T du ( x ) of the Gibbsmeasure is supported on T s< H s ( T ) \ L ( T ). However, the nonlinear part ´ | P N u | dx of the Hamiltonian diverges to ∞ as N → ∞ almost surely on the support of the Wienermeasure ρ . Hence, we need to renormalize the nonlinearity.A typical element in the support of the Wiener measure ρ is given by(3.6) u ( x ; ω ) = X n ∈ Z g n ( ω ) p | n | e in · x , where { g n } n ∈ Z is a family of independent standard complex-valued Gaussian random vari-ables. For simplicity, assume that Var( g n ) = 1. For u of the form (3.6), define a N by a N = E h ˆ −| u N | dx i = X | n |≤ N
11 + | n | . The expression (3.6) is a representation of elements in the support of d e ρ = e Z − exp (cid:0) − ´ | u | − ´ |∇ u | (cid:1) Q x ∈ T du ( x ) due to the problems at the zero Fourier mode for ρ . However, we do not worryabout this issue. UBIC NLS BELOW L We have that a N ∼ log N for large N . We define the Wick ordered truncated Hamiltonian H N by H N ( u N ) = 12 ˆ T |∇ u N | dx + 14 ˆ T : | u N | : dx (3.7) = 12 ˆ T |∇ u N | dx + 14 ˆ T | u N | − a N | u N | + 2 a N dx. (Compare (3.7) with (3.3).) From (3.7), we obtain an Hamiltonian equation that is theWick ordered version of (3.5):(3.8) iu Nt − ∆ u N + P N ( | u N | u N ) − a N u N = 0 . Let c N = ffl | u N | − a N , we see that c ∞ ( ω ) = lim N →∞ c N ( ω ) < ∞ almost surely. Under thechange of variables v N = e − ic N t u N , (3.8) becomes(3.9) iv Nt − ∆ v N + P N (cid:0) | v N | − ffl | v N | (cid:1) v N = 0 . Finally, letting N → ∞ , we obtain the Wick order NLS.(3.10) iv t − ∆ v + ( | v | − ffl | v | (cid:1) v = 0 . On T , one can repeat the same argument. Note the following issue. On the one hand,the assumption that u ( t ) is of the form (1.4) is natural for α ∈ N ∪ { } in view of theconservation laws. On the other hand, c N = ffl | u N | − E (cid:2) ffl | u N | (cid:3) < ∞ for α > . i.e. α = 1 is the smallest integer value of such α . In this case, there is no need for the Wickordered NLS (1.3) since u ∈ H − a.s. for α = 1.4. Weak continuity of the Wick ordered cubic NLS in L ( T )In this section, we present the proof of Theorem 2.1. First, write (1.3) as an integralequation:(4.1) u ( t ) = S ( t ) u ± i ˆ t S ( t − t ′ ) N ( u )( t ′ ) dt ′ where N ( u ) = ( | u | − ffl | u | ) u and S ( t ) = e − i∂ x t . Define N ( u , u , u ) and N ( u , u , u )by N ( u , u , u ) = X n = n − n + n n = n ,n b u ( n ) b u ( n ) b u ( n ) e inx , N ( u , u , u ) = − X n b u ( n ) b u ( n ) b u ( n ) e inx . Moreover, let N j ( u ) := N j ( u, u, u ). Then, we have N ( u ) = N ( u ) + N ( u ) . In [3], Bourgain established global well-posedness of (1.1) (and (1.3)) by introducing anew weighted space-time Sobolev space X s,b whose norm is given by k u k X s,b ( T × R ) = kh n i s h τ − n i b b u ( n, τ ) k L n,τ ( Z × R ) where h · i = 1 + | · | . Define the local-in-time version X s,bδ on [ − δ, δ ] by k u k X s,bδ = inf (cid:8) k e u k X s,b ; e u | [ − δ,δ ] = u (cid:9) . In the following, we list the estimates needed for local well-posedness of (1.3). Let η ( t ) bea smooth cutoff function such that η = 1 on [ − ,
1] and η = 0 on [ − , • Homogeneous linear estimate: for s, b ∈ R , we have(4.2) k η ( t ) S ( t ) f k X s,b ≤ C k f k H s . • Nonhomogeneous linear estimate: for s ∈ R and b > , we have(4.3) (cid:13)(cid:13)(cid:13)(cid:13) η ( t ) ˆ t S ( t − t ′ ) F ( t ′ ) dt ′ (cid:13)(cid:13)(cid:13)(cid:13) X s,bδ . C ( δ ) k F k X s,b − δ . • Periodic L Strichartz estimate: Zygmund [41] proved(4.4) k S ( t ) f k L x,t ( T × [ − , . k f k L , which was improved by Bourgain [3]:(4.5) k u k L x,t ( T × [ − , . k u k X , . These estimates allow us to prove local well-posedness of (1.3) via the fixed point argumentsuch that a solution u exists on the time interval [ − δ, δ ] with δ = δ ( k u k L ). Moreover,we have k u k X ,
12 + δ . k u k L . Such local-in-time solutions can be extended globally in timethanks to the L conservation.Now, fix u ∈ L ( T ), and let u ,n converges weakly to u in L ( T ). Denote by u n and u the unique global solutions of (1.3) with initial data u ,n and u . By the uniformboundedness principle, we have k u ,n k L , k u k L ≤ C for some C >
0. Hence, the localwell-posedness guarantees the existence of the solutions u n , u on the time interval [ − δ, δ ]with δ = δ ( C ), uniformly in n . In the following, we assume δ = 1. i.e. we assume that allthe estimates hold on [ − , − ,
1] by [ − δ, δ ] for some δ > L conservation.)4.1. Proof of Theorem 2.1 (a).
First, we show that u n converges to u as space-timedistributions. • Linear part:
Since u ,n ⇀ u in L ( T ), we have k u ,n − u k H − ε ( T ) → ε > φ ∈ C ∞ c ( T × R ) be a test function. Then, by H¨older inequality and (4.2), we have ¨ η ( t ) S ( t )( u ,n − u ) φ ( x, t ) dxdt ≤ k η ( t ) S ( t )( u ,n − u ) k X − ε,
12 + k φ k X ε, − − . C φ k u ,n − u k H − ε → . Hence, η ( t ) S ( t ) u ,n converges to η ( t ) S ( t ) u as space-time distributions. • Nonlinear part:
Let M ( u ) denote the Duhamel term. i.e. M ( u )( t ) := ± i ˆ t S ( t − t ′ ) N ( u )( t ′ ) dt ′ . Similarly, define M j ( u , u , u ) by M j ( u , u , u )( t ) := ± i ˆ t S ( t − t ′ ) N j ( u , u , u )( t ′ ) dt ′ for j = 1 ,
2. Also, let M j ( u ) := M j ( u, u, u ).From the local theory, we have k u n k X ,
12 +1 . k u ,n k L ≤ C for all n . Thus, there existsa subsequence u n k converging weakly to some v in X , +1 . It follows from [34, Lemmata UBIC NLS BELOW L N j , j = 1 ,
2, is weakly continuous from X , +1 into X , − . Hence, N j ( u k ) ⇀ N j ( v ) in X , − .Recall the following. Given Banach spaces X and Y with a continuous linear operator T : X → Y , we have T ∗ : Y ∗ → X ∗ . If f n ⇀ f in X , then we have, for φ ∈ Y ∗ , h T ( f n − f ) , φ i = h f n − f, T ∗ φ i → T ∗ φ ∈ X ∗ . Hence, T f n ⇀ T f in Y .It follows from (4.3) that the map: F ´ t S ( t − t ′ ) F ( t ′ ) dt ′ is linear and continuous from X , − into X , +1 . Hence, M ( u n k ) ⇀ M ( v ) in X , +1 . In particular, M ( u n k ) convergesto M ( v ) as space-time distributions.Since u n k is a solution to (1.3) with initial data u ,n k , we have u n k = η S ( t ) u ,n k + η M ( u n k ) . By taking the limits of both sides, we obtain v = η S ( t ) u + η M ( v ) , where the equality holds in the sense of space-time distributions. From the uniqueness ofsolutions to (1.3) in X , +1 , we have v = u in X , +1 .In fact, we can show that uniqueness of solutions to (1.3) holds in L x,t ( T × [ − , N ( u ) in (4.1) by | u | u . Then, by (4.4) and (4.5), wehave k η ( t ) u k L x,t ≤ k η ( t ) S ( t ) u k L x,t + (cid:13)(cid:13)(cid:13)(cid:13) η ( t ) ˆ t S ( t − t ′ ) | ηu ( t ′ ) | ηu ( t ′ ) dt ′ (cid:13)(cid:13)(cid:13)(cid:13) L x,t . k u k L x + (cid:13)(cid:13)(cid:13)(cid:13) η ( t ) ˆ t S ( t − t ′ ) | ηu ( t ′ ) | ηu ( t ′ ) dt ′ (cid:13)(cid:13)(cid:13)(cid:13) X , . Moreover, we can use (4.3), duality, L x,t L x,t L x,t L x,t -H¨older inequality, and (4.5) to estimatethe second term by . k| ηu | ηu k X , − = sup k v k X , =1 ¨ v | ηu | ( ηu ) dxdt ≤ sup k v k X , =1 k v k L x,t k ηu k L x,t ≤ k ηu k L x,t . This shows that u is indeed a unique solution in L x,t ( T × [ − , L x,t ( T × [ − , ∗ ⊂ ( X , +1 ) ∗ that weak convergence in X , +1 impliesweak convergence in L x,t ( T × [ − , u n k converges weakly to u in X , +1 and L x,t ( T × [ − , u is the only weaklimit point of u n in X , +1 and L x,t ( T × [ − , u n in X , +1 and L x,t ( T × [ − , u n converges weakly to u .Indeed, suppose that the whole sequence u n does not converge weakly to u . Then, thereexists φ ∈ ( X , +1 ) ∗ such that h u n , φ i h u, φ i . This, in turn, implies that there exists ε > N ∈ N , there exists n ≥ N such that |h u n − u, φ i| > ε . Given ε >
0, we can construct a subsequence u n k with |h u n k − u, φ i| > ε for each k . However, byrepeating the previous argument (from the uniform boundedness of u n k in X , +1 ), u n k hasa sub-subsequence converging to u , which is a contradiction. This establishes Part (a) ofTheorem 2.1 on [ − , Proof of Theorem 2.1 (b).
Recall the following embedding. For b > , we have(4.6) k u k L ∞ ([ − , H s ) ≤ C k u k X s,b . Fix φ ∈ L ( T ) in the following. • Linear part:
Given ε >
0, choose ψ ∈ H ( T ) such that k φ − ψ k L < ε KC C , where K = sup n k u ,n − u k L < ∞ and C , C are as in (4.2), (4.6). Then, by (4.2) and (4.6),we havesup | t |≤ |h S ( t )( u ,n − u , φ i L | ≤ sup | t |≤ |h S ( t )( u ,n − u , ψ i L | + sup | t |≤ |h S ( t )( u ,n − u , φ − ψ i L |≤ k S ( t )( u ,n − u ) k L ∞ ([ − , H − ) k ψ k H + k S ( t )( u ,n − u ) k L ∞ ([ − , L ) k φ − ψ k L ≤ C ψ k S ( t )( u ,n − u ) k X − ,
12 +1 + ε KC k S ( t )( u ,n − u ) k X ,
12 +1 ≤ C k u ,n − u k H − + ε K k u ,n − u k L . Hence, there exists N such that for n ≥ N ,sup | t |≤ |h S ( t )( u ,n − u , φ i L | < ε since u ,n converges strongly u n in H − . • Nonlinear part:
Since u n ⇀ u in X , +1 , we see that N ( u n ) converges strongly to N ( u )in X − , − . See the proof of Lemmata 2.2 and 2.3 in [34]. Then, it follows from (4.3) that M ( u n ) converges strongly to M ( u ) in X − , +1 .Given ε >
0, choose ψ ∈ H ( T ) such that k φ − ψ k L < ε KC , where K = sup n kM ( u n ) −M ( u n ) k X ,
12 +1 < ∞ and C is as in (4.6). Then, by (4.6), we havesup | t |≤ |hM ( u n ) − M ( u ) , φ i|≤ sup | t |≤ |hM ( u n ) − M ( u ) , ψ i L | + sup | t |≤ |hM ( u n ) − M ( u ) , φ − ψ i L |≤ kM ( u n ) − M ( u ) k L ∞ ([ − , H − ) k ψ k H + kM ( u n ) − M ( u ) k L ∞ ([ − , L ) k φ − ψ k L ≤ C ψ kM ( u n ) − M ( u ) k X − ,
12 +1 + ε K kM ( u n ) − M ( u ) k X ,
12 +1 . Hence, there exists N such that for n ≥ N ,sup | t |≤ |hM ( u n ) − M ( u ) , φ i| < ε. Therefore, we have lim n →∞ sup | t |≤ |h u n ( t ) − u ( t ) , φ i L | = 0 . Given [ − T, T ], we can iterate Part 1 and 2 on each [ j, j + 1] and obtain Theorem 2.1.
Acknowledgements
We would like to thank Jim Colliander for helpful discussions and comments and LuisVega for pointing out additional references.
UBIC NLS BELOW L References [1] M. Ablowitz, D. Kaup, David,A. Newell, H. Segur,
The inverse scattering transform-Fourier analysisfor nonlinear problems,
Studies in Appl. Math., 53 (1974), 249–315.[2] M. Ablowitz, Y. Ma,
The periodic cubic Schr¨odinger equation,
Stud. Appl. Math., 65 (1981), 113–158.[3] J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications tononlinear evolution equations I , Geom. Funct. Anal., 3 (1993), no. 2, 107–156.[4] J. Bourgain,
Periodic Korteweg-de Vries equation with measures as initial data,
Sel. Math., New Ser.3 (1997) 115–159.[5] J. Bourgain,
Refinements of Strichartz’ inequality and applications to D-NLS with critical nonlinear-ity,
Int. Math. Res. Not. (1998), no. 5, 253–283.[6] J. Bourgain,
Nonlinear Schr¨odinger equations,
Hyperbolic equations and frequency interactions (ParkCity, UT, 1995), 3–157, IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI, 1999.[7] J. Bourgain,
Invariant measures for the D -defocusing nonlinear Schr¨odinger equation, Comm. Math.Phys. 176 (1996), no. 2, 421–445.[8] N. Burq, P. G´erard, N. Tzvetkov,
An instability property of the nonlinear Schr¨odinger equation on S d , Math. Res. Lett. 9 (2002), no. 2-3, 323–335.[9] R. Carles, E. Dumas, C. Sparber,
Multiphase weakly nonlinear geometric optics for Schr¨odinger equa-tions , arXiv:0902.2468v2 [math.AP].[10] M. Christ,
Power series solution of a nonlinear Schr¨odinger equation,
Mathematical aspects of non-linear dispersive equations, 131–155, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ,2007.[11] M. Christ,
Nonuniqueness of weak solutions of the nonlinear Schroedinger equation, arXiv:math/ 0503366v1 [math.AP].[12] M. Christ, J. Colliander, T. Tao,
Asymptotics, frequency modulation, and low-regularity illposednessof canonical defocusing equations,
Amer. J. Math. 125 (2003), no. 6, 1235–1293.[13] M. Christ, J. Colliander, T. Tao,
Instability of the Periodic Nonlinear Schr¨odinger Equation ,arXiv:math/ 0311227v1 [math.AP].[14] M. Christ, J. Colliander, T. Tao,
A priori bounds and weak solutions for the nonlinear Schr¨odingerequation in Sobolev spaces of negative order,
J. Funct. Anal. 254 (2008), no. 2, 368–395.[15] M. Christ, J. Holmer. D. Tataru,
A priori bounds and instability for the nonlinear Schr¨odinger equa-tion, talk in April workshop “Nonlinear wave and dispersion” at Institut Henri Poincar´e, 2009.[16] J. Colliander, T. Oh,
Almost sure well-posedness of the periodic cubic nonlinear Schr¨odinger equationbelow L , preprint.[17] S. Cui, C. Kenig, Weak Continuity of the Flow Map for the Benjamin-Ono Equation on the Line, arXiv: 0909.0793v2 [math.AP].[18] S. Cui, C. Kenig,
Weak Continuity of Dynamical Systems for the KdV and mKdV Equations, arXiv:0909.0794v2 [math.AP].[19] A. Gr¨unrock,
Bi- and trilinear Schr¨odinger estimates in one space dimension with applications to cubicNLS and DNLS,
Int. Math. Res. Not. (2005), no. 41, 2525–2558.[20] A. Gr¨unrock, S. Herr,
Low regularity local well-posedness of the derivative nonlinear Schr¨odingerequation with periodic initial data,
SIAM J. Math. Anal. 39 (2008), no. 6, 1890–1920.[21] O. Goubet and L. Molinet,
Global weak attractor for weakly damped nonlinear Schr¨odinger equationsin L ( R ) , Nonlinear Anal., 71 (2009), 317–320.[22] B. Gr´ebert, T. Kappeler, J. P¨oschel,
Normal form theory for the NLS equation , arXiv:0907.3938v1[math.AP].[23] S. Guti´errez, J. Rivas, L. Vega,
Formation of singularities and self-similar vortex motion under thelocalized induction approximation,
Comm. Partial Differential Equations 28 (2003), no. 5-6, 927–968.[24] H. Hasimoto,
A soliton on a vortex filament,
J. Fuild Mech., 51 (1972), 477–485.[25] S. Janson,
Gaussian Hilbert Spaces,
Cambridge Tracts in Mathematics, 129. Cambridge UniversityPress, Cambridge, 1997. x+340 pp.[26] T. Kato,
On nonlinear Schr¨odinger equations. II. H s -solutions and unconditional well-posedness, J.Anal. Math. 67 (1995), 281–306.[27] C. Kenig, Y. Martel,
Asymptotic stability of solitons for the Benjamin-Ono equation,
Revista Mat.Iberoamericana, to appear.[28] C. Kenig, G. Ponce, L. Vega,
On the ill-posedness of some canonical dispersive equations,
Duke Math.J. 106 (2001), no.3, 617–633. [29] H. Koch, D. Tataru,
A priori bounds for the 1D cubic NLS in negative Sobolev spaces,
Int. Math. Res.Not. (2007), no. 16, Art. ID rnm053, 36 pp.[30] H. Kuo,
Introduction to stochastic integration,
Universitext. Springer, New York, 2006. xiv+278 pp.[31] M. Ledoux and M. Talagrand,
Probability in Banach spaces. Isoperimetry and Processes,
Ergebnisseder Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23.Springer-Verlag, Berlin, 1991. xii+480 pp[32] Y. Martel, F. Merle,
A Liouville theorem for the critical generalized Korteweg-de Vries equation,
J.Math. Pures Appl., 79 (2000), 339-425.[33] Y. Martel, F. Merle,
Blow up in finite time and dynamics of blow up solutions for the L2-criticalgeneralized KdV equation,
J. Amer. Math. Soc., 15 (2002), 617-664.[34] L. Molinet,
On ill-posedness for the one-dimensional periodic cubic Schr¨odinger equation , Math. Res.Lett. 16 (2009), no. 1, 111–120.[35] T. Oh, J. Quastel, B. Valk´o,
Interpolation of Gibbs measures with the white noise for HamiltonianPDEs, arXiv:1005.3957v1 [math.PR].[36] C. Sulem, P.L. Sulem,
The nonlinear Schr¨odinger equations: Self-focusing and wave collapse,
AppliedMathematical Sciences, 139. Springer-Verlag, New York (1999) 350 pp.[37] Y. Tsutsumi, L -solutions for nonlinear Schr¨odinger equations and nonlinear groups, Funkcial. Ekvac.30 (1987), no. 1, 115–125.[38] N. Tzvetkov,
Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation ,Probab. Theory Relat. Fields 146 (2010), 481–514.[39] A. Vargas, L. Vega,
Global wellposedness for 1 D non-linear Schr¨odinger equation for data with aninfinite L norm, J. Math. Pures Appl. (9) 80 (2001), no. 10, 1029–1044.[40] P. Zhidkov,
Korteweg-de Vries and Nonlinear Schr¨odinger Equations: Qualitative Theory,
Lec. Notesin Math. 1756, Springer-Verlag, 2001.[41] A. Zygmund,
On Fourier coefficients and transforms of functions of two variables,
Stud. Math. 50(1974), 189–201.
Department of Mathematics, University of Toronto, 40 St. George St, Toronto, ON M5S2E4, Canada
E-mail address : [email protected] E-mail address ::