Stochastic nonlinear Schrödinger equations: no blow-up in the non-conservative case
aa r X i v : . [ m a t h . P R ] S e p Stochastic nonlinear Schr¨odinger equations:no blow-up in the non-conservative case
Viorel Barbu , Michael R¨ockner , Deng Zhang Abstract.
This paper is devoted to the study of noise effects onblow-up solutions to stochastic nonlinear Schr¨odinger equations. It isa continuation of our recent work [2], where the (local) well-posednessis established in H , also in the non-conservative critical case. Here weprove that in the non-conservative focusing mass-(super)critical case,by adding a large multiplicative Gaussian noise, with high probabilityone can prevent the blow-up on any given bounded time interval [0 , T ],0 < T < ∞ . Moreover, in the case of spatially independent noise, theexplosion even can be prevented with high probability on the wholetime interval [0 , ∞ ). The noise effects obtained here are completelydifferent from those in the conservative case studied in [5]. Keywords : (stochastic) nonlinear Schr¨odinger equation, Wienerprocess, noise effect, blow-up. Octav Mayer Institute of Mathematics (Romanian Academy) and Al.I. Cuza Univer-sity and, 700506, Ia¸si, Romania. This work was supported by the DFG through CRC701. Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, D-33501 Bielefeld, Germany. Thisresearch was supported by the DFG through CRC 701. Department of Mathematics, Shanghai Jiao Tong University, 200240 Shanghai, China. Introduction and main results.
We consider the stochastic nonlinear Schr¨odinger equation with linear mul-tiplicative noise, idX ( t, ξ ) = ∆ X ( t, ξ ) dt + λ | X ( t, ξ ) | α − X ( t, ξ ) dt − iµ ( ξ ) X ( t, ξ ) dt + iX ( t, ξ ) dW ( t, ξ ) , t ∈ (0 , T ) , ξ ∈ R d , (1.1) X (0) = x ∈ H . Here, the exponents of particular interest lie in the focusing mass-(super)criticalrange, namely, λ = 1 , α ∈ [1 + 4 d , d − + ) . (1.2) W is the colored Wiener process W ( t, ξ ) = N X j =1 µ j e j ( ξ ) β j ( t ) , t ≥ , ξ ∈ R d , (1.3)where N < ∞ , µ j ∈ C , e j are real-valued functions, and β j ( t ) are independentreal Brownian motions on a probability space (Ω , F , P ) with natural filtation( F t ) t ≥ , 1 ≤ j ≤ N . Moreover, as required by the physical context (see [3]and [4]), µ is of the form µ = N X j =1 | µ j | e j . (1.4)Hence | X ( t ) | is a martingale, which allows to define the so-called ”physicalprobability law”. In particular, in the conservative case (i.e. Reµ j = 0,1 ≤ j ≤ N ), the last two terms in (1.1) coincide with the Stratonovitchintegration. We also refer to [1] for discussions on the physical background. Definition 1.1
A solution X to (1.1) on [0 , τ ] , where τ is an ( F t ) -stoppingtime, is an H -valued continuous ( F t ) -adapted process, such that | X | α − X ∈ L (0 , τ ; H − ) , P − a.s , and it satisfies P − a.sX ( t ) = x − Z t ( i ∆ X ( s ) + µX ( s ) + λi | X ( s ) | α − X ( s )) ds + Z t X ( s ) dW ( s ) , t ∈ [0 , τ ] , (1.5) as an equation in H − . µ j = 0, 1 ≤ j ≤ N )and to [5] and [8] for the stochastic conservative case (i.e. Reµ j = 0, 1 ≤ j ≤ N ).The main interest of this article is to study the noise effects on blow-upin the focusing mass-(super)critical case. Our motivations mainly come fromtwo aspects. On the one hand, the blow-up phenomenon in the determin-istic case is extensively studied in the literature, and it is well known thatthere exist blow-up solutions in the focusing mass-(super)critical case (1.2),especially for initial data with negative Hamiltonian (cf. e.g. [9], [16]). Onthe other hand, when there is noise in the system, it is of great interest toinvestigate the noise effects on the formation of singularities. For example,in the conservative case, it is proved in [6] in the supercritical case that noisecan accelerate blow-up with positive probability. But in the critical case nu-merical results suggest that noise has the effect to delay explosion (cf. [7],[10] and [11])Here, we focus on the noise effects on blow-up, but in the non-conservativecase, i.e., ∃ j : 1 ≤ j ≤ N, such that Reµ j = 0 (1.6)(Without loss of generality, we assume that Reµ = 0). Surprisingly, thenoise effects here are completely different from those in the conservativecases. We will prove that, in the non-conservative case by adding a largenoise, with high probability one can prevent blow-up on any given boundedtime interval [0 , T ], 0 < T < ∞ . Moreover, when the noise is spatially inde-pendent, the explosion even can be prevented with high probability on thewhole time interval [0 , ∞ ).To state our resutls precisely, we assume for the spatial functions in thenoise that(H) e j = f j + c j , 1 ≤ j ≤ N , where c j are real constants and f j are real-valued functions, such that f j ∈ C ∞ b andlim | ξ |→∞ ζ ( ξ ) X ≤| γ |≤ | ∂ γ f j ( ξ ) | = 0 , γ is a multi-index and ζ = (cid:26) | ξ | , if d = 2;(1 + | ξ | )(ln(1 + | ξ | )) , if d = 2.(In Section 3 we will take c large enough such that c > | f | ∞ . Hence,without loss of generality, we assume that f is positive.)The main result is then as follows: Theorem 1.2
Consider (1.1) in the non-conservative case (1.6) . Let λ and α satisfy (1.2) . Assume ( H ) with f j , ≤ j ≤ N , and c k , ≤ k ≤ N beingfixed. Then for any x ∈ H and < T < ∞ , P ( X ( t ) does not blow up on [0 , T ]) → , as c → ∞ . (where we recall that by (1.6) we have Reµ = 0 .)Furthermore, if f j , ≤ j ≤ N , are also constants, then for any x ∈ H , P ( X ( t ) does not blow up on [0 , ∞ )) → , as c → ∞ . Remark 1.3
Theorem 1.2 can be viewed as a complement to [6]. It wasproved there that in the conservative supercritical case, i.e.,
Reµ j = 0 , ≤ j ≤ N , α ∈ (1 + d , ∞ ) if d = 1 , and α ∈ ( , if d = 3 , the non-degenerate multiplicative noise can accelerate blow-up with positive probability(see Theorem . in [6]). In contrast to [6], Theorem 1.2 reveals that in thenon-conservative supercritical and also critical cases specified in (1.2) with d ≥ , the large multiplicative noise has the effect to stabilize the system. Similar phenomena happen for the deterministic damped nonlinear Schr¨odingerequation, i∂ t u + ∆ u + | u | α − u + iau = 0 , a > . (1.7)Note that, this equation is analogous to (2.2) below in the special case wherethe noise W ( t ) is spatially independent and µ k ∈ R , 1 ≤ k ≤ N , i.e. i∂ t y − ∆ y − e ( α − ReW ( t ) | y | α − y + i b µy = 0 , b µ > . This similarity indeed indicates the dissipative effects produced by the mul-tiplicative noise in the non-conservative case.4he global well-posedness of (1.7) is proved in [18, Theorem 1] (see also[19, p.98]), provided a is large enough, and the proof is based on the decayestimate of e it ∆ (see [18, Lemma 4]).However, since the decay estimates do not necessarily hold for the gen-eral Schr¨odinger-type operator A ( t ) in (2.3), we employ here quite differentarguments based on the contraction mapping arguments as in [1, 2], involv-ing a second transformation (see (2.8) below) and the Strichartz estimatesestablished in [17]. The advantage of this proof is that it is also applicableto the case of spatially dependent noise.This article is structured as follows. In Section 2 we apply two transfor-mations to reduce the original stochastic equation (1.1) to a random equation(2.9) below, which reveals the dissipative effect produced by the noise in thenon-conservative case. Then the non-explosion results in Theorem 1.2 areestablished in Section 3. Furthermore, we also show that these results donot generally hold with probability 1. Finally, the Appendix contains Itˆo-formulas for the Hamiltonian, variance and momentum that are used in theproof. Following [1] and [2], we apply the rescaling transformation X = e W y (2.1)to (1.1) and obtain the random equation ∂y∂t ( t, ξ ) = A ( t ) y ( t, ξ ) − ie ( α − ReW ( t,ξ ) | y ( t, ξ ) | α − y ( t, ξ ) , (2.2) y (0) = x, where A ( t ) = − i (∆ + b ( t ) · ∇ + c ( t )) , (2.3) b ( t ) = 2 ∇ W ( t ) , (2.4) c ( t ) = d X j =1 ( ∂ j W ( t )) + ∆ W ( t ) − i b µ, (2.5)5nd b µ := N X j =1 ( | µ j | + µ j ) e j . (2.6)We stress that on a heuristic level (2.2) follows easily by Itˆo’s product rule.The rigorous proof is more involved. We refer to [1, Lemma 6 .
1] for the L -case and [20, Theorem 2 . .
3] for the H -case.Note that the real part of the damped term b µ is positive in the non-conservative case, namely, Re b µ = N X j =1 ( Reµ j ) e j ≥ ( Reµ ) c > , (2.7)but it vanishes in the conservative case, which indicates the different noiseeffects between the two cases.To explore this damped term, we apply to (2.2) a second transformation z ( t, ξ ) = e b µt y ( t, ξ ) , (2.8)and derive that ∂z ( t ) ∂t = b A ( t ) z ( t ) − ie − ( α − Re b µt − ReW ( t )) | z ( t ) | α − z ( t ) , (2.9) z (0) = x ∈ H , where b A ( t ) = − i (∆ + b b ( t ) · ∇ + b c ( t )) (2.10)with b b ( t ) = − t ∇ b µ + 2 ∇ W ( t ) , (2.11)and b c ( t ) = t N X j =1 ( ∂ j b µ ) − t ∆ b µ − t ∇ W ( t ) · ∇ b µ + " N X j =1 ( ∂ j W ( t )) + ∆ W ( t ) . (2.12)6he key fact here is that, an exponential decay term e − ( α − Re b µt appearsin (2.9), which weakens the nonlinearity and thus can be expected to preventblow-up, provided that µ is sufficiently large (or the noise is sufficiently largein some other appropriate sense). For this purpose, let us rewrite equation(2.9) in the mild form z ( t ) = V ( t, x + Z t ( − i ) V ( t, s ) (cid:2) h ( s ) | z ( s ) | α − z ( s ) (cid:3) ds, (2.13)where h ( s ) := e − ( α − Re b µs − ReW ( s )) (2.14)and V ( t, s ) is the evolution operator generated by the homogenous part of(2.2), namely, V ( t, s ) x = z ( t ), s ≤ t ≤ T , solves dz ( t ) dt = b A ( t ) z ( t ) , a.e t ∈ ( s, T ) , (2.15) z ( s ) = x ∈ H . (The existence and uniqueness of the evolution operator V ( t, s ) follow mainlyfrom [12, 13]. For more details, we refer to [1, 2].) Remark 2.1
The solutions to (2.9) are understood analogously to Definition1.1, and Assumption ( H ) is sufficient to establish the local existence anduniqueness of solutions for (2.9) , hence also for (1.1) , by the transformations (2.1) and (2.8) . Indeed, the proofs follow by similar arguments as in [2,Proposition . ] (see also [1, Lemma . ]), and one can remove the additionaldecay assumption lim | ξ |→ ζ ( ξ ) | e j ( ξ ) | = 0 in [2], due to the fact that b b, b c in (2.9) only involve the gradient of b µ and W ( t ) . This fact allows us later to take c very large to prevent blow-up. As in [2, Lemma 2 .
7] one can check from [17] and Assumption ( H ) thatStrichartz estimates hold for V ( t, s ), Lemma 2.2
Assume ( H ) . Then for any T > , u ∈ H and f ∈ L q ′ (0 , T ; W ,p ′ ) ,the solution of u ( t ) = V ( t, u + Z t V ( t, s ) f ( s ) ds, ≤ t ≤ T, (2.16)7 atisfies the estimates k u k L q (0 ,T ; L p ) ≤ C T ( | u | + k f k L q ′ (0 ,T ; L p ′ ) ) , (2.17) and k u k L q (0 ,T ; W ,p ) ≤ C T ( | u | H + k f k L q ′ (0 ,T ; W ,p ′ ) ) , (2.18) where ( p , q ) and ( p , q ) are Strichartz pairs, i.e., ( p i , q i ) ∈ [2 , ∞ ] × [2 , ∞ ] : 2 q i = d − dp i , if d = 2 , or ( p i , q i ) ∈ [2 , ∞ ) × (2 , ∞ ] : 2 q i = d − dp i , if d = 2 , Furthermore, the process C t , t ≥ , can be taken to be ( F t ) -progressivelymeasurable, increasing and continuous. Proof of Theorem 1.2. ( i ). For convenience, let us first consider the easiercase of spatially independent noise to illustrate the main idea.By the transformations (2.1) and (2.8), it is equivalent to prove the asser-tion for the random equation (2.9). Note that in this case b b = b c = 0, hence V ( t, s ) = e − i ( t − s )∆ and the Strichartz coefficient C t ≡ C is independent of t .Choose the Strichartz pair ( p, q ) = ( α + 1 , α +1) d ( α − ). Set Z τM = { u ∈ C (0 , τ ; L ) ∩ L q (0 , τ ; L p ) : k u k L ∞ (0 ,τ ; H ) + k u k L q (0 ,τ ; W ,p ) ≤ M } , (3.1)and define the integral operator G on Z τM by G ( u )( t ) = V ( t, x + Z t ( − i ) V ( t, s ) (cid:2) h ( s ) | u ( s ) | α − u ( s ) (cid:3) ds, u ∈ Z τM . (3.2)We claim that, for u ∈ Z τM , k G ( u ) k L ∞ (0 ,τ ; H ) + k G ( u ) k L q (0 ,τ ; W ,p ) ≤ C | x | H + 2 CD ( τ ) M α , (3.3)8here D ( t ) = αD α − k h k L v (0 ,t ) (3.4)with D the Sobolev coefficient such that k u k L p ≤ D | u | H , v > v =1 − q > k G ( u ) k L ∞ (0 ,τ ; H ) + k G ( u ) k L q (0 ,τ ; W ,p ) ≤ C | x | H + 2 C k h | u | α − u k L q ′ (0 ,τ ; W ,p ′ ) . (3.5)Moreover, H¨older’s inequality and Sobolev’s imbedding theorem yield k h | u | α − u k L q ′ (0 ,τ ; L p ′ ) ≤| h | L v (0 ,τ ) k| u | α − u k L q (0 ,τ ; L p ′ ) ≤ D α − | h | L v (0 ,τ ) k u k α − L ∞ (0 ,τ ; H ) k u k L q (0 ,τ ; L p ) , (3.6)and k h ∇ ( | u | α − u ) k L q ′ (0 ,τ ; L p ′ ) ≤ α k h | u | α − |∇ u |k L q ′ (0 ,τ ; L p ′ ) ≤ αD α − | h | L v (0 ,τ ) k u k α − L ∞ (0 ,τ ; H ) k∇ u k L q (0 ,τ ; L p ) . (3.7)Hence, plugging (3.6) and (3.7) into (3.5) implies (3.3), as claimed.Similarly to (3.3), for u , u ∈ Z τM , k G ( u ) − G ( u ) k L ∞ (0 ,τ ; L ) + k G ( u ) − G ( u ) k L q (0 ,τ ; L p ) ≤ CD ( τ ) M α − k u − u k L q (0 ,τ ; L p ) . (3.8)Now, let M = 3 C | x | H , choose the ( F t )-stopping time τ = τ ( c ), τ := inf (cid:8) t > · α | x | α − H C α D ( t ) > (cid:9) . (3.9)Then, as in the proof of Proposition 2 . z of (2.9) on [0 , τ ].Next we show that P ( τ = ∞ ) →
1, as c → ∞ . As the definition of τ involves the term D ( t ), we shall use (3.4) to estimate k h k L v (0 , ∞ ) .9et φ k = µ k e k , 1 ≤ k ≤ N . By the scaling property of Brownian motion,i.e. P ◦ [ Reφ k β k ( · )] − = P ◦ [ β k (( Reφ k ) · )] − , for any c ≥ P ( k h k vL v (0 , ∞ ) ≥ c )= P Z ∞ N Y k =1 e − ( α − v [( Reφ k ) s − Reφ k β k ( s )] ds ≥ c ! = P Z ∞ N Y k =1 e − ( α − v [( Reφ k ) s − β k (( Reφ k ) s )] ds ≥ c ! . (3.10)Note that, by the law of the iterated logarithm of Brownian motion, C ∗ := Z ∞ e − ( α − v [ s − β ( s )] ds < ∞ , a.s, (3.11)and C := 1 ∨ max ≤ k ≤ N sup s ≥ e − ( α − v [( Reφ k ) s − β k (( Reφ k ) s )] < ∞ , a.s. (3.12)Then P -a.s., Z ∞ N Y k =1 e − ( α − v [( Reφ k ) s − β k (( Reφ k ) s )] ds ≤ C N Z ∞ e − ( α − v [( Reφ ) s − β (( Reφ ) s )] ds ≤ Reφ ) C N C ∗ . (3.13)Hence, plugging (3.13) into (3.10), since C N C ∗ < ∞ a.s. and ( Reφ ) → ∞ as c → ∞ , we deduce that for any fixed c ≥ P ( k h k vL v (0 , ∞ ) ≥ c ) ≤ P (cid:16) C N e C ∗ ≥ c ( Reφ ) (cid:17) → , as c → ∞ . (3.14)Consequently, choose c = (cid:2) · α α | x | α − H C α D α − (cid:3) − v >
0. By the defini-10ion of τ in (3.9) and (3.14), we then derive that P ( τ = ∞ )= P (cid:0) · α | x | α − H C α D ( t ) < , ∀ t ∈ [0 , ∞ ) (cid:1) ≥ P (cid:18) · α α | x | α − H C α D α − k h k L v (0 , ∞ ) ≤ (cid:19) ≥ − P (cid:0) k h k vL v (0 , ∞ ) ≥ c (cid:1) → , as c → ∞ , which completes the proof for spatially independent noise.( ii ). Now, we consider the general case when the noise W ( t ) is space-dependent. Again it is equivalent to prove the assertion for the randomequation (2.9).Let Z τM , G be as in (3.1) and (3.2) respectively. Similarly to (3.3), for u ∈ Z τM , k G ( u ) k L ∞ (0 ,τ ; H ) + k G ( u ) k L q (0 ,τ ; W ,p ) ≤ C τ | x | H + 2 C τ D ( τ ) M α , (3.15)where C t is the Strichartz coefficient, and D ( t ) = αD α − k h k L v (0 ,t ; W , ∞ ) . (3.16)with v > v = 1 − q > u , u ∈ Z τM , k G ( u ) − G ( u ) k L ∞ (0 ,τ ; L ) + k G ( u ) − G ( u ) k L q (0 ,τ ; L p ) ≤ C τ D ( τ ) M α − k u − u k L q (0 ,τ ; L p ) . (3.17)Set M = 3 C τ | x | H , choose the ( F t )-stopping time τ = τ ( c ), τ := inf { t ∈ [0 , T ] , · α | x | α − H C αt D ( t ) > } ∧ T. (3.18)It follows from (3.15) and (3.17) that G ( Z τM ) ⊂ Z τM and G is a contrac-tion on C ([0 , τ ]; L ) ∩ L q (0 , τ ; L p ). Therefore, using the same arguments asin [2], we obtain a local solution z on [0 , τ ].11o show that P ( τ = T ) →
1, as c → ∞ , using (3.18) and (3.16), weshall estimate k h k L v (0 ,t ; W , ∞ ) below. For simplicity, set | f | ∞ := | f | L ∞ for any f ∈ L ∞ ( R d ) and φ k := µ k e k , 1 ≤ k ≤ N .As regards the norm k h k L v (0 ,t ; L ∞ ) , by (2.14) and (2.7), | h ( t ) | L ∞ ≤ e − ( α − N P k =1 [ ( Reµ c N t −| Reφ k | ∞ | β k ( t ) | ] . (3.19)Analogously to (3.12), e C := 1 ∨ max ≤ k ≤ N sup t ≥ e − ( α − v [ ( Reµ c N t −| β k ( | Reφ k | ∞ t ) | ] < ∞ , a.s. (3.20)Moreover, choosing c large enough such that c > | f | ∞ , we have Z T e − ( α − v [ ( Reµ c N t −| β ( | Reφ | ∞ t ) | ] dt = 1 | Reφ | ∞ Z | Reφ | ∞ T e − ( α − v [ ( Reµ c N | Reφ | ∞ t −| β ( t ) | ] dt ≤ | Reφ | ∞ e C ∗ , (3.21)where e C ∗ := R ∞ e − ( α − v [ N t −| β ( t ) | ] dt < ∞ P -a.s.Thus, as in (3.14), it follows from (3.19)-(3.21) and the scaling propertyof β k , 1 ≤ k ≤ N , that for any c > P (cid:0) C αvT k h k vL v (0 ,T ; L ∞ ) ≥ c (cid:1) ≤ P (cid:16) C αvT e C N e C ∗ ≥ | Reφ | ∞ c (cid:17) → , as c → ∞ , P − a.s., (3.22)where C T is the Strichartz coefficient.Similar arguments can also be applied to the norm k∇ h k L v (0 ,t ; L ∞ ) . Indeed,from (2.14) and (2.7), ∇ h ( t ) = h ( t ) " − ( α − N X k =1 (2 Reφ k ( Reµ k ∇ f k ) t − Reµ k ∇ f k β k ( t )) , |∇ h ( t ) | ∞ ≤ ( α − | h ( t ) | ∞ N X k =1 (2 | Reφ k | ∞ | Reµ k ∇ f k | ∞ t + | Reµ k ∇ f k | ∞ | β k ( t ) | ) . Hence, for any c > P ( C αvT k∇ h k vL v (0 ,T ; L ∞ ) ≥ c ) ≤ P (cid:0) C αvT Z T ( α − v | h ( t ) | vL ∞ " N X k =1 | Reφ k | ∞ | Reµ k ∇ f k | ∞ t + | Reµ k ∇ f k | ∞ | β k ( t ) | v dt ≥ c (cid:1) ≤ P (cid:18) C αvT Z T ( α − v " N Y k =1 e − ( α − v [ ( Reµ c N t −| β k ( | Reφ k | ∞ t ) | ] N X k =1 | Reφ k | ∞ | Reµ k ∇ f k | ∞ t + | Reµ k ∇ f k | ∞ | β k ( t ) | v dt ≥ c (cid:19) ≤ P (cid:18) C αvT e C N | Reφ | ∞ Z ∞ e − ( α − v [ N t −| β ( t ) | ] " N X k =1 | Reφ k | ∞ | Reµ k ∇ f k | ∞ | Reφ | ∞ t + | Reµ k ∇ f k | ∞ (cid:12)(cid:12)(cid:12)(cid:12) β k ( t | Reφ | ∞ ) (cid:12)(cid:12)(cid:12)(cid:12) v dt ≥ c ( α − v (cid:19) . Choosing c large enough, such that N P k =1 2 | Reφ k | ∞ | Reµ k ∇ f k | ∞ | Reφ | ∞ < | Reµ k ∇ f k | ∞ | Reφ | ∞ <
1, we have as c → ∞ , P ( C αvT k∇ h k vL v (0 ,T ; L ∞ ) ≥ c ) ≤ P ( C αvT e C N e C ′ ≥ c ( α − v | Reφ | ∞ ) → . (3.23)where C T is the Stichartz coefficient and e C ′ := R ∞ e − ( α − v [ N t −| β ( t ) | ] (cid:20) t + N P k =1 β k ( t ) (cid:21) v dt < ∞ P -a.s. 13ow we come back to the definition of τ in (3.18). Choosing c = [4 · α αD α − | x | α − H ] − v > , we deduce from (3.22) and (3.23) that P ( τ = T ) ≥ P (2 · α | x | α − H C αt D ( t ) < , ∀ t ∈ [0 , T ]) ≥ P (2 · α αD α − | x | α − H C αT k h k L v (0 ,T,W , ∞ ) <
12 ) ≥ − P ( C αvT k h k vL v (0 ,T,W , ∞ ) ≥ c ) ≥ − P ( C αvT k h k vL v (0 ,T,L ∞ ) ≥ c ) − P ( C αvT k∇ h k vL v (0 ,T,L ∞ ) ≥ c ) → , as c → ∞ . Therefore, we complete the proof of Theorem 1.2. (cid:3)
One may further ask whether the non-explosion results in Theorem 1.2hold with probability 1. This is, unfortunately, not generally true. In fact,define the Hamiltonian H ( z ) = 12 |∇ z | − α + 1 | z | α +1 α +1 , z ∈ H , and set P = { u ∈ H , R | ξ | | u ( ξ ) | dξ < ∞ . } . We have the following result Proposition 3.1
Consider (1.1) in the non-conservative case (1.6) . Let λ and α satisfy (1.2) . Assume ( H ) with f j , ≤ j ≤ N , and c k , ≤ k ≤ N being fixed. Furthermore, assume µ k ∈ R , ≤ k ≤ N . Let x ∈ P with H ( x ) < ,Then there exists ǫ > , such that for < ǫ < ǫ and ≤ P ≤ k ≤ N |∇ f k | L ∞ <ǫ , the solution to (1.1) blows up in finite time with positive probability.In particular, in the case that f j , ≤ j ≤ N , are fixed constants, thesolution to (1.1) blows up in finite time with positive probability. The proof follows from the standard virial analysis (see e.g [14]). For any u ∈ P , define the variance V ( u ) = Z | ξ | | u ( ξ ) | dξ, (3.24)14nd the momentum G ( u ) = Im Z ξu ( ξ ) · ∇ u ( ξ ) dξ. (3.25) Proof of Proposition 3.1 . We prove the assertion by contradiction.Assume that the solution X ( t ) to (1.1) exists globally in H P − a.s .By Lemmas 4 .
1, 4.2 and 4.3 in the Appendix, V ( X ( t )) = V ( x ) + 4 G ( x ) t + 8 H ( x ) t + 4 N X k =1 Z t ( t − s ) |∇ φ k X ( s ) | ds − α − N X k =1 Z t ( t − s ) Z φ k | X ( s ) | α +1 dξds + 16 α + 1 (cid:20) − d ( α − (cid:21) Z t ( t − s ) | X ( s ) | α +1 α +1 ds (3.26)+ M t , where φ k = µ k e k , 1 ≤ k ≤ N , and M t :=8 N X k =1 Z t ( t − s ) (cid:20) Re h∇ ( φ k X ( s )) , ∇ X ( s ) i − Z φ k | X ( s ) | α +1 dξ (cid:21) dβ k ( s ) − N X k =1 Z t ( t − s ) Im Z ξ · ∇ X ( s ) X ( s ) φ k dξdβ k ( s )+ 2 N X k =1 Z t Z | ξ | | X ( s ) | φ k dξdβ k ( s ) . Fix t > r ∈ [0 , ∞ ), f M ( t, r ) :=8 N X k =1 Z r ( t − s ) (cid:20) Re h∇ ( φ k X ( s )) , ∇ X ( s ) i − Z φ k | X ( s ) | α +1 dξ (cid:21) dβ k ( s ) − N X k =1 Z r ( t − s ) Im Z ξ · ∇ X ( s ) X ( s ) φ k dξdβ k ( s )+ 2 N X k =1 Z r Z | ξ | | X ( s ) | φ k dξdβ k ( s ) . (3.27)15et σ m := inf { s ∈ [0 , t ] , |∇ X m ( s ) | > m } ∧ t . Then σ m → t , as m → ∞ .Direct computations show that f M ( t, · ∧ σ m ) is a square integrable mar-tingale, in particular, E [ f M ( t, t ∧ σ m )] = 0 . (3.28)Indeed, e.g. in regard to the second term in the right hand side of (3.27) ,we note that E Z r ∧ σ m N X k =1 (cid:12)(cid:12) ( t − s ) Im Z ξ · ∇ X ( s ) X ( s ) φ k dξ (cid:12)(cid:12) ds ≤ C E Z r ∧ σ m ( t − s ) V ( X ( s )) |∇ X ( s ) | ds ≤ mC E sup s ∈ [0 ,σ m ] V ( X ( s )) Z r ( t − s ) ds, (3.29)where C = N P k =1 | φ k | L ∞ < ∞ . Then, as in the proof of (4.10) below, we deducethat the right hand side in (3.29) is finite. The other terms can be estimatedeven more easily.Now, take the expectation in (3.26). Since the fifth and sixth terms inthe right hand side of (3.26) are non-positive for α satisfying (1.2), it followsthat E V ( X ( σ m ∧ t )) ≤ V ( x ) + 4 G ( x )( σ m ∧ t ) + 8 H ( x )( σ m ∧ t ) + 4 E Z σ m ∧ t ( σ m ∧ t − s ) N X k =1 |∇ φ k X ( s ) | ds, t < ∞ . Then, taking m → ∞ , by Fatou’s lemma, and since ∇ φ k = µ k ∇ f k and E | X ( t ) | = | x | , we obtain E V ( X ( t )) ≤ V ( x ) + 4 G ( x ) t + 8 H ( x ) t + at (3.30)with a = 43 N X k =1 | µ k ||∇ f k | L ∞ | x | . Let f ( t ) denote the right hand side of (3.30), i.e., f ( t ) := V ( x ) + 4 G ( x ) t + 8 H ( x ) t + at .
16e claim that, if N P k =1 |∇ f k | L ∞ is small enough, then there exists T > f ( T ) <
0. But, taking into account E V ( X ( t )) ≥ f ′ ( t ) = 3 at + 16 H ( x ) t + 4 G ( x ) , for N P k =1 |∇ f k | L ∞ small enough, the discriminant is positive and the largestroot of f ( t ) is t ∗ := 2 G ( x ) − H ( x ) − p H ( x )) − aG ( x ) > . (3.31)Note that, proving the claim is equivalent to showing that f ( t ∗ ) < f ′ ( t ∗ ) = 0, simple computations show that f ( t ∗ ) = 83 H ( x ) t ∗ + 83 G ( x ) t ∗ + V ( x ) . Since the largest roof of g ( t ) := 83 H ( x ) t + 83 G ( x ) t + V ( x )is e t ∗ := − G ( x ) − q ( G ( x )) − H ( x ) V ( x )2 H ( x ) , which is independent of a . But by (3.31), t ∗ → ∞ as a →
0, yielding that e t ∗ < t ∗ for a small enough, thereby implying f ( t ∗ ) < (cid:3) This appendix contains the Itˆo-formulas for the Hamiltonian, variance andmomentum. As mentioned in Remark 2.1, one can obtain a local solution X
17o (1.1) on [0 , τ n ], n ∈ N , where τ n are ( F t )-stopping times, and X satisfies P -a.s. for any Strichartz pair ( ρ, γ ), X | [0 ,t ] ∈ C ([0 , t ]; H ) ∩ L γ (0 , t ; W ,ρ ) , t < τ ∗ ( x ) (4.1)with τ ∗ ( x ) = lim n →∞ τ n .Let us start with the Itˆo-formula for the Hamiltonian H ( X ( t )) proved in[2, Theorem 3 . Theorem 4.1
Let α satisfy (1.2) . Set φ j := µ j e j , j = 1 , ..., N . Then P -a.s H ( X ( t ))= H ( x ) + Z t Re h−∇ ( µX ( s )) , ∇ X ( s ) i ds + 12 N X j =1 Z t |∇ ( X ( s ) φ j ) | ds − λ ( α − N X j =1 Z t Z ( Reφ j ) | X ( s ) | α +1 dξds + N X j =1 Z t Re h∇ ( φ j X ( s )) , ∇ X ( s ) i dβ j ( s ) − λ N X j =1 Z t Z Reφ j | X ( s ) | α +1 dξdβ j ( s ) , ≤ t < τ ∗ ( x ) . The following lemma is concerned with the Itˆo-formula for the variance.
Lemma 4.2
Let P be as in Proposition 3.1 and x ∈ P . Then P -a.s. for t < τ ∗ ( x ) , V ( X ( t )) = V ( x ) + 4 Z t G ( X ( s )) ds + M ( t ) , (4.2) where G is as in (3.25) and M ( t ) := 2 N X k =1 Z t Z | ξ | | X ( s ) | Reφ k dξdβ k ( s ) with φ k := µ k e k , ≤ k ≤ N . roof. The proof is similar to that in [2, Lemma 5 .
1] (see also [15]), hencewe just give a sketch of it below.Set ϕ ǫ := ϕ ∗ φ ǫ for any locally integrable function ϕ mollified by φ ǫ , where φ ǫ = ǫ − d φ ( xǫ ) and φ ∈ C ∞ c ( R d ) is a real-valued nonnegative function withunit integral. Set V η ( u ) = R e − η | ξ | | ξ | | u ( ξ ) | dξ and V ( u ) = R | ξ | | u ( ξ ) | dξ for u ∈ P .By (1.1) it follows that P -a.s. for every ξ ∈ R d , t < τ ∗ ( x ),( X ( t )) ǫ ( ξ ) = x ǫ ( ξ ) + Z t [ − i ∆( X ( s )) ǫ ( ξ ) − ( µX ( s )) ǫ ( ξ ) − i ( g ( X ( s ))) ǫ ( ξ )] ds + N X k =1 Z t ( X ( s ) φ j ) ǫ ( ξ ) dβ j ( s ) , (4.3)where g ( X ( s )) := | X ( s ) | α − X ( s ). For simplicity, we set X ǫ ( t ) := ( X ( t )) ǫ ( ξ )and correspondingly for the other arguments.Applying the product rule yields P -a.s. | X ǫ ( t ) | = | x ǫ | − Re Z t X ǫ ( s ) i ∆ X ǫ ( s ) ds − Re Z t X ǫ ( s )( µX ( s )) ǫ ds − Re Z t X ǫ ( s ) i [ g ( X ( s ))] ǫ ds + N X k =1 Z t | ( X ( s ) φ k ) ǫ | ds + 2 N X k =1 Re Z t X ǫ ( s )( X ( s ) φ k ) ǫ dβ k ( s ) , t < τ ∗ ( x ) . Then, integration over R d with e − η | ξ | | ξ | , interchanging integrals and inte-19rating by parts, we have P -a.s. for t < τ ∗ ( x ), V η ( X ǫ ( t )) = V η ( x ǫ ) + 4 Im Z t Z e − η | ξ | (1 − η | ξ | ) X ǫ ( s ) ξ · ∇ X ǫ ( s ) dξds − Re Z t Z e − η | ξ | | ξ | X ǫ ( s )( µX ( s )) ǫ dξds − Re Z t Z e − η | ξ | | ξ | X ǫ ( s ) i [ g ( X ( s ))] ǫ dξds + N X k =1 Z t Z e − η | ξ | | ξ | | ( X ( s ) φ k ) ǫ | dξds + 2 N X k =1 Re Z t Z e − η | ξ | | ξ | X ǫ ( s )( X ( s ) φ k ) ǫ dξdβ k ( s ) . (4.4)As sup ξ ∈ R d e − η | ξ | [ | (1 − η | ξ | ) ξ | + | ξ | ] < ∞ , one can take the limit ǫ → V η ( X ( t )) = V η ( x ) + 4 Im Z t Z e − η | ξ | (1 − η | ξ | ) X ( s ) ξ · ∇ X ( s ) dξds + 2 N X k =1 Z t Z e − η | ξ | | ξ | | X ( s ) | Reφ k dξdβ k ( s ) , t < τ ∗ ( x ) . (4.5)To pass to the limit η →
0, we shall prove thatsup s ∈ [0 ,τ n ] V ( X ( s )) ≤ e C ( n ) < ∞ , P − a.s. (4.6)Then by (4.5), (4.6), sup η> sup ξ ∈ R d | e − η | ξ | (1 − η | ξ | ) | = 1 and Lebesque’s dom-inated theorem, we obtain (4.2) for t ≤ τ n , n ∈ N . Consequently, since τ n → τ ∗ ( x ), as n → ∞ , we conclude (4.2) for t < τ ∗ ( x ).It remains to prove (4.6). For every n ∈ N , set σ n,m := inf { s ∈ [0 , τ n ] : |∇ X ( s ) | > m } ∧ τ n . E sup s ∈ [0 ,t ∧ σ n,m ] V η ( X ( s )) ≤ E Z t ∧ σ n,m Z e − η | ξ | | − η | ξ | || X ( s ) ξ · ∇ X ( s ) | dξds + c E vuutZ t ∧ σ n,m N X k =1 (cid:18)Z e − η | ξ | | ξ | | X ( s ) | Reφ k dξ (cid:19) ds = J + J , (4.7)where c is independent of n , m and η .Since sup η> sup ξ ∈ R d | e − η | ξ | (1 − η | ξ | ) | = 1 and E sup s ∈ [0 ,σ n,m ] |∇ X ( s ) | ≤ m < ∞ , J ≤ E Z t ∧ σ n,m p V ( X ( s )) |∇ X ( s ) | ds ≤ Z t E sup r ∈ [0 ,s ∧ σ n,m ] V ( X ( r )) ds + 4 mT. (4.8)Moreover, J ≤ C E sZ t ∧ σ n,m [ V η ( X ( s ))] ds ≤ ǫC E sup s ∈ [0 ,t ∧ σ n,m ] V η ( X ( s )) + CC ǫ Z t E sup r ∈ [0 ,s ∧ σ n,m ] V η ( X ( r )) ds, (4.9)where C depends on | φ k | L ∞ , 1 ≤ k ≤ N , and is independent of n, m and η .Hence, plugging (4.8) and (4.9) into (4.7), taking ǫ small enough, andnoting that V η ( X ) ≤ V ( X ), we derive that E sup s ∈ [0 ,t ∧ σ n,m ] V η ( X ( s )) ≤ C Z t E sup r ∈ [0 ,s ∧ σ n,m ] V ( X ( r )) ds + C ( m, T ) , with C and C ( m, T ) independent of η . Then letting η → E sup s ∈ [0 ,t ∧ σ n,m ] V ( X ( s )) ≤ C Z t E sup r ∈ [0 ,s ∧ σ n,m ] V ( X ( r )) ds + C ( m, T ) , t ∈ [0 , T ] , E sup t ∈ [0 ,σ n,m ] V ( X ( t )) ≤ C ( m, T ) < ∞ , (4.10)hence sup t ∈ [0 ,σ n,m ] V ( X ( t )) ≤ e C ( m, T ) < ∞ , P -a.s. But, since sup t ∈ [0 ,τ n ] |∇ X ( t ) | < ∞ , P -a.s, for P -a.e. ω ∈ Ω, ∃ m ( ω ) < ∞ such that σ n,m ( ω ) ( ω ) = τ n ( ω ). Then P (cid:18) S m ∈ N { σ n,m = τ n } (cid:19) = 1. This implies (4.6) and completes the proof ofLemma 4.2. (cid:3) We conclude this section with the Itˆo-formula for the momentum.
Lemma 4.3
Let x ∈ P . Then P -a.s for t < τ ∗ ( x ) , G ( X ( t )) = G ( x ) + 4 Z t P ( X ( s )) ds − N X k =1 Z t Im Z ξ · ∇ φ k | X ( s ) | φ k dξds + M ( t ) , (4.11) where P ( X ) := 12 |∇ X | − d ( α − α + 1) | X | α +1 L α +1 = H ( X ) + 1 α + 1 [1 − d ( α − | X | α +1 L α +1 ,φ k = µ k e k , ≤ k ≤ N , and M ( t ) := d N X k =1 Z t Z | X ( s ) | Imφ k dξdβ k ( s ) − N X k =1 Z t Im Z ξ · ∇ X ( s ) X ( s ) φ k dξdβ k ( s ) . Here, d is the dimension of the space. Proof.
The proof is similar to that in Lemma 4.2 but involves morecomplicated computations. For simplicity of exposition, we omit the proofhere and refer to [20, Lemma 3 . .
2] for details. (cid:3)
Acknowledge.
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