Bilinear control and growth of Sobolev norms for the nonlinear Schrödinger equation
aa r X i v : . [ m a t h . A P ] J a n Bilinear control and growth of Sobolev norms forthe nonlinear Schr¨odinger equation
Alessandro Duca ∗ Vahagn Nersesyan † January 29, 2021
Abstract
We consider the nonlinear Schr¨odinger equation (NLS) on a torusof arbitrary dimension. The equation is studied in presence of an ex-ternal potential field whose time-dependent amplitude is taken as con-trol. Assuming that the potential satisfies a saturation property, we showthat the NLS equation is approximately controllable between any pair ofeigenstates in arbitrarily small time. The proof is obtained by develop-ing a multiplicative version of a geometric control approach introducedby Agrachev and Sarychev. We give an application of this result to thestudy of the large time behavior of the NLS equation with random poten-tial. More precisely, we assume that the amplitude of the potential is arandom process whose law is 1-periodic in time and non-degenerate. Com-bining the controllability with a stopping time argument and the Markovproperty, we show that the trajectories of the random equation are almostsurely unbounded in regular Sobolev spaces.
AMS subject classifications:
Keywords:
Nonlinear Schr¨odinger equation, approximate controllability,geometric control theory, growth of Sobolev norms, random perturbation ∗ Universit´e Paris-Saclay, UVSQ, CNRS, Laboratoire de Math´ematiques de Versailles,78000, Versailles, France; e-mail: [email protected] † Universit´e Paris-Saclay, UVSQ, CNRS, Laboratoire de Math´ematiques de Versailles,78000, Versailles, France; e-mail: [email protected] ontents In this paper, we study the controllability and the growth of Sobolev norms forthe following nonlinear Schr¨odinger (NLS) equation on the torus T d = R d / π Z d : i∂ t ψ = − ∆ ψ + V ( x ) ψ + κ | ψ | p ψ + h u ( t ) , Q ( x ) i ψ. (0.1)We assume that V : T d → R is an arbitrary smooth potential, Q : T d → R q isa given smooth external field subject to some geometric condition, d, p ≥ κ is an arbitrary real number. The role of the control(or the random perturbation) is played by R q -valued function (or random pro-cess) u which is assumed to depend only on time. Eq. (0.1) is equipped with theinitial condition ψ (0 , x ) = ψ ( x ) (0.2)belonging to a Sobolev space H s = H s ( T d ; C ) of order s > d/
2, so that theproblem is locally well-posed.The purpose of this paper is to study the NLS equation (0.1) when the drivingforce u acts multiplicatively through only few low Fourier modes. Referring thereader to the subsequent sections for the general setting, let us formulate in thisIntroduction particular cases of our main results. Let K ⊂ Z d ∗ be the set of d vectors defined by K = { (1 , , . . . , , (0 , , . . . , , . . . , (0 , , . . . , , , (1 , . . . , } , (0.3)and assume that the field Q = ( Q , . . . , Q q ) is such that { , sin h x, k i , cos h x, k i : k ∈ K} ⊂ span { Q j : j = 1 , . . . , q } . (0.4)Let s d be the least integer strictly greater than d/ heorem A. The problem (0.1) , (0.2) is approximately controllable in the fol-lowing sense: for any s ≥ s d , ε > , κ > , ψ ∈ H s , and θ ∈ C ∞ ( T d ; R ) ,there is a time T ∈ (0 , κ ) , a control u ∈ L ([0 , T ]; R q ) , and a unique solution ψ ∈ C ([0 , T ]; H s ) of (0.1) , (0.2) such that k ψ ( T ) − e iθ ψ k H s < ε. More general formulation of this result is given in Theorem 2.2, wherethe controllability is proved under an abstract saturation condition for thefield Q (see Condition (H )). Note that the time T may depend on the ini-tial condition ψ , the target e iθ ψ , and the parameters in the equation. In thesecond result, we show that, when V = 0 and ψ is an eigenstate φ l ( x ) =(2 π ) − d/ e i h x,l i , l ∈ Z d of the Laplacian, the system can be approximately con-trolled in any fixed time T > e iθ φ m with m ∈ Z d . Theorem B.
For any s ≥ s d , ε > , l, m ∈ Z d , θ ∈ C ∞ ( T d ; R ) , and T > ,there is a control u ∈ L ([0 , T ]; R q ) and a unique solution ψ ∈ C ([0 , T ]; H s ) of (0.1) , (0.2) with V = 0 and ψ = φ l such that k ψ ( T ) − e iθ φ m k L < ε. The controllability of the Schr¨odinger equation with time-dependent bilin-ear (multiplicative) control has attracted a lot of attention during the last fif-teen years. In the one-dimensional case, local exact controllability results areestablished by Beauchard, Coron, and Laurent [Bea05, BC06, BL10]. There isa vast literature on the approximate controllability in the multidimensionalcase. For the first achievements, we refer the reader to the papers by Boscainet al. [CMSB09, BCCS12], Mirrahimi [Mir09], and the second author [Ner10].Except the paper [BL10], all the other works consider the linear Schr¨odingerequation, i.e., the one obtained by taking κ = 0 in Eq. (0.1); note that in thatcase the control problem is still nonlinear in u .Theorems A and B are the first to deal with the problem of bilinear approxi-mate controllability of the NLS equation. Let us emphasise that the controllabil-ity between any pair of eigenstates in arbitrarily small time is new even in thelinear case κ = 0. It is interesting to note that Theorem B complements a resultby Beauchard et al. [BCT18], which proves that, for some choices of the field Q ,there is a minimal time for the approximate controllability to some particularstates in the phase space.The approach adopted in the proofs of Theorems A and B is quite differ-ent from those used in the literature for bilinear control systems. We pro-ceed by developing Agrachev–Sarychev type arguments which were previouslyemployed in the case of additive controls. Let us recall that Agrachev andSarychev [AS05, AS06] considered the global approximate controllability of the2D Navier–Stokes and Euler systems. Their approach has been further extendedby many authors to different equations. Let us mention, for example, the pa-pers [Shi06, Shi07] by Shirikyan who considered the approximate controllabilityof the 3D Navier–Stokes system and Sarychev [Sar12] who considered the case3f the 2D defocusing cubic Schr¨odinger equation. The configuration we use inthe present paper is closer to the one elaborated in the recent paper [Ner21],where parabolic PDEs are studied with polynomial nonlinearities. We refer thereader to the reviews [AS08, Shi18] and the paper [Ner21] for more referencesand discussions.The present paper is the first to deal with Agrachev–Sarychev type argu-ments in a bilinear setting. To explain the scheme of the proof of Theorem A,let us denote by R t ( ψ , u ) the solution of problem (0.1), (0.2) defined up tosome maximal time. A central role in the proof is played by the limit e − iδ − / ϕ R δ ( e iδ − / ϕ ψ , δ − u ) → e − i ( B ( ϕ )+ h u,Q i ) ψ in H s as δ → + (0.5)which holds for any ψ ∈ H s , ϕ ∈ C ∞ ( T d ; R ), and constant u ∈ R q . Here wedenote B ( ϕ )( x ) = P dj =1 (cid:0) ∂ x j ϕ ( x ) (cid:1) . Applying this limit with ϕ = 0 and usingthe assumption (0.4), we see that the equation can be controlled in small timefrom initial point ψ arbitrarily close to e iθ ψ for any θ in the vector space H = span { , sin h x, k i , cos h x, k i : k ∈ K} . By applying again the limit (0.5) with functions ϕ = θ j ∈ H , j = 1 , . . . , n ,we add more directions in θ . That is, we show that the system can be steeredfrom ψ close to e iθ ψ , where θ now belongs to a larger vector space H whoseelements are of the form θ − n X j =1 B ( θ j ) . We iterate this argument and construct an increasing sequence of subspaces {H j } such that the equation can be approximately controlled to any target e iθ ψ withany θ ∈ H j and j ≥
1. Using trigonometric computations, we show that theunion ∪ j =1 H j is dense in C k ( T d , R ) for any k ≥ H is asaturating space for the NLS equation, see Definition 2.1). This completes theproof of Theorem A.Theorem B is derived from Theorem A by noticing that the eigenstate φ l canbe approximated in L by functions of the form e iθ φ m and that the eigenstatesare constant solutions of Eq. (0.1) corresponding to some control. This allowsto appropriately adjust the controllability time and choose it the same for anyinitial condition and target.As an application of Theorem A, we study the large time behavior of thetrajectories of the random NLS equation. We show that if a random process per-turbes the same Fourier modes as in the above controllability results, then theenergy is almost surely transferred to higher modes resulting in the unbounded-ness of the trajectories in regular Sobolev spaces. More precisely, we replace thecontrol u by an R q -valued random process η of the form η ( t ) = + ∞ X k =1 I [ k − ,k ) ( t ) η k ( t − k + 1) , (0.6) This follows immediately from the assumptions that V = 0 and ∈ H . I [ k − ,k ) is the indicator function of the interval [ k − , k ) and { η k } areindependent identically distributed random variables in L ([0 , R q ) with non-degenerate law (see Condition (H )). The solution ψ of the problem (0.1), (0.2),(0.6) will be itself a random process in H s . We prove the following result. Theorem C.
For any s > s d and any non-zero ψ ∈ H s , the trajectory of (0.1) , (0.2) , (0.6) is almost surely unbounded in H s . The idea of constructing unbounded solutions by using random perturbationsis not new. Such results have been obtained by Bourgain [Bou99] and Erdoganet al. [EgKS03] for linear one-dimensional Schr¨odinger equations. They also pro-vided polynomial lower bounds for the growth. Unboundedness of trajectoriesfor multidimensional linear Schr¨odinger equations is obtained in [Ner09]. In thatpaper, the assumptions on the law of the random perturbation are rather gen-eral and no estimates for the growth are given; Theorem C is a generalisationof that result to the case of the NLS equations. There are also examples oflinear Schr¨odinger equations with various deterministic time-dependent poten-tials which admit unbounded trajectories: e.g., see the papers by Bambusi etal. [BGMR18], Delort [Del14], Haus and Maspero [HM20, Mas19], and the ref-erences therein.There are only few results in the case of unperturbed NLS equations. For cu-bic defocusing Schr¨odinger equations on bounded domains or manifolds, theexistence of unbounded trajectories in regular Sobolev spaces is a challengingopen problem (see Bourgain [Bou00]). In different situations, existence of tra-jectories with arbitrarily large finite growth has been shown by Kuksin [Kuk97],Colliander et al. [CKS + R × T d . In thecase of the cubic Szeg˝o equation on the circle, G´erard and Grellier [GG17] showthat the trajectories are generically unbounded in Sobolev spaces. Moreover,they exhibit the existence of a family of solutions with superpolynomial growth.Let us give a brief (and not entirely accurate) description of the main ideas ofthe proof of Theorem C. By starting from any initial point ψ ∈ H s , Theorem Aallows to increase the Sobolev norms by choosing appropriately the control.This, together with a compactness argument and the assumption that the lawof the process η is non-degenerate, leads to a uniform estimate of the form c M = sup ψ ∈ H s P ( sup t ∈ [0 , k ψ ( t ) k H s > M ) < M >
0. By combining the latter with the Markov property, we showthat P ( sup t ∈ [0 ,n ] k ψ ( t ) k H s > M ) ≤ c nM for any ψ ∈ H s . Then, the Borel–Cantelli lemma implies that the norm of anytrajectory becomes almost surely larger than M in some random time that isalmost surely finite. As M is arbitrary, this proves the required result.5he paper is organised as follows. In Section 1, we discuss the local well-posedness and some stability properties of the NLS equation. In Section 2,we formulate more general versions of Theorems A and B and give their proofs.Section 3 is devoted to the derivation of limit (0.5). In Section 4, we establish ageneral criterion for the validity of the saturation property. Finally, in Section 5,we prove Theorem C. Acknowledgement
The authors thank Armen Shirikyan for his valuable comments. The researchof AD was supported by the ANR grant ISDEEC ANR-16-CE40-0013. Theresearch of VN was supported by the ANR grant NONSTOPS ANR-17-CE40-0006-02.
Notation
In what follows, we use the following notation. h· , ·i is the Euclidian scalar product in R q and k · k is the corresponding norm.We write m ⊥ l when the vectors m, l ∈ R q are orthogonal and m l when theyare not. H s = H s ( T d ; C ) , s ≥ L p = L p ( T d ; C ) , p ≥ f : T d → C endowed with the norms k · k s and k · k L p . The space L is endowed with the scalar product h f, g i L = Z T d f ( x ) g ( x )d x.C s = C s ( T d ; C ), s ∈ N ∪ {∞} is the space of s -times continuously differentiablefunctions f : T d → C .Let X be a Banach space. We denote by B X ( a, r ) the closed ball of radius r > a ∈ X .We write J T instead of [0 , T ] and J instead of [0 , C ( J T ; X ) is the space of continuous functions f : J T → X with the norm k f k C ( J T ; X ) = max t ∈ J T k f ( t ) k X .L p ( J T ; X ) , ≤ p < ∞ is the space of Borel-measurable functions f : J T → X with k f k L p ( J T ; X ) = Z T k f ( t ) k pX d t ! /p < ∞ . ⌈ x ⌉ is the least integer greater than or equal to x ∈ R . s d is the least integer strictly greater than d/ is the function identically equal to 1 on T d .6 Preliminaries
In this section, we consider the NLS equation (0.1), where u is a deterministic R q -valued function and V : T d → R and Q : T d → R q are arbitrary smoothfunctions. In what follows, we shall always assume that the parameters d ≥ p ≥
1, and κ ∈ R are arbitrary. Here we formulate two propositions that willbe used in the proofs of our main results. The first one gathers some well-knownfacts about the local well-posedness and stability of the NLS equation in regularSobolev spaces. Proposition 1.1.
For any s > d/ , ˆ ψ ∈ H s , and ˆ u ∈ L loc ( R + ; R q ) , there isa maximal time T = T ( ˆ ψ , ˆ u ) > and a unique solution ˆ ψ of the problem (0.1) , (0.2) with ( ψ , u ) = ( ˆ ψ , ˆ u ) whose restriction to the interval J T belongsto C ( J T ; H s ) for any T < T . If T < ∞ , then k ˆ ψ ( t ) k s → + ∞ as t → T − .Furthermore, for any T < T , there are constants δ = δ ( T, Λ) > and C = C ( T, Λ) > , where Λ = k ˆ ψ k C ( J T ; H s ) + k ˆ u k L ( J T ; R q ) , such that the following two properties hold.(i) For any ψ ∈ H s and u ∈ L ( J T ; R q ) satisfying k ψ − ˆ ψ k s + k u − ˆ u k L ( J T ; R q ) < δ, (1.1) the problem (0.1) , (0.2) has a unique solution ψ ∈ C ( J T ; H s ) . (ii) Let R be the resolving operator for Eq. (0.1) , i.e., the mapping taking acouple ( ψ , u ) satisfying (1.1) to the solution ψ . Then kR ( ψ , u ) − R ( ˆ ψ , ˆ u ) k C ( J T ; H s ) ≤ C (cid:16) k ψ − ˆ ψ k s + k u − ˆ u k L ( J T ; R q ) (cid:17) . The proof of this proposition is rather standard, so we omit it (e.g., seeSection 3.3 in [Tao06] or Section 4.10 in [Caz03] for similar results). Let S bethe unit sphere in L . As the functions V, Q , and u are real-valued, the solution ψ belongs to S throughout its lifespan, provided that ψ ∈ S ∩ H s .Before formulating the second proposition, let us introduce some notation.For any ψ ∈ H s and T >
0, let Θ( ψ , T ) be the set of functions u ∈ L ( J T ; R q )such that the problem (0.1), (0.2) has a solution in C ( J T ; H s ). By the previousproposition, the set Θ( ψ , T ) is open in L ( J T ; R q ). For any ϕ ∈ C ( T d ; R ), let B ( ϕ )( x ) = d X j =1 (cid:0) ∂ x j ϕ ( x ) (cid:1) . (1.2)We have the following asymptotic property in small time.7 roposition 1.2. For any s ≥ s d , ψ ∈ H s , u ∈ R q , and ϕ ∈ C r ( T d ; R ) ,where r = ⌈ s ⌉ + 2 , there is a constant δ > such that δ − u ∈ Θ( e iδ − / ϕ ψ , δ ) for any δ ∈ (0 , δ ) and the following limit holds e − iδ − / ϕ R δ ( e iδ − / ϕ ψ , δ − u ) → e − i ( B ( ϕ )+ h u,Q i ) ψ in H s as δ → + , (1.3) where R δ is the restriction of the solution at time t = δ . The proof of this proposition is postponed to Section 4. Limit (1.3) is amultiplicative version of a limit established in Proposition 2 in [Ner21] in thecase of parabolic PDEs with additive controls.
In what follows, we assume that s ≥ s d and denote r = ⌈ s ⌉ + 2 as in Propo-sition 1.2. We start this section with a definition of a saturation property in-spired by the papers [AS06, Shi06]. Let H be a finite-dimensional subspaceof C r ( T d ; R ), and let F ( H ) be the largest subspace of C r ( T d ; R ) whose elementscan be represented in the form θ − n X j =1 B ( θ j )for some integer n ≥ θ j ∈ H , j = 0 , . . . , n , where B isgiven by (1.2). As B is quadratic, F ( H ) is well-defined and finite-dimensional.Let us define a non-decreasing sequence {H j } of finite-dimensional subspacesby H = H and H j = F ( H j − ), j ≥
1, and denote H ∞ = + ∞ [ j =1 H j . (2.1) Definition 2.1.
A finite-dimensional subspace
H ⊂ C r ( T d ; R ) is said to be satu-rating if H ∞ is dense in C r ( T d ; R ).We assume that the following condition is satisfied. (H ) The field Q = ( Q , . . . , Q q ) is saturating, i.e., the subspace H = span { Q j : j = 1 , . . . , q } is saturating in the sense of Definition 2.1In this section, we prove the following result. As we will see below, it impliesTheorems A and B formulated in the Introduction. For any vector u ∈ R q , with a slight abuse of notation, we denote by the same letter theconstant function equal to u . heorem 2.2. Assume that Condition (H ) is satisfied. Then for any ε > , κ > , ψ ∈ H s , and θ ∈ C r ( T d ; R ) , there is a time T ∈ (0 , κ ) and a control u ∈ Θ( ψ , T ) such that kR T ( ψ , u ) − e iθ ψ k s < ε. Proof.
By using an induction argument in N , we show that the approximate con-trollability property in this theorem is true for any θ ∈ H N and N ≥
0. Com-bined with the saturation hypothesis, this will lead to approximate controllabil-ity with any θ ∈ C r ( T d ; R ). Step 1. Case N = 0 . Let us show that, for any ε > κ > ψ ∈ H s ,and θ ∈ H , there is a time T ∈ (0 , κ ) and a control u ∈ Θ( ψ , T ) such that kR T ( ψ , u ) − e iθ ψ k s < ε. (2.2)By applying Proposition 1.2 with ϕ = 0 and u ∈ R q such that θ = −h u, Q i ,we obtain R δ ( ψ , δ − u ) → e iθ ψ in H s as δ → + . This implies (2.2) with sufficiently small time T = δ and control δ − u . Step 2. Case N ≥ . We assume that the result is true for any θ ∈ H N − .Let ˜ θ ∈ H N be of the form ˜ θ = θ − n X j =1 B ( θ j ) , where n ≥ θ j ∈ H N − , j = 0 , . . . , n . By applying Proposition 1.2 with ϕ = θ and u = 0, we get e − iδ − / θ R δ ( e iδ − / θ ψ , → e − i B ( θ ) ψ in H s as δ → + . The induction hypothesis, the assumption that θ ∈ H N − , and Proposition 1.1imply that, for any ε > κ >
0, there is a time T ∈ (0 , κ ) and a con-trol u ∈ Θ( ψ , T ) such that kR T ( ψ , u ) − e − i B ( θ ) ψ k s < ε. By iterating this argument with θ j ∈ H N − , j = 0 , . . . , n , we obtain that forany ε > κ >
0, there is T n ∈ (0 , κ ) and u n ∈ Θ( ψ , T n ) such that kR T n ( ψ , u n ) − e i ( θ − P nj =1 B ( θ j ) ) ψ k s = kR T n ( ψ , u n ) − e i ˜ θ ψ k s < ε. As ˜ θ ∈ H N is arbitrary, this proves the required property for N . Step 3. Conclusion.
Finally, let θ ∈ C r ( T d ; R ) be arbitrary. By the satu-ration hypothesis, H ∞ is dense in C r ( T d ; R ). Hence, we can find N ≥ θ ∈ H N such that k e iθ ψ − e i ˜ θ ψ k s < ε. Applying the controllability property proved in the previous steps for ˜ θ ∈ H N ,we complete the proof. 9s a consequence of this result, we have the following two theorems. Theorem 2.3.
Under the conditions of Theorem 2.2, for any
M > , κ > ,and non-zero ψ ∈ H s , there is a time T ∈ (0 , κ ) and a control u ∈ Θ( ψ , T ) such that kR T ( ψ , u ) k s > M. Proof.
It suffices to apply Theorem 2.2 by choosing θ ∈ C r ( T d ; R ) such that k e iθ ψ k s > M. To find such θ , we take any θ ∈ C r ( T d ; R ) verifying k e iθ ψ k = 0, put θ = λθ with sufficiently large λ >
0, and use the inequality k · k ≤ k · k s . Theorem 2.4.
Assume that the conditions of Theorem 2.2 are satisfied and ∈ span { Q j : j = 1 , . . . , q } and V = 0 . (2.3) Then, for any ε > , l, m ∈ Z d , θ ∈ C r ( T d ; R ) , and T > , there is a control u ∈ Θ( φ l , T ) such that kR T ( φ l , u ) − e iθ φ m k L < ε. Proof.
Let us take any θ ∈ C r ( T d ; R ). Applying Theorem 2.2, we find atime T ∈ (0 , T ) and a control u ∈ Θ( φ l , T ) such that kR T ( φ l , u ) − e iθ φ l k s < ε . Choosing θ ∈ C r ( T d ; R ) such that k e iθ φ l − e iθ φ m k L < ε , we arrive at kR T ( φ l , u ) − e iθ φ m k L < ε. Now, notice that φ l is a stationary solution of Eq. (0.1) corresponding a control u ∈ L loc ( R + ; R q ) satisfying the relation h u ( t ) , Q ( x ) i = −| l | − κ (2 π ) − dp for any t ≥ x ∈ T d . Such a choice of u is possible in view of assumption (2.3). Thus, u ∈ Θ( φ l , t )and φ l = R t ( φ l , u ) for any t ≥
0. Setting u ( t ) = ( u ( t ) for t ∈ [0 , T − T ] ,u ( t − T + T ) for t ∈ ( T − T , T ] , we complete the proof of the theorem.10et us close this section with an example of a saturating subspace. Let I ⊂ Z d ∗ be a finite set and let H = H ( I ) = span { , sin h x, k i , cos h x, k i : k ∈ I} . (2.4)Recall that I is a generator if any vector of Z d is a linear combination of vectorsof I with integer coefficients. The following proposition is proved in Section 4. Proposition 2.5.
The subspace H ( I ) is saturating in the sense of Definition 2.1 ifand only if I is a generator and for any l, m ∈ I , there are vectors { n j } kj =1 ⊂ I such that l n , n j n j +1 , j = 1 , . . . , k − , and n k m . Clearly, the set
K ⊂ Z d ∗ defined by (0.3) satisfies the condition in this proposi-tion. Therefore, the subspace H ( K ) is saturating, and Theorems A and B followfrom Theorems 2.2 and 2.4, respectively. We start by proving the result in the case when s > d/ r = s +2.Let us fix any R > ψ ∈ H s , ϕ ∈ C r ( T d ; R ), and u ∈ R q aresuch that k ψ k s + k ϕ k C r + k u k R q ≤ R. (3.1)For any δ >
0, we denote φ ( t ) = e − iδ − / ϕ R t ( e iδ − / ϕ ψ , δ − u ). According toProposition 1.1, φ ( t ) exists up to some maximal time T δ = T ( e iδ − / ϕ ψ , δ − u ),and k e iδ − / ϕ φ ( t ) k s → + ∞ as t → T δ − , if T δ < ∞ .We need to show that( a ) there is a constant δ > T δ > δ for any δ < δ ;( b ) the following limit holds φ ( δ ) → e − i ( B ( ϕ )+ h u,Q i ) ψ in H s as δ → + . To prove these properties, we introduce the functions w ( t ) = e − i ( B ( ϕ )+ h u,Q i ) t ψ δ , (3.2) v ( t ) = φ ( δt ) − w ( t ) , where ψ δ ∈ H r is such that k ψ δ k s ≤ C for δ ≤ , (3.3) k ψ δ k r ≤ Cδ − / for δ ≤ , (3.4) k ψ − ψ δ k s → δ → + . In what follows, C denotes positive constants which may change from line to line. Theseconstants depend on the parameters R, V, Q, κ, p, d, s , but not on δ . ψ δ by using the heat semigroup: ψ δ = e δ / ∆ ψ .In view of (3.1)-(3.4), we have k w ( t ) k s ≤ C, t ≥ , (3.5) k w ( t ) k r ≤ Cδ − / , t ≥ . (3.6)Furthermore, v ( t ) is well-defined for t < δ − T δ and satisfies the equation i∂ t v = − δ ∆( v + w ) + δV ( v + w ) + δκ | v + w | p ( v + w ) − iδ D ( v + w, ϕ ) + B ( ϕ ) v + h u, Q i v, (3.7)and the initial condition v (0) = ψ − ψ δ , (3.8)where D ( v + w, ϕ ) = ( v + w )∆ ϕ + 2 d X j =1 ∂ x j ( v + w ) ∂ x j ϕ. Let α = ( α , . . . , α d ) ∈ N d be such that | α | = | α | + . . . + | α d | ≤ s . We take thescalar product of Eq. (3.7) with ∂ α v in L and integrating by parts, we obtain ∂ t k ∂ α v k L ≤ C (cid:16) δ |h ∆ w, ∂ α v i L | + δ |h V ( v + w ) , ∂ α v i L | + δ |h| v + w | p ( v + w ) , ∂ α v i L | + δ / |h D ( v + w, ϕ ) , ∂ α v i L | + |h B ( ϕ ) v + h u, Q i v, ∂ α v i L | (cid:17) = X j =1 I j . (3.9)We estimate the terms I , I , I , and I by integrating by parts and by using(3.1), (3.5), and (3.6): | I | ≤ Cδ k w k r k v k s ≤ Cδ / k v k s , | I | ≤ Cδ k v + w k s k v k s ≤ Cδ k v k s + Cδ k v k s , | I | ≤ Cδ k v + w k p +1 s k v k s ≤ Cδ k v k p +1) s + Cδ k v k s , | I | ≤ C k v k s . We estimate I as follows | I | ≤ Cδ / k v k s + Cδ / k w k s +1 k v k s ≤ Cδ / k v k s + Cδ / k v k s , In the last relation, we used again the integration by parts, the identities (3.1), (3.5)and (3.6), and the equality h ∂ x j ϕ ∂ x j ∂ α v, ∂ α v i L = 12 h ∂ x j ϕ, ∂ x j | ∂ α v | i L = −h ∂ x j ϕ, | ∂ α v | i L . α ∈ N d , | α | ≤ s , combining the resultinginequality with the estimates for I j and the Young inequality, and recallingthat δ ≤
1, we obtain ∂ t k v k s ≤ Cδ / + C (1 + δ / ) k v k s + Cδ k v k p +1) s , t ≤ δ − T δ . This inequality, together with (3.8) and the Gronwall inequality, implies that k v ( t ) k s ≤ e C (1+ δ / ) t (cid:18) Cδ / t + k ψ − ψ δ k s + Cδ Z t k v ( y ) k p +1) s d y (cid:19) (3.10)for t ≤ δ − T δ . Let us take δ ∈ (0 ,
1) so small that, for δ < δ , k ψ − ψ δ k s < , (3.11) e C (1+ δ / ) (cid:16) Cδ / + k ψ − ψ δ k s (cid:17) < , (3.12)and denote τ δ = sup (cid:8) t < δ − T δ : k v ( t ) k s < (cid:9) . From (3.8) and (3.11) it follows that τ δ > δ < δ . Let us show that τ δ > δ < (cid:0) Ce C (cid:1) − . (3.13)Assume, by contradiction, that τ δ ≤
1. Let t = τ δ in (3.10). By using (3.12)and (3.13), we obtain1 = k v ( τ δ ) k s <
12 + 12 Z τ δ k v ( y ) k p +1) s d y ≤ . This contradiction shows that τ δ > δ < δ , hence also 1 < δ − T δ . Thus,property (a) is proved. Taking t = 1 in (3.10), we arrive at k v (1) k s ≤ e C (1+ δ / ) (cid:16) Cδ / + k ψ − ψ δ k s + Cδ (cid:17) → δ → + . This implies (b) and completes the proof in the case when s > d/ s ≥ s d isan arbitrary number, we use inequality (3.10) for integer values of s and aninterpolation argument. Proof of Proposition 2.5.
The proof is divided into four steps.
Step 1.
First, let us assume that
I ⊂ Z d ∗ is an arbitrary finite set, H ( I ) = H ( I ) is the subspace defied by (2.4), H j ( I ) = F ( H j − ( I )) for j ≥
1, and H ∞ ( I )is defined by (2.1). 13 tep 1.1. Let us show that, ifcos h x, m i , sin h x, m i ∈ H ∞ ( I ) for some m ∈ Z d ∗ , then B (cos h x, m i ) , B (sin h x, m i ) ∈ H ∞ ( I ) . Indeed, assume thatcos h x, m i , sin h x, m i ∈ H N ( I ) for some N ≥ . (4.1)The equalitiescos h x, m i = 1 − | m | B (cos h x, m i ) = 2 | m | B (sin h x, m i ) − , (4.2)the assumptions ∈ H ( I ) and (4.1), and the definition of F imply thatcos h x, m i ∈ H N +1 ( I ) . (4.3)As a consequence of (4.2) and (4.3), we have B (cos h x, m i ) = | m | − cos h x, m i ) ∈ H N +1 ( I ) , B (sin h x, m i ) = | m | h x, m i ) ∈ H N +1 ( I ) , which imply the required result. Step 1.2.
Let us show that, ifcos h x, m i , sin h x, m i , cos h x, l i , sin h x, l i ∈ H ∞ ( I )for some m, l ∈ Z d ∗ such that m l , thencos h x, m + l i , sin h x, m + l i ∈ H ∞ ( I ) . Indeed, this follows immediately from the equalitiescos h x, m + l i = ± h m, l i (cid:16) B (sin h x, m i ± sin h x, l i ) + B (cos h x, m i ∓ cos h x, l i ) − B (sin h x, m i ) − B (sin h x, l i ) − B (cos h x, m i ) − B (cos h x, l i ) (cid:17) , sin h x, m + l i = ± h m, l i (cid:16) B (sin h x, m i ∓ cos h x, l i ) + B (cos h x, m i ∓ sin h x, l i ) − B (sin h x, m i ) − B (sin h x, l i ) − B (cos h x, m i ) − B (cos h x, l i ) (cid:17) and the result of step 1.1. Step 2.
Now, let us suppose that
I ⊂ Z d ∗ is a finite set such that, forany l, m ∈ I , there are vectors { n j } kj =1 ⊂ I satisfying l n , n j n j +1 ,14 = 1 , . . . , k − , and n k m . Let N = card( I ) and I = { m , . . . , m N } . Arguingby induction on N , we show in this step thatcos h x, a m + . . . + a N m N i , sin h x, a m + . . . + a N m N i ∈ H ∞ ( I ) (4.4)for any a , . . . , a N ∈ Z . Step 2.1.
Let I = { m , m } ⊂ Z d ∗ with m m . By the result of step 1.2,we havecos h x, a m i , sin h x, a m i , cos h x, a m i , sin h x, a m i ∈ H ∞ ( I )for any a , a ∈ Z . Again, in view of step 1.2, this implies thatcos h x, a m + a m i , sin h x, a m + a m i ∈ H ∞ ( I )for any a , a ∈ Z . Step 2.2.
Assume that the required property is true if the cardinality ofthe set I is less or equal to N −
1. Let
I ⊂ Z d ∗ be such that N = card( I )and I = { m , . . . , m N } . Without loss of generality, we can assume m N − m N and the set { m , . . . , m N − } satisfies the condition formulated in the beginningof step 2. Let us take any a , . . . , a N ∈ Z and k ≥ a m + . . . + a N m N = ( a m + . . . + a N − m N − + ( a N − − k ) m N − )+ ( km N − + a N m N ) . (4.5)Then, h a m + . . . + ( a N − − k ) m N − , km N − + a N m N i = ( a N − − k ) k k m N − k + O ( k ) as k → + ∞ .As m N − = 0, for sufficiently large k ≥
1, we have a m + . . . + a N − m N − + ( a N − − k ) m N − km N − + a N m N . (4.6)Relation (4.4) is proved by combining (4.5) and (4.6), the induction hypothesis,and the assumption that m N − m N . Step 3.
We conclude from step 2 that, if
I ⊂ Z d ∗ is a set satisfying theconditions of Proposition 2.5, thencos h x, m i , sin h x, m i ∈ H ∞ ( I ) for any m ∈ Z d ∗ .This implies that H ∞ ( I ) is dense in C r ( T d ; R ) for any r ≥
0, hence H ( I ) issaturating. Step 4.
Finally, let us assume that the conditions of the proposition are notsatisfied for
I ⊂ Z d ∗ . We distinguish between two cases. Step 4.1. If I is not a generator, we can find a vector n ∈ Z d ∗ which does notbelong to the set ˜ I of linear combinations of vectors of I with integer coefficients.It is easy to see that H ∞ ( I ) ⊂ span { sin h x, m i , cos h x, m i : m ∈ ˜ I} . h x, n i and cos h x, n i are orthogonal to the vector space H ∞ ( I )in the Sobolev spaces H j ( T d ; R ) for any j ≥
0. We conclude that H ∞ ( I ) is notdense in C r ( T d ; R ), thus the subspace H ( I ) is not saturating. Step 4.2. If I does not satisfy the second condition in the theorem, then itis of the form I = ∪ kj =1 { m j , . . . , m jn j } , where k ≥ m j i ⊥ m j i for any integers 1 ≤ j < j ≤ k, ≤ i ≤ n j , and 1 ≤ i ≤ n j . By using the arguments of the steps 1 and 2, it is easy toverify that the function cos h x, m j + m j i is orthogonal to H ∞ ( I ) in H j ( T d ; R )for any j ≥
0. Thus, the space H ∞ ( I ) is not dense in C r ( T d ; R ). Let us consider the NLS equation i∂ t ψ = − ∆ ψ + V ( x ) ψ + κ | ψ | p ψ + h η ( t ) , Q ( x ) i ψ, (5.1) ψ (0) = ψ (5.2)with potential V and parameters d, p, κ as in the previous sections. We as-sume that the field Q satisfies Condition (H ) and η is a random process of theform (0.6) with the following condition satisfied for the random variables { η k } .We denote J = [0 ,
1] and E = L ( J ; R q ). (H ) { η k } are independent random variables in E with common law ℓ such that Z E k y k E ℓ (d y ) < ∞ and supp ℓ = E . For example, this condition is satisfied if the random variables { η k } are ofthe form η k ( t ) = + ∞ X j =1 b j ξ jk e j ( t ) , t ∈ J, where { b j } are non-zero real numbers verifying P + ∞ j =1 b j < ∞ , { e j } is an or-thonormal basis in E , and { ξ jk } are independent real-valued random variableswhose law has a continuous density ρ j with respect to the Lebesgue measuresuch that Z + ∞−∞ x ρ j ( x ) d x = 1 , ρ j ( x ) > x ∈ R and j ≥ H s forany s > d/ T = T ( ψ , η ) >
0. Let P ψ be the probability measure corresponding to the trajectories issued from ψ (e.g., see Section 1.3.1 in [KS12]). Recall that S is the unit sphere in L .16 heorem 5.1. Under the Conditions (H ) and (H ) , for any s > s d and any ψ ∈ H s ∩ S , we have P ψ (cid:26) lim sup t →T − k ψ ( t ) k s = + ∞ (cid:27) = 1 . (5.3)By the blow-up alternative, equality (5.3) gives new information in thecase T ( ψ , η ) = + ∞ . Proof. Step 1. Reduction.
Together with Eq. (5.1), let us consider the followingtruncated NLS equation: i∂ t ψ = − ∆ ψ + V ( x ) ψ + κχ R ( k ψ k s ) | ψ | p ψ + h η ( t ) , Q ( x ) i ψ, (5.4)where R > χ R ∈ C ∞ ( R ) is such that 0 ≤ χ R ( x ) ≤ x ∈ R and χ R ( x ) = 1 for | x | ≤ R . Let F k , k ≥ σ -algebra generated by thefamily { η j } kj =1 . The problem (5.4), (5.2) is globally well-posed. The followingproposition is proved at the end of this section. Proposition 5.2.
For any ψ ∈ H s and R > , the problem (5.4) , (5.2) has aunique solution ψ R ∈ C ( R + ; H s ) . Moreover, the family (cid:8) ψ R ( k + · ) : J → H s (cid:9) k ≥ defines an C ( J ; H s ) -valued Markov process with respect to the filtration F k +1 . Let us fix any 0 < M < R and consider the stopping time τ M,R = 1 + min (cid:8) k ≥ k ψ R ( k + · ) k C ( J ; H s ) > M (cid:9) , ψ ∈ H s , where the minimum over an empty set is equal to + ∞ . Assume we have shown that P ψ { τ M,R < ∞} = 1 , ψ ∈ H s ∩ S . (5.5)Since R > M , this implies that P ψ { τ M < ∞} = 1 , ψ ∈ H s ∩ S , (5.6)where τ M = min ( k ≥ t ∈ J, k + t< T k ψ ( k + t ) k s > M ) and again the minimum over an empty set is + ∞ . As M >
Step 2. Proof of (5.5) . Assume that there is an integer l ≥ c = c ( M, R ) = sup ψ ∈ H s ∩S P ψ { τ M,R > l } < . (5.7)Combining this with the Markov property, we obtain P ψ { τ M,R > nl } = E ψ (cid:0) I { τ M,R > ( n − l } P φ { τ M,R > l }| φ = ψ R (( n − l ) (cid:1) ≤ c P ψ { τ M,R > ( n − l } , E ψ is the expectation corresponding to P ψ . Iterating this inequality,we get P ψ { τ M,R > nl } ≤ c n . This, together with the Borel–Cantelli lemma, implies (5.5).
Step 3. Proof of (5.7) . By Theorem 2.3, for any ψ ∈ H s d ∩ S , there is acontrol u ∈ E such that sup t ∈ J, t< T k ψ ( t ) k s d > M. (5.8)On the other hand, Condition (H ) implies that P {k u − η k E < δ } > δ >
0. Combining this with Proposition 1.1 and inequality (5.8), we seethat there is a number δ > ψ ′ ∈ B Hsd ( ψ ,δ ) ∩S P ψ ′ (cid:26) sup t ∈ J, t< T ′ k ψ ( t ) k s d > M (cid:27) > , where T ′ = T ( ψ ′ , η ). As R > M , we also haveinf ψ ′ ∈ B Hsd ( ψ ,δ ) ∩S P ψ ′ (cid:26) sup t ∈ J k ψ R ( t ) k s d > M (cid:27) > . Since the ball B H s (0 , M ) is compact in H s d and k · k s d ≤ k · k s , we derive thatinf ψ ∈ B Hs (0 ,M ) ∩S P ψ (cid:26) sup t ∈ J k ψ R ( t ) k s > M (cid:27) > . The latter and the fact that P ψ { τ M,R = 1 } = 1 if k ψ k s > M imply (5.7) with l = 1 and c = 1 − inf ψ ∈ B Hs (0 ,M ) ∩S P ψ (cid:26) sup t ∈ J k ψ R ( t ) k s > M (cid:27) . This completes the proof of the theorem.
Proof of Proposition 5.2.
The local well-posedness of (5.4), (5.2) is proved bystandard arguments. As the H s -norm of the solution remains bounded on anybounded interval, it can be extended to any t >
0. For any k ≥
1, let usdenote by ψ k ( ψ , η , . . . , η k ) the restriction of the solution of (5.4), (5.2) to theinterval [ k − , k ] (we skip the dependence on R ). Then { ψ k ( ψ , η , . . . , η k ) } k ≥ is a Markov process in C ( J, H s ). Indeed, we have ψ k + n ( ψ , η , . . . , η k + n ) = ψ n ( ψ k ( ψ , η , . . . , η k ) , η k +1 , . . . , η k + n ) . { η j } j ≥ k +1 is independent of F k and ψ k is F k -measurable, the following equal-ity holds E ( f ( ψ k + n ( ψ , η , . . . , η k + n )) |F k ) = E f ( ψ n ( ψ, η k +1 , . . . , η k + n )) (5.9)for any bounded measurable function f : C ( J, H s ) → R . Here, ψ is the value attime-1 of ψ k ( ψ , η , . . . , η k ). The vectors ( η , . . . , η n ) and ( η k +1 , . . . , η k + n ) havethe same law, so E f ( ψ n ( ψ, η k +1 , . . . , η k + n )) = E f ( ψ n ( ψ, η , . . . , η n )) . Combining this and (5.9), we arrive at the required result.
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