On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities
aa r X i v : . [ m a t h . A P ] A ug ON THE PROBABILISTIC WELL-POSEDNESS OF THE NONLINEARSCHR ¨ODINGER EQUATIONS WITH NON-ALGEBRAICNONLINEARITIES
TADAHIRO OH, MAMORU OKAMOTO, AND OANA POCOVNICU
Abstract.
We consider the Cauchy problem for the nonlinear Schr¨odinger equations(NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we studythe energy-critical NLS on R d , d = 5 ,
6, and energy-critical NLS without gauge invarianceand prove that they are almost surely locally well-posed with respect to randomized initialdata below the energy space. We also study the long time behavior of solutions to theseequations: (i) we prove almost sure global well-posedness of the (standard) energy-criticalNLS on R d , d = 5 ,
6, in the defocusing case, and (ii) we present a probabilistic constructionof finite time blowup solutions to the energy-critical NLS without gauge invariance belowthe energy space.
Contents
1. Introduction 21.1. Nonlinear Schr¨odinger equations 21.2. Almost sure global well-posedness of the defocusing energy-critical NLS belowthe energy space 71.3. Probabilistic construction of finite time blowup solutions below the criticalregularity 102. Probabilistic lemmas 123. Function spaces and their basic properties 134. Nonlinear estimates 165. Proof of Theorems 1.1 and 1.2 226. A variant of almost sure local well-posedness 227. Almost sure global well-posedness of the defocusing energy-critical NLS belowthe energy space 257.1. Energy estimate for the perturbed NLS 267.2. Long time existence of solutions to the perturbed NLS 357.3. Proof of Theorem 1.5 378. Probabilistic construction of finite time blowup solutions below the energy space 38References 41
Mathematics Subject Classification.
Key words and phrases. nonlinear Schr¨odinger equation; almost sure local well-posedness; almost sureglobal well-posedness; finite time blowup. Introduction
Nonlinear Schr¨odinger equations.
We consider the Cauchy problem for the fol-lowing energy-critical nonlinear Schr¨odinger equation (NLS) on R d , d = 5 , ( i∂ t u + ∆ u = ±| u | d − uu | t =0 = φ, ( t, x ) ∈ R × R d . (1.1)This equation enjoys the following dilation symmetry: u ( t, x ) u µ ( t, x ) := µ d − u ( µ t, µx )for µ >
0. This dilation symmetry preserves the ˙ H -norm of the initial data φ , thus inducingthe scaling critical Sobolev regularity s crit = 1. Moreover, the energy (= Hamiltonian) of asolution u remains invariant under this dilation symmetry. For this reason, we refer to (1.1)as energy-critical and ˙ H ( R d ) as the energy space.The Cauchy problem (1.1) in a general dimension has been at the core of the study ofdispersive equations for several decades and has been studied extensively. In particular, for d ≥
5, it is known that (1.1) is (i) locally well-posed in the energy space [12] and (ii) globallywell-posed in the defocusing case [53] and also in the focusing case under some assumptionon the (kinetic) energy [33]. On the other hand, (1.1) is known to be ill-posed in H s ( R d ), s < s crit = 1, in the sense of norm inflation [14]; there exists a sequence { u n } n ∈ N of (smooth)solutions to (1.1) and { t n } n ∈ N ⊂ R + such that k u n (0) k H s < n but k u n ( t n ) k H s > n with t n < n . This in particular shows that the solution map to (1.1) can not be extended tobe a continuous map on H s ( R d ), s <
1, thus violating one of the important criteria forwell-posedness.Despite the ill-posedness below the energy space, one may still hope to construct uniquelocal-in-time solutions in a probabilistic manner, thus establishing almost sure local well-posedness in some suitable sense. Such an approach first appeared in the work by McK-ean [39] and Bourgain [6] in the study of invariant Gibbs measures for the cubic NLS on T d , d = 1 ,
2. In particular, they established almost sure local well-posedness with respectto particular random initial data. This random initial data in [39, 6] can be viewed as arandomization of the Fourier coefficients of a particular function (basically the antideriv-ative of the Dirac delta function) via the multiplication by independent Gaussian randomvariables. Such randomization of the Fourier series is classical and well studied [46, 31]. In[9], Burq-Tzvetkov elaborated this idea further. In particular, in the context of the cubicnonlinear wave equation (NLW) on a three dimensional compact Riemannian manifold,they considered a randomization via the Fourier series expansion as above for any roughinitial condition below the scaling critical Sobolev regularity and established almost surelocal well-posedness with respect to the randomization. Such randomization via the Fourierseries expansion is natural on compact domains and more generally in situations where theassociated elliptic operators have discrete spectra [52, 19, 16].Our main focus is to study NLS (1.1) on the Euclidean space R d . In this setting, the ran-domization via the Fourier series expansion does not quite work as the frequency space R dξ is These local-in-time solutions were then extended globally in time by invariance of the Gibbs measures.In the following, however, we do not use any invariant measure.
ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , not discrete. We instead consider a randomization associated to the Wiener decomposition R dξ = S n ∈ Z d ( n + ( − , ] d ). See [57, 37, 2, 3, 25]. Let ψ ∈ S ( R d ) satisfysupp ψ ⊂ [ − , d and X n ∈ Z d ψ ( ξ − n ) = 1 for any ξ ∈ R d . Then, given a function φ on R d , we have φ = X n ∈ Z d ψ ( D − n ) φ. This replaces the role of the Fourier series expansion on compact domains. We then definethe Wiener randomization of φ by φ ω := X n ∈ Z d g n ( ω ) ψ ( D − n ) φ, (1.2)where { g n } is a sequence of independent mean zero complex-valued random variables on aprobability space (Ω , F , P ). In the following, we assume that the real and imaginary partsof g n are independent and endowed with probability distributions µ (1) n and µ (2) n , satisfyingthe following exponential moment bound: ˆ R e κx dµ ( j ) n ( x ) ≤ e cκ for all κ ∈ R , n ∈ Z d , j = 1 ,
2. This condition is satisfied by the standard complex-valuedGaussian random variables and the standard Bernoulli random variables.On the one hand, the randomization does not improve differentiability just like therandomization via the Fourier series expansion [9, 1]. On the other hand, it improvesintegrability as for the classical random Fourier series [46, 31]. From this point of view,the randomization makes the problem subcritical in some sense, at least for local-in-timeproblems.In the following, we study the Cauchy problem (1.1) with random initial data given bythe Wiener randomization φ ω of a given function φ ∈ H s ( R d ), d = 5 ,
6. In view of thedeterministic well-posedness result for s ≥
1, we only consider s < s crit = 1.
Theorem 1.1.
Let d = 5 , and − d < s < . Given φ ∈ H s ( R d ) , let φ ω be its Wienerrandomization defined in (1.2) . Then, the Cauchy problem (1.1) is almost surely locallywell-posed with respect to the random initial data φ ω .More precisely, there exist C, c, γ > such that for each < T ≪ , there exists Ω T ⊂ Ω with P (Ω cT ) ≤ C exp (cid:16) − cT γ k φ k Hs (cid:17) such that for each ω ∈ Ω T , there exists a unique solution u = u ω ∈ C ([ − T, T ]; H s ( R d )) to (1.1) with u | t =0 = φ ω in the class S ( t ) φ ω + X T ⊂ S ( t ) φ ω + C ([ − T, T ]; H ( R d )) ⊂ C ([ − T, T ]; H s ( R d )) , where S ( t ) = e it ∆ and X T is defined in Section 3 below. Almost sure local well-posedness with respect to the Wiener randomization has beenstudied in the context of the cubic NLS and the quintic NLS on R d [2, 3, 8] which areenergy-critical in dimensions 4 and 3, respectively. Note that when d = 5 ,
6, the energy-critical nonlinearity | u | d − u is no longer algebraic, presenting a new difficulty in applyingthe argument in [2, 3, 8]. T. OH, M. OKAMOTO, AND O. POCOVNICU
Let z ( t ) = z ω ( t ) := S ( t ) φ ω denote the random linear solution with φ ω as initial data. If u is a solution to (1.1), then the residual term v := u − z satisfies the following perturbedNLS: ( i∂ t + ∆ v = N ( v + z ω ) v | t =0 = 0 , (1.3)where N ( u ) = ±| u | d − u . In terms of the Duhamel formulation, (1.3) reads as v ( t ) = − i ˆ t S ( t − t ′ ) N ( v + z ω )( t ′ ) dt ′ . (1.4)Then, the main objective is to solve the fixed point problem (1.4). In fact, the first andthird authors (with B´enyi) [2, 3] studied this problem for the residual term v in the contextof the cubic NLS on R d by carrying out case-by-case analysis and estimating terms of theform vvv , vvz , vzz , etc. In [8], Brereton carried out similar analysis for the quintic NLSon R d . Such case-by-case analysis is possible only for algebraic, i.e. smooth, nonlinearitiesand thus is not applicable to our problem at hand. In this paper, we adjust the analysisfrom [3] in order to handle non-algebraic nonlinearities. Moreover, our analysis in this paperis simpler than that in [2, 3] in the sense that we avoid thorough case-by-case analysis.There is, however, a price to pay: (i) While our approach for non-algebraic nonlinearitiesin this paper can be applied to the energy-critical cubic NLS on R , this would yield aworse regularity range s ∈ ( ,
1) than the regularity range s ∈ ( ,
1) obtained in [3]. Thisis due to the fact that we adjust our calculation to a non-smooth nonlinearity. (ii) Theconstants in the nonlinear estimates in Section 4 depend on the local existence time
T > V p of functions of bounded p -variation and their pre-duals U p , the bilinear refinement of the Strichartz estimate, and the probabilistic Strichartzestimates thanks to the gain of integrability via the Wiener randomization. In order to avoidthe use of fractional derivatives, we focus on the energy-critical NLS and solve the fixedpoint problem (1.4) in X T at the critical regularity (for the residual term) by performinga precise computation. Namely, it is important that we use this refined version of theFourier restriction norm method, since if we were to use the usual X σ,b -spaces introducedin [5], then we would need to study the problem at the subcritical regularity σ = 1 + ε asin [2], creating a further difficulty. Moreover, in proving almost sure global well-posednessof (1.1), it is essential that we only use the X σT -norm, σ ≤
1, for the residual part v . SeeTheorem 1.5 below. In the field of stochastic parabolic PDEs, this change of viewpoint and solving the fixed point problemfor the residual term v is called the Da Prato-Debussche trick [17, 18]. In the context of deterministicdispersive PDEs with random initial data, this goes back to the work by McKean [39] and Bourgain [6],which precedes [17, 18]. ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , Next, we consider the following energy-critical NLS without gauge invariance on R d , d = 5 , ( i∂ t u + ∆ u = λ | u | d +2 d − u | t =0 = φ, (1.5)where λ ∈ C \ { } . As in the case of the standard NLS (1.1), one can prove local well-posedness of (1.5) in H s ( R d ), s ≥
1, via the Strichartz estimates. On the other hand,Ikeda-Inui [27] showed that (1.5) is ill-posed in H s ( R d ) with s <
1. More precisely, theyproved non-existence of solutions for rough initial data, satisfying a certain condition. Thisill-posedness result by non-existence is much stronger than the norm inflation proved forthe standard NLS (1.1). The non-existence result in [27] studies a rough initial conditionand exhibits a pathological behavior in a direct manner, while the norm inflation resultin [14] is proved by studying the behavior of a sequence of smooth solutions; in particularit does not say anything about rough solutions.
Theorem 1.2.
Let d = 5 , and − d < s < . Given φ ∈ H s ( R d ) , let φ ω be its Wienerrandomization defined in (1.2) . Then, the Cauchy problem (1.5) is almost surely locallywell-posed with respect to the random initial data φ ω in the sense of Theorem 1.1. Theorem 1.2, in particular, states that upon the randomization, we can avoid thesepathological initial data constructed in [27] for which no solution exists. Compare this withthe “standard” almost sure local well-posedness results such as Theorem 1.1 above, wherethe only known obstruction to well-posedness below a threshold regularity is discontinuityof the solution map. In this sense, Theorem 1.2 provides a more striking role of randomiza-tion, overcoming the non-existence result below the scaling critical regularity, and it seemsthat Theorem 1.2 is the first such result.The proof of Theorem 1.2 follows the same lines as that of Theorem 1.1. When d = 6,the nonlinearity | u | = uu in (1.5) is algebraic. Hence, one may also perform case-by-caseanalysis as in [3]. We, however, do not pursue this direction since our purpose is to presenta unified approach to the problem.Next, let us state an almost sure local well-posedness result with slightly more generalinitial data. Fix φ ∈ H s ( R d ) \ H ( R d ). Then, we consider the following Cauchy problemfor given v ∈ H ( R d ): ( i∂ t u + ∆ u = N ( u ) u | t =0 = v + φ ω , (1.6)where N ( u ) = ±| u | d − u or λ | u | d +2 d − and φ ω is the Wiener randomization of φ . Then, as acorollary to (the proof of) Theorems 1.1 and 1.2, we have the following proposition. Proposition 1.3.
Let d = 5 , and − d < s < . Given φ ∈ H s ( R d ) , let φ ω be its Wienerrandomization defined in (1.2) . Then, given v ∈ H ( R d ) , the Cauchy problem (1.6) isalmost surely locally well-posed with respect to the Wiener randomization φ ω , where the(random) local existence time T = T ω is assumed to be sufficiently small, depending on the Namely, the pathological behavior of the standard NLS (1.1) below the scaling critical regularity s crit = 1is about the solution map (stability under perturbation) and is not about individual solutions (such asexistence). On the contrary, in the case of (1.5), there are individual initial data, each of which is responsiblefor the pathological behavior (non-existence of solutions). T. OH, M. OKAMOTO, AND O. POCOVNICU deterministic part v of the initial data. Moreover, the following blowup alternative holds;let T ∗ = T ∗ ( ω, v ) be the forward maximal time of existence. Then, either T ∗ = ∞ or lim T → T ∗ k u − S ( t ) φ ω k L qdt ([0 ,T ); W ,rdx ) = ∞ , (1.7) where ( q d , r d ) is a particular admissible pair given by ( q d , r d ) := (cid:0) dd − , d d − d +4 (cid:1) . (1.8)Namely, this is an almost sure local well-posedness result with the initial data of theform: “a fixed smooth deterministic function + a rough random perturbation”. See, forexample, [44]. The proof of Proposition 1.3 is based on studying the equation for theresidual term v = u − z ω as above: ( i∂ t v + ∆ v = N ( v + z ω ) v | t =0 = v ∈ H ( R d ) , (1.9)where we now have a non-zero initial condition. For this fixed point problem, the criticalnature of the problem appears through the deterministic initial condition v . In particular,the local existence time T = T ( v ) depends on the profile of the (deterministic) initial data v . We point out that the good set of probability 1 on which almost sure local well-posednessholds does not depend on the choice of v ∈ H ( R d ).In the next two subsections, we state results on the long time behavior of solutions to (1.1)and (1.5), using Proposition 1.3. In particular, we prove almost sure global well-posednessof the defocusing energy-critical NLS (1.1) below the energy space (Theorem 1.5). As forNLS (1.5) without gauge invariance, we use Proposition 1.3 to construct finite time blowupsolutions below the critical regularity in a probabilistic manner (Theorem 1.7). Remark 1.4.
When s <
1, the solution mapΦ : u ∈ H s ( R d ) u ∈ C ([ − T, T ]; H s ( R d ))is not continuous for (1.1) and is not even well defined for (1.5); see [14, 27]. Once we view z ω = S ( t ) φ ω as a probabilistically pre-defined data, we can factorize the solution map for(1.6) as u = v + φ ω ∈ H s ( R d ) ( v , z ω ) v ∈ C ([ − T ω , T ω ]; H ( R d )) , where the first map can be viewed as a universal lift map and the second map is thesolution map Ψ to (1.9), which is in fact continuous in ( v , z ω ) ∈ H ( R d ) × S s ([0 , T ]),where S s ([0 , T ]) ⊂ C ([0 , T ]; H s ( R d )) is the intersection of suitable space-time functionspaces. See (4.3) below for example. We also point out that under this factorization, it isclear that the probabilistic component appears only in the first step while the second stepis entirely deterministic.One can go further and introduce more probabilistically pre-defined objects in order toimprove the regularity threshold. In the context of the cubic NLS on R [4], the first andthird authors (with B´enyi) decomposed u as u = z ω + z ω + v , where z ω = S ( t ) φ ω and z ω = − i ´ t S ( t − t ′ ) | z | z ( t ′ ) dt ′ , thus leading to the following factorization: u = v + φ ω ∈ H s ( R ) ( v , z ω , z ω ) v ∈ C ([ − T ω , T ω ]; H ( R )) . ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , The introduction of the higher order pre-defined object z allowed us to lower the regularitythreshold from the previous work [3]. For NLS with non-algebraic nonlinearities suchas (1.1) and (1.5), it is not clear how to introduce a further decomposition at this point.This is due to the non-smoothness of the nonlinearities. If one has an algebraic (or analytic)nonlinearity, then a Picard iteration yields analytic dependence, thus enabling us to writea solution as a power series in terms of initial data, at least in theory. See [13, 42]. On theother hand, if a nonlinearity is non-smooth, then a Picard iteration does not yield analyticdependence, which makes it hard to find a higher order term.More recently, the first author (with Tzvetkov and Wang) proved invariance of the whitenoise for the (renormalized) cubic fourth order NLS on the circle [45]. In this work, weintroduced an infinite sequence { z j − } j ∈ N of pre-defined objects of order 2 j − u = P ∞ j =1 z j − + v , thus considering thefollowing factorization: u ω ∈ H s ( T ) ( z ω , z ω , z ω , . . . ) v ∈ C ( R ; H s ( T )) , for s < − , where u ω is the Gaussian white noise on the circle. We conclude this remarkby pointing out an analogy of this factorization of the ill-posed solution map to that in therough path theory [20] and more recent studies on stochastic parabolic PDEs [22, 24].1.2. Almost sure global well-posedness of the defocusing energy-critical NLSbelow the energy space.
In this subsection, we consider the energy-critical NLS (1.1) inthe defocusing case (i.e. with the + sign). Let us first recall the known related result in thisdirection. In [3], the first and third authors (with B´enyi) studied the global-in-time behaviorof solutions to the defocusing energy-critical cubic NLS (1.1) on R . By implementing theprobabilistic perturbation theory, we proved conditional almost sure global well-posednessof the defocusing energy-critical cubic NLS on R , assuming the following energy bound onthe residual part v = u − z : Energy bound:
Given any
T, ε >
0, there exists R = R ( T, ε ) and Ω
T,ε ⊂ Ω such that(i) P (Ω cT,ε ) < ε , and(ii) If v = v ω is the solution to (1.3) for ω ∈ Ω T,ε , then the following a priori energyestimate holds: k v ( t ) k L ∞ ([0 ,T ]; H ( R d )) ≤ R ( T, ε ) . (1.10)The main ingredient in this conditional almost sure global well-posedness result in [3] isa perturbation lemma (see Lemma 7.4 below). Assuming the energy bound (1.10) above,we iteratively applied the perturbation lemma in the probabilistic setting to show that asolution can be extended to a time depending only on the H -norm of the residual part v .Such a perturbative approach was previously used by Tao-Vi¸san-Zhang [51] and Killip-Vi¸san with the first and third authors [34]. The main novelty in [3] was an application ofsuch a technique in the probabilistic setting, allowing us to study the long time behaviorof solutions when there is no invariant measure available for the problem. It is worthwhile to mention that the conditional almost sure global well-posedness in [3] and Theorem 1.5below exploit certain “invariance” property of the distribution of the linear solution S ( t ) φ ω ; the distributionof S ( t ) φ ω on an interval [ t , t + τ ∗ ] (measured in a suitable space-time norm) depends only on the length τ ∗ of the interval. In [16], similar invariance of the distribution of the random linear solution played anessential role in proving almost sure global well-posedness. T. OH, M. OKAMOTO, AND O. POCOVNICU
This probabilistic perturbation method can be easily adapted to other critical equations.In [47, 43], by establishing the energy bound (1.10), we implemented the probabilisticperturbation theory in the context of the defocusing energy-critical NLW on R d , d = 3 , , d = 5 , Theorem 1.5.
Let d = 5 , and set s ∗ = s ∗ ( d ) by (i) s ∗ = 6368 when d = 5 and (ii) s ∗ = 2023 when d = 6 .Given φ ∈ H s ( R d ) , s ∗ < s < , let φ ω be its Wiener randomization defined in (1.2) . Then,the defocusing energy-critical NLS (1.1) on R d is almost surely globally well-posed withrespect to the random initial data φ ω .More precisely, there exists a set Σ ⊂ Ω with P (Σ) = 1 such that, for each ω ∈ Σ , thereexists a (unique) global-in-time solution u to (1.1) with u | t =0 = φ ω in the class S ( t ) φ ω + C ( R ; H ( R d )) ⊂ C ( R ; H s ( R d )) . In a recent preprint [35], Killip-Murphy-Vi¸san studied the defocusing energy-criticalcubic NLS with randomized initial data when d = 4. In particular, under the radialassumption, they proved almost sure global well-posedness and scattering below the energyspace by implementing a double bootstrap argument intertwining the energy and Morawetzestimates.Our main goal in Theorem 1.5 is to simply prove almost sure global well-posedness(without scattering) by establishing the energy bound (1.10). In particular, Theorem 1.5establishes the first almost sure global well-posedness result of the defocusing energy-criticalNLS (1.1) below the energy space without the radial assumption. As mentioned above, themain difficulty in proving Theorem 1.5 is to establish the a priori energy bound (1.10). Forthis purpose, let us recall the following conservation laws for (1.1):Mass: M ( u )( t ) = ˆ R d | u ( t, x ) | dx, Energy: E ( u )( t ) = 12 ˆ R d |∇ u ( t, x ) | dx + d − d ˆ R d | u ( t, x ) | dd − dx. The main task is to control the growth of the energy E ( v ) for the residual part v = u − z byestimating the time derivative of E ( v ). We first point out that while M ( v ) is not conserved,one can easily establish a global-in-time bound on M ( v ). See Lemma 7.1.By a direct computation with (1.3), we have ∂ t E ( v ) = Re i ˆ (cid:8) | v + z | d − ( v + z ) − | v | d − v (cid:9) ∆ vdx − Re i ˆ | v + z | d − ( v + z ) | v | d − vdx =: I + II . (1.11) ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , We need to estimate ∂ t E ( v ) by E ( v ) and various norms of the random linear solution z = S ( t ) φ ω . Moreover, we are allowed to use at most one power of E ( v ) in order to closea Gronwall-type argument. Note that the energy E ( v ) consists of two parts. On the onehand, while the kinetic part controls the derivative of v , its homogeneity (= degree) is lowand hence can not be used to control a nonlinear term of a high degree (in v ). On the otherhand, the potential part has a higher homogeneity but it can not be used to control anyderivative. Hence, we need to combine the kinetic and potential parts of the energy in anintricate manner.The main contribution to I in (1.11) is given by a term of the form: ˆ | v | d − |∇ v · ∇ z | dx . ˆ |∇ v | dx + (cid:13)(cid:13) | v | d − ∇ z (cid:13)(cid:13) L x . (1.12)In order to estimate the second term on the right-hand side, we integrate in time andperform multilinear space-time analysis. More precisely, we divide the second term on theright-hand side of (1.12) into a θ -power and a (1 − θ )-power for some θ = θ ( s ) ∈ (0 ,
1) andestimate them in different manners. As for the θ -power, we apply the refinement of thebilinear Strichartz estimate (Lemma 3.6), substitute the Duhamel formula for v (yielding ahigher order term in v ), and control the resulting contribution (by ignoring the derivativeon v ) by the potential part of the energy. We then use the (1 − θ )-power to absorb thederivative on v from the θ -power and control the resulting contribution by the kinetic partof the energy and the mass. See Propositions 7.2 and 7.3.When d = 6, the main contribution to the second term II in (1.11) is given by ´ | v | | z | dx ,which can be controlled by (the potential part of) the energy E ( v ). On the other hand,when d = 5, the main contribution to the second term II in (1.11) is given by ´ | v | | z | dx ,which we can not control by the energy E ( v ). In order to overcome this problem, we usethe following modified energy when d = 5: E ( v ) = 12 ˆ |∇ v | dx + 310 ˆ | v + z | dx. (1.13)The use of this modified energy E ( v ) eliminates the contribution II in (1.11) at the expenseof introducing ∆ z in I . It turns out, however, the worst term is still given by the secondterm on the right-hand side of (1.12) and hence there is no loss in using the modified energy E ( v ).Lastly, we point out the following. On the one hand, the regularity for almost sure localwell-posedness in Theorem 1.1 is worse when d = 6. On the other hand, the regularity foralmost sure global well-posedness in Theorem 1.5 is worse when d = 5:2023 ≈ . < ≈ . . This is due to the fact that the main contribution (1.12) to the energy estimate comes witha higher order term in v when d = 5. In fact, when d = 4, our argument completely breaksdown. In this case, the left-hand side of (1.12) becomes ˆ | v | |∇ v · ∇ z | dx . ˆ |∇ v | dx + ˆ | v | |∇ z | dx. Recalling that the potential energy is given by ´ | v | dx , it is easy to see that we can notpass a part of the derivative on z to | v | in the second term on the right-hand side and hence it is not possible to bound it by (cid:0) E ( v ) (cid:1) α , α ≤
1, since z / ∈ W ,p ( R ) for any p , almostsurely. For this problem, some other space-time control such as the (interaction) Morawetzestimate is required. Remark 1.6.
In [38], L¨uhrmann-Mendelson used a modified energy with the potentialpart given by p +1 ´ | v + z | p +1 dx in studying the defocusing energy-subcritical NLW on R (3 < p < ∂ t u − ∆ u + | u | p − u = 0with randomized initial data below the scaling critical regularity. In particular, theyadapted the technique from [43] and proved almost sure global well-posedness in H s ( R ) × H s − ( R ) for p − p +1 < s < with the standard energy and proved almost sure global well-posedness with a betterregularity threshold: p − p − < s <
1, which interpolates the almost sure global well-posednessresults by Burq-Tzvetkov ( p = 3) in [10] and the first and third authors ( p = 5) in [43].While our use of the modified energy E ( v ) in (1.13) removes the issue with the timederivative of the potential part of the energy (i.e. II in (1.11)), it does not worsen theregularity threshold in the sense that the worst term is still given by (1.12).1.3. Probabilistic construction of finite time blowup solutions below the criticalregularity.
In this subsection, we focus on NLS (1.5) without gauge invariance. As com-pared to the standard NLS (1.1) with the gauge invariant nonlinearity, the equation (1.5)is less understood, in particular due to lack of structures such as conservation laws.In recent years, starting with the work by Ikeda-Wakasugi [29], there has been somedevelopment in the construction of finite time blowup solutions for (1.5), including the caseof small initial data. See also [40, 41, 28]. While there are some variations, the criteria forfinite time blowup solutions are very different from those for the standard NLS (1.1) andthey are given in terms of a condition on the sign of the product of the real part (and theimaginary part, respectively) of the coefficient λ ∈ C \ { } in (1.5) and the imaginary part(and the real part, respectively) of (the spatial integral of) an initial condition. We nowrecall the result of particular interest due to Ikeda-Inui [27, Theorem 2.3 and Remark 2.1].Given v ∈ H ( R d ), consider NLS (1.5) without gauge invariance equipped with an initialcondition of the form φ = αv , α ≥
0. Moreover, assume that v satisfies(Im λ )(Re v )( x ) ≥ | x |≤ | x | − k for all x ∈ R d , (1.14)or − (Re λ )(Im v )( x ) ≥ | x |≤ | x | − k for all x ∈ R d (1.15)for some positive k < d −
1. Then, there exists α = α ( d, k, | λ | ) > α > α , the solution u = u ( α ) to (1.5) with u | t =0 = αv blows up forward in finite time. See [51] for the interaction Morawetz estimate for NLS with a perturbation. In a recent preprint [35], Killip-Murphy-Vi¸san proved almost sure global well-posedness and scatteringbelow the energy space for the defocusing energy-critical cubic NLS on R in the radial setting, where theMorawetz estimate (among other tools available in the radial setting) played an important role. While the main result in [49] is stated on the three-dimensional torus T , the same result holds on R by the same proof. ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , If we denote T ∗ ( α ) > T ∗ ( α ) ≤ Cα − κ (1.16)for all α > α , where κ = d − − k . Moreover, we havelim T → T ∗ k u k L qd ([0 ,T ); W ,rdx ) = ∞ , where ( q d , r d ) is as in (1.8). A similar statement holds for the negative time direction ifwe replace (1.14) and (1.15) by − (Im λ )(Re v )( x ) ≥ | x |≤ | x | − k and (Re λ )(Im v )( x ) ≥ | x |≤ | x | − k , respectively.In the following, we fix v satisfying (1.14) or (1.15) and consider (1.5) with u | t =0 = αv + εφ ω , where φ ω is the Wiener randomization of some fixed φ ∈ H s ( R d ) \ H ( R d ), s < d = 5 ,
6. Namely, we study stability of the finite time blowup solution constructedin [27] under a rough perturbation in a probabilistic manner.
Theorem 1.7.
Let d = 5 , , − d < s < , and k < d − . Given φ ∈ H s ( R d ) , let φ ω beits Wiener randomization defined in (1.2) . Fix v ∈ H ( R d ) , satisfying (1.14) or (1.15) .Then, for each R > and ε > , there exists Ω R,ε ⊂ Ω with P (Ω cR,ε ) ≤ C exp (cid:18) − c R ε k φ k L (cid:19) and α = α ( d, k, | λ | , R, ε ) > such that for each ω ∈ Ω R,ε and any α > α , the solution u = u ω to (1.5) with initial data u | t =0 = αv + εφ ω blows up forward in finite time with the forward maximal time T ∗ ( α ) of existence satisfy-ing (1.16) , where the implicit constant depends only on R > . Moreover, we have lim T → T ∗ k u − εz ω k L qd ([0 ,T ); W ,rdx ) = ∞ , (1.17) where z ω = S ( t ) φ ω . This result in particular allows us to construct finite time blowup solutions below thecritical regularity s crit = 1. Moreover, it can be viewed as a probabilistic stability result ofthe finite time blowup solutions in H ( R d ) constructed in [27] under random and roughperturbations. Note that P (Ω R,ε ) → ε → u = εz + v and considering the equation for the residual term v : ( i∂ t v + ∆ v = λ | v + εz ω | d +2 d − ,v | t =0 = αv , (1.18)where z ω = S ( t ) φ ω as before. In view of Proposition 1.3, the equation (1.18) is almostsurely locally well-posed with a blowup alternative (1.7). This allows us to show that thesolution v is a weak solution in the sense of Definition 8.1 and hence to carry out theanalysis in [27] with a small modification coming from the random perturbation term. Onecrucial point to note is that once we reduce our analysis to the weak formulation in (8.1), we only require space-time integrability of the random perturbation z ω and its differentiabilityplays no role. This enables us to prove Theorem 1.7.We now give a brief outline of this article. In Sections 2 and 3, we recall probabilisticand deterministic lemmas along with the definitions of the basic function spaces. We thenprove the crucial nonlinear estimates in Section 4, and present the proof of the almostsure local well-posedness (Theorems 1.1 and 1.2) in Section 5. In Section 6, we prove avariant of almost sure local well-posedness (Proposition 1.3). In Section 7, we establish thecrucial energy bound (1.10) and present the proof of almost sure global well-posedness ofthe defocusing energy-critical NLS (1.1) (Theorem 1.5). In Section 8, we use Proposition1.3 to construct finite time blowup solutions below the critical regularity in a probabilisticmanner.In view of the time reversibility of the equations, we only consider positive times in thefollowing. Moreover, in the local-in-time theory, the defocusing/focusing nature of (1.1)does not play any role, so we assume that it is defocusing (with the +-sign in (1.1)).Similarly, we simply set λ = 1 in (1.5).2. Probabilistic lemmas
In this section, we state the probabilistic lemmas used in this paper. See [2, 43] for theirproofs. The first lemma states that the Wiener randomization almost surely preserves thedifferentiability of a given function.
Lemma 2.1.
Given φ ∈ H s ( R d ) , let φ ω be its Wiener randomization defined in (1.2) .Then, there exist C, c > such that P (cid:0) k φ ω k H s > λ (cid:1) ≤ C exp (cid:18) − c λ k φ k H s (cid:19) for all λ > . In fact, one can also show that there is almost surely no smoothing upon randomizationin terms of differentiability (see, for example, Lemma B.1 in [9]). We, however, do not needsuch a non-smoothing result in the following.Next, we state the probabilistic Strichartz estimates. Before doing so, we first recallthe usual Strichartz estimates on R d for readers’ convenience. We say that a pair ( q, r ) isadmissible if 2 ≤ q, r ≤ ∞ , ( q, r, d ) = (2 , ∞ , q + dr = d . (2.1)Then, the following Strichartz estimates are known to hold. See [48, 54, 21, 32]. Lemma 2.2.
Let ( q, r ) be admissible. Then, we have k S ( t ) φ k L qt L rx . k φ k L . As a corollary, we obtain k S ( t ) φ k L pt,x . (cid:13)(cid:13) |∇| d − d +2 p φ (cid:13)(cid:13) L . (2.2)for p ≥ d +2) d , which follows from Sobolev’s inequality and Lemma 2.2. ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , The following lemma shows an improvement of the Strichartz estimates upon the ran-domization of initial data. The improvement appears in the form of integrability and notdifferentiability. Note that such a gain of integrability is classical in the context of randomFourier series [46]. The first estimate (2.3) follows from Minkowski’s integral inequalityalong with Bernstein’s inequality. As for the L ∞ T -estimate (2.4), see [43] for the proof (inthe context of the wave equation). Lemma 2.3.
Given φ on R d , let φ ω be its Wiener randomization defined in (1.2) . Then,given finite q, r ≥ , there exist C, c > such that P (cid:16) k S ( t ) φ ω k L qt L rx ([0 ,T ) × R d ) > λ (cid:17) ≤ C exp (cid:18) − c λ T q k φ k H s (cid:19) (2.3) for all T > and λ > with (i) s = 0 if r < ∞ and (ii) s > if r = ∞ . Moreover, when q = ∞ , given ≤ r ≤ ∞ , there exist C, c > such that P (cid:16) k S ( t ) φ ω k L ∞ t L rx ([0 ,T ) × R d ) > λ (cid:17) ≤ C (1 + T ) exp (cid:18) − c λ k φ k H s (cid:19) (2.4) for all λ > with s > . Function spaces and their basic properties
In this section, we go over the basic definitions and properties of the functions spacesused for the Fourier restriction norm method adapted to the space of functions of bounded p -variation and its pre-dual, introduced and developed by Tataru, Koch, and their collab-orators [36, 23, 26]. We refer readers to Hadac-Herr-Koch [23] and Herr-Tataru-Tzvetkov[26] for proofs of the basic properties. See also [3].Let Z be the set of finite partitions −∞ < t < t < · · · < t K ≤ ∞ of the real line. Byconvention, we set u ( t K ) := 0 if t K = ∞ . Definition 3.1.
Let 1 ≤ p < ∞ . We define a U p -atom to be a step function a : R → L ( R d )of the form a = K X k =1 φ k − χ [ t k − ,t k ) , where { t k } Kk =0 ∈ Z and { φ k } K − k =0 ⊂ L ( R d ) with P K − k =0 k φ k k pL = 1. Furthermore, we definethe atomic space U p = U p ( R ; L ( R d )) by U p := (cid:26) u : R → L ( R d ) : u = ∞ X j =1 λ j a j for U p -atoms a j , { λ j } j ∈ N ∈ ℓ ( N ; C ) (cid:27) with the norm k u k U p := inf (cid:26) ∞ X j =1 | λ j | : u = ∞ X j =1 λ j a j for U p -atoms a j , { λ j } j ∈ N ∈ ℓ ( N ; C ) (cid:27) , where the infimum is taken over all possible representations for u . Definition 3.2.
Let 1 ≤ p < ∞ . (i) We define V p = V p ( R ; L ( R d )) to be the space of functions u : R → L ( R d ) of bounded p -variation with the standard p -variation norm k u k V p := sup { t k } Kk =0 ∈Z (cid:18) K X k =1 k u ( t k ) − u ( t k − ) k pL (cid:19) p . By convention, we impose that the limits lim t →±∞ u ( t ) exist in L ( R d ).(ii) Let V p rc be the closed subspace of V p of all right-continuous functions u ∈ V p withlim t →−∞ u ( t ) = 0.Recall the following inclusion relation; for 1 ≤ p < q < ∞ , U p ֒ → V p rc ֒ → U q ֒ → L ∞ ( R ; L ( R d )) . (3.1)The space V p is the classical space of functions of bounded p -variation and the space U p appears as the pre-dual of V p ′ with p + p ′ = 1. Their duality relation and the atomicstructure of the U p -space turned out to be very effective in studying dispersive PDEs incritical settings.Next, we define the U p - and V p -spaces adapted to the Schr¨odinger flow. Definition 3.3.
Let 1 ≤ p < ∞ . We define U p ∆ := S ( t ) U p and ( V p ∆ := S ( t ) V p , respectively)to be the space of all functions u : R → L ( R d ) such that t → S ( − t ) u ( t ) is in U p (and in V p , respectively) with the norms k u k U p ∆ := k S ( − t ) u k U p and k u k V p ∆ := k S ( − t ) u k V p . The closed subspace V p rc , ∆ is defined in an analogous manner.Next, we define the dyadically defined versions of U p ∆ and V p ∆ . We use the conventionthat capital letters denote dyadic numbers, e.g., N = 2 n for n ∈ N := N ∪ { } . Fix anonnegative even function ϕ ∈ C ∞ (( − , , ϕ ( r ) = 1 for | r | ≤
1. Then, we set ϕ N ( r ) := ϕ ( r/N ) − ϕ (2 r/N ) for N ≥ ϕ ( r ) := ϕ ( r ). Given N ∈ N , let P N denotethe Littlewood-Paley projection operator with the Fourier multiplier ϕ N ( | ξ | ), i.e. P N f := F − [ ϕ N ( | ξ | ) b f ( ξ )]. We also define P ≤ N := P ≤ M ≤ N P M and P >N := Id − P ≤ N . Definition 3.4.
Let s ∈ R . We define X s and Y s as the closures of C ( R ; H s ( R d )) ∩ U and C ( R ; H s ( R d )) ∩ V , ∆ with respect to the norms k u k X s := (cid:18) X N ∈ N N s k P N u k U (cid:19) and k u k Y s := (cid:18) X N ∈ N N s k P N u k V (cid:19) , respectively.The transference principle ([23, Proposition 2.19]) and the interpolation lemma [23,Proposition 2.20] applied on the Strichartz estimates (Lemma 2.2 and (2.2)) imply thefollowing estimate for the Y -space. Lemma 3.5.
Let d ≥ . Then, given any admissible pair ( q, r ) with q > and p ≥ d +2) d ,we have k u k L qt L rx . k u k Y , k u k L pt,x . (cid:13)(cid:13) |∇| d − d +2 p u (cid:13)(cid:13) Y . ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , Similarly, the bilinear refinement of the Strichartz estimate [7, 15] implies the followingbilinear estimate.
Lemma 3.6.
Let N , N ∈ N with N ≤ N . Then, given any ε > , we have k P N u P N u k L t,x . N d − (cid:18) N N (cid:19) − ε k P N u k Y k P N u k Y for all u , u ∈ Y . For our analysis, we need to introduce the local-in-time versions of the spaces definedabove.
Definition 3.7.
Let B be a Banach space consisting of continuous H -valued functions (in t ∈ R ) for some Hilbert space H . We define the corresponding restriction space B ( I ) to agiven time interval I ⊂ R as B ( I ) := (cid:8) u ∈ C ( I ; H ) : there exists v ∈ B such that v | I = u (cid:9) . We endow B ( I ) with the norm k u k B ( I ) := inf (cid:8) k v k B : v | I = u (cid:9) , where the infimum is taken over all possible extensions v of u onto the real line. When I = [0 , T ), we simply set B T := B ( I ) = B ([0 , T )).Recall that the space B ( I ) is a Banach space. As a consequence of (3.1), we have thefollowing inclusion relation; for any interval I ⊂ R , we have X s ( I ) ֒ → Y s ( I ) ֒ → h∇i − s V ( I ) ∩ C ( I ; H s ( R d )) . We conclude this section by stating the linear estimates. Given a ∈ R , we define theintegral operator I a on L ([ a, ∞ ); L ( R d )) by I a [ F ]( t ) := ˆ ta S ( t − t ′ ) F ( t ′ ) dt ′ (3.2)for t ≥ a and I a [ F ]( t ) = 0 otherwise. When a = 0, we simply set I = I a . Given an interval I = [ a, b ), we set the dual norm N s ( I ) controlling the nonhomogeneous term on I by k F k N s ( I ) = (cid:13)(cid:13) I a [ F ] (cid:13)(cid:13) X s ( I ) , we have the following linear estimates. Lemma 3.8.
Let s ∈ R and T ∈ (0 , ∞ ] . Then, the following linear estimates hold: k S ( t ) φ k X sT ≤ k φ k H s , k F k N sT ≤ sup w ∈ Y − sT k w k Y − sT =1 (cid:12)(cid:12)(cid:12)(cid:12) ˆ T h F ( t ) , w ( t ) i L x dt (cid:12)(cid:12)(cid:12)(cid:12) for any φ ∈ H s ( R d ) and F ∈ L ([0 , T ); H s ( R d )) . The first estimate is immediate from the definition of the space X sT . The second esti-mate basically follows from the duality relation between U and V ([23, Proposition 2.10,Remark 2.11]). See also Proposition 2.11 in [26]. Nonlinear estimates
As in Section 1, let z ( t ) = z ω ( t ) = S ( t ) φ ω denote the linear solution with the randomizedinitial data φ ω in (1.2). If u is a solution to (1.1), then the residual term v = u − z satisfiesthe perturbed NLS (1.3). In this section, we establish relevant nonlinear estimates in solvingthe fixed point problem (1.4) for the residual term v .Given d = 5 ,
6, fix an admissible pair:( q d , r d ) := (cid:18) dd − , d d − d + 4 (cid:19) = ((cid:0) , (cid:1) , d = 5 , (cid:0) , (cid:1) , d = 6 . (4.1)Note that d + 2 d − q ′ d = q d , where q ′ d denotes the H¨older conjugate of q d . By Sobolev’s inequality, we have W ,r d ( R d ) ֒ → L ρ d ( R d ) , ρ d := 2 d ( d − = ( , d = 5 , , d = 6 . (4.2)Before we state the main probabilistic nonlinear estimates, let us define the set of indices: S δ := (cid:26)(cid:16) q d − δq d , r d (cid:17) , (cid:16) q d − δq d , d + 2 d − r ′ d (cid:17) , (cid:16) q d − δq d , ρ d (cid:17) , (cid:16) − δ , (cid:17) , (cid:16) , δδ (cid:17)(cid:27) for small δ >
0. Given an interval I ⊂ R and δ >
0, we define S s ( I ) = S s ( I ; δ ) by k u k S s ( I ) := max (cid:8) kh∇i s u k L qt L rx ( I × R d ) : ( q, r ) ∈ S δ (cid:9) . (4.3)Furthermore, given M > I , define the set E M ( I ) ⊂ Ω by E M ( I ) := (cid:8) ω ∈ Ω : k φ ω k H s + k S ( t ) φ ω k S s ( I ) ≤ M (cid:9) . (4.4)When I = [0 , T ), we simply write E M,T = E M ([0 , T )). Proposition 4.1.
Let d = 5 , , − d < s < , and N ( u ) = | u | d − u or N ( u ) = | u | d +2 d − . Given φ ∈ H s ( R d ) , let φ ω be its Wiener randomization defined in (1.2) and z = S ( t ) φ ω .Then, there exist sufficiently small δ = δ ( d, s ) > and θ = θ ( d, s ) > such that kN ( v + z ) k N T ≤ C n k v k d +2 d − Y T + T θ M d +2 d − o , (4.5) kN ( v + z ) − N ( v + z ) k N T ≤ C n k v k d − Y T + k v k d − Y T + T θ M d − o k v − v k Y T , (4.6) for any T > , v, v , v ∈ Y T , and ω ∈ E M,T . As we see below, we fix δ = δ ( d, s ) > δ for simplicity of thepresentation. A similar comment applies to E M ( I ) and e E M ( I ) defined in (4.4) and (6.2). ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , Note that we have k u k X T ∼ k u k X T + k∇ u k X T . (4.7)It is crucial that we handle a regular gradient ∇ rather than h∇i for our purpose. We alsopoint out that once we fix the set E M,T , the nonlinear estimates are entirely deterministic . Proof.
Part 1:
We first prove (4.5). In view of (4.7), Lemma 3.8 and Definition 3.7 of thetime restriction norm, it suffices to show (cid:12)(cid:12)(cid:12)(cid:12) ˆ [0 ,T ] × R d N ( v + z ) · wdxdt (cid:12)(cid:12)(cid:12)(cid:12) . k v k d +2 d − Y + T θ M d +2 d − , (4.8) (cid:12)(cid:12)(cid:12)(cid:12) ˆ [0 ,T ] × R d ∇N ( v + z ) · wdxdt (cid:12)(cid:12)(cid:12)(cid:12) . k v k d +2 d − Y + T θ M d +2 d − , (4.9)for all w ∈ Y with k w k Y = 1 and any ω ∈ E M,T ,.Let us first consider (4.8). H¨older’s inequality and the embedding W d +2 ,r d ( R d ) ֒ → L d +2 d − r ′ d ( R d ) yieldLHS of (4.8) . (cid:13)(cid:13) | v + z | d +2 d − (cid:13)(cid:13) L q ′ dT L r ′ dx k w k L qdT L rdx . k v + z k d +2 d − L qdT L d +2 d − r ′ dx . k v k d +2 d − Y + k z k d +2 d − L qdT L d +2 d − r ′ dx . k v k d +2 d − Y + ( T δ M ) d +2 d − (4.10)for any ω ∈ E M,T , where we used k z k L qdT L d +2 d − r ′ dx ≤ T δ k z k L qd − δqdT L d +2 d − r ′ dx ≤ T δ M. Next, we consider (4.9). The contribution from P ≤ w can be estimated in an analogousmanner to the computation above. Hence, without loss of generality, we assume w = P > w in the following.We first prove (4.9) for N ( u ) = | u | d +2 d − . With ∇ ( | f | α ) = α | f | α − Re( f ∇ f ) , (4.11)the estimate (4.9) is reduced to showing (cid:12)(cid:12)(cid:12)(cid:12) ˆ [0 ,T ] × R d ( ∇ w )( v + z ) | v + z | − dd − wdxdt (cid:12)(cid:12)(cid:12)(cid:12) . k v k d +2 d − Y + T θ M d +2 d − (4.12)for w = v or z . A small but important observation is that a derivative does not fall on thethird factor with the absolute value. In the following, we preform analysis on the relativesizes of the frequencies of the first two factors. Strictly speaking, we need to work with a truncated nonlinearity as in [3] so that Lemma 3.8 is applicable.This modification, however, is standard and we omit details. See [3] for the details. • Case 1: w = v . In this case, from Lemma 3.5 with (4.2) and (4.4), we haveLHS of (4.12) . k∇ v k L qdT L rdx k v + z k L qdT L ρdx (cid:13)(cid:13) | v + z | − dd − (cid:13)(cid:13) L d − dT L d − d )( d − x k w k L qdT L rdx . k v k Y k v + z k d − L qdT L ρdx . k v k Y (cid:8) k v k L qdT W ,rdx + k z k L qdT L ρdx (cid:9) d − . k v k Y n k v k d − Y + ( T δ M ) d − o (4.13)for any ω ∈ E M,T . Then, (4.12) follows from Young’s inequality. • Case 2: w = z . Using the Littlewood-Paley decomposition, we haveLHS of (4.12) . X N ,N ∈ N (cid:12)(cid:12)(cid:12)(cid:12) ˆ [0 ,T ] × R d N P N z P N ( v + z ) | v + z | − dd − wdxdt (cid:12)(cid:12)(cid:12)(cid:12) . Subcase 2.a: We first consider the contribution from N & N d − . Note that we have k z k L qdT ( W s,rdx ∩ L ρdx ) ≤ T δ k z k L qd − δqdT ( W s,rdx ∩ L ρdx ) ≤ T δ M on E M,T . Then, proceeding as in Case 1 with Lemma 3.5, (4.2), and (4.4), we haveLHS of (4.12) . X N ,N ∈ N N & N d − N k P N z k L qdT L ρdx k P N ( v + z ) k L qdT L rdx × (cid:13)(cid:13) | v + z | − dd − (cid:13)(cid:13) L d − dT L d − d )( d − x k w k L qdT L rdx . X N ,N ∈ N N & N d − N − s +11 N − s k P N z k L qdT W s,ρdx (cid:8) k P N v k L qdT W s,rdx + k P N z k L qdT W s,rdx (cid:9) × (cid:8) k v k L qdT W ,rdx + k z k L qdT L ρdx (cid:9) − dd − k w k L qdT L rdx . X N ,N ∈ N N & N d − N − s +11 N − s T δ M n k v k d − Y + ( T δ M ) d − o . k v k d +2 d − Y + ( T δ M ) d +2 d − (4.14)for any ω ∈ E M,T , provided that s > − d .Subcase 2.b: Next, we estimate the contribution from N ≪ N d − . Noting that (cid:0) d (6 − d )( d − , d d − d +6 (cid:1) is an admissible pair, H¨older’s inequality and Lemma 3.5 yield k w k L dd − T L d d − d +6 x ≤ T d − k w k L d (6 − d )( d − T L d d − d +6 x . T d − k w k Y . (4.15) ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , Then, by applying Lemma 3.6 with Lemmas 3.5 and 3.8 and (4.4), we obtainLHS of (4.12) . X N ,N ∈ N N ≪ N d − N k P N z P N ( v + z ) k L T,x × (cid:13)(cid:13) | v + z | − dd − (cid:13)(cid:13) L d − dT L d − d )( d − x k w k L dd − T L d d − d +6 x . T d − X N ,N ∈ N N ≪ N d − N N d − (cid:18) N N (cid:19) − ε k P N z k Y T k P N ( v + z ) k Y T × (cid:8) k v k L qdT W ,rdx + k z k L qdT L ρdx (cid:9) − dd − k w k Y . T d − X N ,N ∈ N N ≪ N d − N − s + + ε N − s + d − − ε M × ( k v k Y s + M ) n k v k − dd − Y + ( T δ M ) − dd − o . T θ ′ M n k v k d − Y + M d − o . k v k d +2 d − Y + T θ M d +2 d − (4.16)for any ω ∈ E M,T , provided that s > − d . This proves (4.5) for N ( u ) = | u | d +2 d − .We now prove (4.9) for N ( u ) = | u | d − u . In this case, we have ∇ ( | f | α − f ) = ( α − | f | α − f | f | Re( f ∇ f ) + | f | α − ∇ f. (4.17)Noting that (cid:12)(cid:12) | f | α − f (cid:12)(cid:12) = | f | α − , we can estimate the first term in (4.17) using (4.12). Itremains to estimate the contribution from the second term in (4.17). Namely, we prove (cid:12)(cid:12)(cid:12)(cid:12) ˆ [0 ,T ] × R d ( ∇ w ) | v + z | d − wdxdt (cid:12)(cid:12)(cid:12)(cid:12) . k v k d +2 d − Y + T θ M d +2 d − (4.18)for w = v or z . When w = v , (4.18) follows from Case 1 above. Hence, we assume that w = z in the following. By writing ( ∇ z ) | v + z | d − = ( ∇ z ) | v + z | · | v + z | − dd − , it follows from Here, we assumed that ∂ { x ∈ R d : f ( x ) = 0 } has measure 0. This assumption can be verified forsmooth truncated P ≤ N z and smooth v N . Then, we can establish the desired estimates for smooth P ≤ N z and v N and take a limit as N → ∞ . Lemma 3.5 and (4.15) with (4.4) thatLHS of (4.18) . k ( ∇ z )( v + z ) k L T,x (cid:13)(cid:13) | v + z | − dd − (cid:13)(cid:13) L d − dT L d − d )( d − x k w k L dd − T L d d − d +6 x . X N ,N ∈ N N k P N z P N ( v + z ) k L T,x × (cid:13)(cid:13) | v + z | − dd − (cid:13)(cid:13) L d − dT L d − d )( d − x k w k L dd − T L d d − d +6 x . T d − n k v k − dd − Y + ( T δ M ) − dd − o X N ,N ∈ N N k P N z P N ( v + z ) k L T,x (4.19)for any ω ∈ E M,T . When N ≪ N d − , we can apply Lemma 3.6 as in Subcase 2.b andestablish (4.18).Let us consider the remaining case N & N d − . As in Subcase 2.a, we have X N ,N ∈ N N & N d − N k P N z P N z k L T,x . X N ,N ∈ N N & N d − N − s +11 N − s k P N z k L T W s, x k P N z k L T W s, x . ( T δ M ) for any ω ∈ E M,T , provided that s > − d . Similarly, it follows from Sobolev’s inequality(with sufficiently small δ > − sd ≥ − δ ) and (4.4) that X N ,N ∈ N N & N d − N k P N z P N v k L T,x . X N ,N ∈ N N & N d − N − s +11 N − s k P N z k L T W s, δδx k P N v k L T W s, δx . T k v k Y M for any ω ∈ E M,T , provided that s > − d . This proves (4.5) for N ( u ) = | u | d − u . Part 2:
Next, we prove the difference estimates (4.6). Our main goal is to prove (cid:12)(cid:12)(cid:12)(cid:12) ˆ [0 ,T ] × R d (cid:8) N ( v + z ) − N ( v + z ) (cid:9) wdxdt (cid:12)(cid:12)(cid:12)(cid:12) . n k v k d − Y + k v k d − Y + T θ M d − o k v − v k Y , (4.20)and (cid:12)(cid:12)(cid:12)(cid:12) ˆ [0 ,T ] × R d (cid:8) ∇N ( v + z ) − ∇N ( v + z ) (cid:9) wdxdt (cid:12)(cid:12)(cid:12)(cid:12) . n k v k d − Y + k v k d − Y + T θ M d − o k v − v k Y (4.21)for all w ∈ Y with k w k Y = 1. In the following, we only consider (4.21) and discuss how toapply the computations in Part 1. The first difference estimate (4.20) follows in a similar,but simpler manner. ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , • Case 3: N ( u ) = | u | d +2 d − . Let F ( ζ ) = F ( ζ, ζ ) = | ζ | − dd − ζ . Then, we have ∂ ζ F = d d − | ζ | − dd − and ∂ ζ F = − d d − | ζ | − dd − ζ | ζ | . (4.22)By Fundamental Theorem of Calculus, we have F ( v + z ) − F ( v + z ) = ˆ ∂ ζ F ( v + z + θ ( v − v ))( v − v )+ ∂ ζ F ( v + z + θ ( v − v ))( v − v ) dθ. (4.23)Then, from (4.11) and (4.23), we have ∇ ( | v + z | d +2 d − ) − ∇ ( | v + z | d +2 d − )= d +2 d − Re (cid:8) F ( v + z ) ∇ ( v + z ) − F ( v + z ) ∇ ( v + z ) (cid:9) = d +2 d − Re (cid:26) F ( v + z ) ∇ ( v − v )+ ˆ ∂ ζ F ( v + z + θ ( v − v ))( v − v ) dθ · ∇ ( v + z )+ ˆ ∂ ζ F ( v + z + θ ( v − v ))( v − v ) dθ · ∇ ( v + z ) (cid:27) . (4.24)The contribution to (4.21) from the first term on the right-hand side of (4.24) canbe estimated as in (4.12). As for the second term on the right-hand side of (4.24), theestimate (4.21) is reduced to ˆ (cid:12)(cid:12)(cid:12)(cid:12) ˆ [0 ,T ] × R d ( ∇ w )( v − v ) · (cid:12)(cid:12) v + z + θ ( v − v ) (cid:12)(cid:12) − dd − wdxdt (cid:12)(cid:12)(cid:12)(cid:12) dθ . n k v k d − Y + k v k d − Y + T θ M d − o k v − v k Y for w = v or z , which once again follows from (4.12) in Part 1. In view of (4.22), we have | ∂ ζ F | ∼ | ζ | − dd − . Hence, the third term on the right-hand side of (4.24) can be estimated ina similar manner. • Case 4: N ( u ) = | u | d − u . In view of (4.17), there are two contributions to ∇N ( v + z ) − ∇N ( v + z ) . Let G ( ζ ) = G ( ζ, ζ ) = | ζ | − dd − ζ . Then, we have ∂ ζ G = dd − | ζ | − dd − ζ | ζ | and ∂ ζ G = − dd − | ζ | − dd − ζ | ζ | . (4.25)Next, let H ( z ) = H ( ζ, ζ ) = | ζ | d − . Then, we have ∂ ζ H = d − | ζ | − dd − ζ | ζ | and ∂ ζ H = d − | ζ | − dd − ζ | ζ | . (4.26)Then, from (4.17), (4.25), and (4.26), we have ∇N ( v + z ) − ∇N ( v + z )= d − Re (cid:8) G ( v + z ) ∇ ( v + z ) − G ( v + z ) ∇ ( v + z ) (cid:9) + H ( v + z ) ∇ ( v + z ) − H ( v + z ) ∇ ( v + z ) . Noting that | ∂ ζ G | ∼ | ∂ ζ G | ∼ | ∂ ζ H | ∼ | ∂ ζ H | ∼ | ζ | − dd − , we can use (4.23) with G and H replacing F and repeat the computation in Part 1 toestablish (4.21). This completes the proof of Proposition 4.1. (cid:3) Proof of Theorems 1.1 and 1.2
We present the proof of Theorems 1.1 and 1.2. Namely, we solve the following fixed pointproblem: v = − i I [ N ( v + z )] , where N ( u ) = | u | d − u or N ( u ) = | u | d +2 d − . Let η > C η d − ≤ C η d − ≤ , where C and C are the constants in (4.5) and (4.6). Given M >
0, we set T := min n(cid:0) ηM (cid:1) d +2 d − , (cid:0) ηM (cid:1) d − o θ . (5.1)Then, it follows from Proposition 4.1 with X T ֒ → Y T that for each ω ∈ E M,T , the mapping v
7→ − i I [ N ( v + z )] is a contraction on the ball B η ⊂ X T defined by B η := { v ∈ X T : k v k X T ≤ η } . Moreover, it follows from Lemmas 2.1 and 2.3 with (5.1) imply the following tail estimate: P (Ω \ E M,T ) ≤ C exp (cid:18) − c M k φ k H s (cid:19) + C exp (cid:18) − c M T γ k φ k H s (cid:19) ≤ C exp (cid:18) − cT γ k φ k H s (cid:19) for some γ >
0. This proves almost sure local well-posedness of (1.1) and (1.5).6.
A variant of almost sure local well-posedness
In this section, we briefly discuss the proof of Proposition 1.3. In particular, we considerthe perturbed NLS (1.9) with a non-zero initial condition v . This will be useful in provingTheorems 1.5 and 1.7. As in [3], we consider the following Cauchy problem for NLS witha perturbation: ( i∂ t v + ∆ v = N ( v + f ) ,v | t =0 = v ∈ H ( R d ) , (6.1)where f is a given deterministic function, satisfying certain regularity conditions. Thisallows us to separate the probabilistic and deterministic components of the argument in aclear manner.First, note that, since our initial condition is not 0, the Y T -norm of the solution v doesnot tend to 0 even when T →
0. Hence, we need to use an auxiliary norm that tends to 0 as T →
0. As a corollary to (the proof of) Proposition 4.1, we obtain the following nonlinearestimates, which are stated for a general time interval I ⊂ R . Note that all the terms onthe right-hand side in the first estimate (6.3) have (i) two factors of the L q d t ( I ; W ,r d x )-norm ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , of v (which is weaker than the X ( I )-norm) or (ii) a factor of | I | θ , which can be made smallby shrinking the interval I .In the following, let ( q d , r d ) be the admissible pair defined in (4.1). Given δ > M > I , define e E M ( I ) by e E M ( I ) := (cid:8) f ∈ Y s ( I ) ∩ S s ( I ) : k f k Y s ( I ) + k f k S s ( I ) ≤ M (cid:9) , (6.2)where S s ( I ) = S s ( I ; δ ) is as in (4.3). When I = [0 , T ), we simply write e E M,T = e E M ([0 , T )). Corollary 6.1.
Let d = 5 , , − d < s < , and N ( u ) = | u | d − u or N ( u ) = | u | d +2 d − . Then, there exist sufficiently small δ = δ ( d, s ) > and θ = θ ( d, s ) > such that k [ N ( v + f ) k N ( I ) . k v k d +2 d − L qdt ( I ; W ,rdx ) + | I | θ M d +2 d − + | I | θ M k v k − dd − L qdt ( I ; W ,rdx ) k v k Y ( I ) , (6.3) kN ( v + f ) − N ( v + f ) k N ( I ) . n k v k d − L qdt ( I ; W ,rdx ) + k v k d − L qdt ( I ; W ,rdx ) + | I | θ M d − o k v − v k Y ( I ) , (6.4) for any interval I ⊂ R , v, v , v ∈ Y ( I ) , and f ∈ e E M ( I ) .Proof. This corollary follows from the proof of Proposition 4.1 simply by not applying theStrichartz estimates (Lemma 3.5). In particular, a small modification to (4.10), (4.13),and (4.14) yields (6.3) for the corresponding cases, where the left-hand side is controlledby the first two terms on the right-hand side of (6.3). In (4.16) and (4.19), the subcriticalnature of the perturbation f allows us to gain a small power of | I | through (4.15). Hence, weobtain (6.3), where the left-hand side is controlled by the last two terms on the right-handside of (6.3). The difference estimate (6.4) also follows from a similar modification. (cid:3) By following the proof of Proposition 6.3 in [3], we obtain the following almost sure localwell-posedness of the perturbed NLS (6.1) with non-zero initial data. Proposition 1.3 inSection 1 then follows from this lemma with Lemmas 2.1 and 2.3 by setting f = z ω = S ( t ) φ ω . Lemma 6.2.
Assume the hypotheses of Corollary 6.1. Given
M > , let e E M ( · ) be asin (6.2) and let θ > be as in Corollary 6.1. Then, there exists small η = η ( k v k H , M ) > such that if k S ( t − t ) v k L qdt ( I ; W ,rdx ) ≤ η and | I | ≤ η θ for some η ≤ η and some time interval I = [ t , t ] ⊂ R , then for any f ∈ e E M ( I ) , thereexists a unique solution v ∈ X ( I ) ∩ C ( I ; H ( R d )) to (1.9) with v | t = t = v , satisfying k v k L qdt ( I ; W ,rdx ) ≤ η, k v − S ( t − t ) v k X ( I ) . η. Proof.
As mentioned above, one can prove Lemma 6.2 by following the proof of Proposi-tion 6.3 in [3]. More precisely, by applying Corollary 6.1 and choosing η ≪ e R − d +2 d − with e R := max( k v k H , M ), a straightforward computation shows that the map Γ definedby Γ v ( t ) := S ( t − t ) v − i ˆ tt S ( t − t ′ ) N ( v + f )( t ′ ) dt ′ is a contraction on B R,M,η = (cid:8) v ∈ X ( I ) ∩ C ( I ; H ) : k v k X ( I ) ≤ e R, k v k L qdt ( I ; W ,rdx ) ≤ η (cid:9) , provided that f ∈ e E M ( I ). (cid:3) Lastly, note that Lemma 6.2 yields the following blowup alternative. Suppose that thereexists M ( t ) such that f ∈ e E M ( t ) ([0 , t )) for each t >
0. Then, given v ∈ H ( R d ), let v bethe solution to the perturbed NLS (6.1) with v | t =0 = v on a forward maximal time interval[0 , T ∗ ) of existence. Then, either T ∗ = ∞ orlim T → T ∗ k v k L qdt ([0 ,T ); W ,rdx ) = ∞ . (6.5)In view of Lemma 6.2, this blowup alternative follows from a standard argument as in [11].In fact, suppose T ∗ < ∞ and A ∗ := lim T → T ∗ k v k L qdt ([0 ,T ); W ,rdx ) < ∞ . Then, we will derive a contradiction in the following.Without loss of generality, assume that M ( t ) is non-decreasing and set M ∗ := sup t ∈ [0 ,T ∗ +1] M ( t ) < ∞ . (6.6)Partition the interval [0 , T ∗ ] as [0 , T ∗ ] = J [ j =0 I j ∩ [0 , T ∗ ]where I j = [ t j , t j +1 ] with t = 0 and t J +1 = T ∗ . From (6.3) in Corollary 6.1 withLemma 3.8, we have k v k X ( I j ) ≤ k v ( t j ) k H + kN ( v + z ) k N ( I j ) ≤ k v ( t j ) k H + C ( T ∗ , A ∗ , M ∗ ) + | I j | θ M ∗ ( A ∗ ) − dd − k v k X ( I j ) . Hence by imposing that the lengths of the subintervals I j are sufficiently small, dependingonly on A ∗ and M ∗ , we obtainsup t ∈ I j k v ( t ) k H . k v k X ( I j ) . k v ( t j ) k H + C ( T ∗ , A ∗ , M ∗ ) , (6.7)where the implicit constants are independent of j = 0 , , . . . , J . By iteratively applying theestimate (6.7), we obtain R ∗ := sup t ∈ [0 ,T ∗ ] k v ( t ) k H ≤ C ( T ∗ , A ∗ , M ∗ ) < ∞ . (6.8)Then, combining (6.7) and (6.8), we obtain k v k X ( I j ) ≤ C ( T ∗ , A ∗ , M ∗ ) < ∞ (6.9)uniformly in j = 0 , , . . . , J . ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , Given e η > k v k L qdt ( I j ; W ,rdx ) < e η. (6.10)Fix η = η ( R ∗ , M ∗ ) >
0, where η is as in Lemma 6.2 and R ∗ and M ∗ are as in (6.8)and (6.6). Then, by taking the L q d t ( I j ; W ,r d x )-norm of the Duhamel formulation: S ( t − t j ) v ( t j ) = v ( t ) + i ˆ tt j S ( t − t ′ ) N ( v + f ) dt ′ , applying Corollary 6.1 with (6.9) and the smallness condition (6.10), and taking e η = e η ( η ) = e η ( R ∗ , M ∗ ) > | I j | = | I j | ( T ∗ , A ∗ , M ∗ , η ) sufficiently small, we have k S ( t − t j ) v ( t j ) k L qdt ( I j ; W ,rdx ) ≤ e η + C e η d +2 d − + C ( T ∗ , A ∗ , M ∗ ) | I j | θ ≤ η . In particular, with j = J , this implies that there exists some ε > k S ( t − t J ) v ( t J ) k L qdt ([ t J ,T ∗ + ε ]; W ,rdx ) ≤ η . By further imposing that | I J | ≤ η θ , we conclude from Lemma 6.2 that the solution v can be extended to [0 , T ∗ + ε ] for some ε >
0, which is a contradiction to the assumption T ∗ < ∞ . Therefore, if T ∗ < ∞ , then we must have (6.5). Remark 6.3.
Suppose T ∗ < ∞ . Then, it follows from the argument above with Lemma 6.2and the subadditivity of the X -norm over disjoint intervals (Lemma A.4 in [3]) that v ∈ X ([0 , T ∗ − δ )) for any δ >
0. If T ∗ = ∞ , we have v ∈ X ([0 , T )) for any finite T >
Almost sure global well-posedness of the defocusing energy-criticalNLS below the energy space
In this section, we present the proof of Theorem 1.5. Namely, we prove almost sureglobal well-posedness of the defocusing energy-critical NLS on R d , d = 5 , ( i∂ t u + ∆ u = | u | d − u,u | t =0 = φ ω , ( t, x ) ∈ R × R d . (7.1)where φ ω is the Wiener randomization of a given function φ ∈ H s ( R d ) for some s < ( i∂ t v + ∆ v = | v + f | d − ( v + f ) v | t =0 = 0 . (7.2)Under a suitable regularity assumption on f , Lemma 6.2 guarantees local existence ofsolutions to (7.2). In the following, we assume(i) f is a linear solution f = S ( t ) ψ for some deterministic initial condition ψ ,(ii) f satisfies certain space-time integrability conditions. Under these assumptions, we first establish crucial energy estimates (Proposition 7.2 for d = 6 and Proposition 7.3 for d = 5) for a solution v to the perturbed NLS (7.2). This is themain new ingredient in this paper as compared to [3]. Once we have these energy estimates,we can proceed as in [3] and hence we only sketch the argument. Fix an interval [0 , T ).Given t ∈ [0 , T ), we iteratively apply the perturbation lemma (Lemma 7.4) on short timeintervals I j = [ t j , t j +1 ] and approximate a solution v to the perturbed NLS (7.2) by theglobal solution w to the original NLS (7.1) with w | t = t = v ( t ). This allows us to show thatthe solution v to the perturbed NLS (7.2) exists on [ t , t + τ ], where τ is independent of t ∈ [0 , T ) (Proposition 7.5). By iterating this “good” local well-posedness, we can extendthe solution v to the entire interval [0 , T ]. Since the choice of T > f for long time existence are satisfied with a large probabilityby setting f ( t ) = z ( t ) = S ( t ) φ ω . This yields Theorem 1.5.7.1. Energy estimate for the perturbed NLS.
First, we discuss the following a priori control on the mass. Lemma 7.1.
Let v be a solution to (7.2) with f = S ( t ) ψ . Then, we have ˆ | v ( t ) | dx . ˆ | ψ | dx, (7.3) where the implicit constant is independent of t ∈ R .Proof. Note that u = v + f satisfies (7.1). Hence, by the mass conservation for (7.1), wehave ˆ | ψ | dx = ˆ | v ( t ) + f ( t ) | dx = ˆ | v ( t ) | dx + 2 Re ˆ v ( t ) f ( t ) dx + ˆ | f ( t ) | dx. By the unitarity of the linear solution operator, we obtain ˆ | v ( t ) | dx = − ˆ v ( t ) f ( t ) dx ≤ ˆ | v ( t ) | dx + 2 ˆ | f ( t ) | dx. By invoking the unitarity of the linear solution operator once again, we obtain (7.3). (cid:3)
Next, we establish an energy estimate when d = 6. Recall the following conserved energyfor NLS (7.1): E ( u ) = 12 ˆ |∇ u | dx + 13 ˆ | u | dx. In the following, we estimate the growth of E ( v ) for a solution v to the perturbed NLS (7.2). Proposition 7.2.
Let d = 6 and s > . Then, the following energy estimate holds for asolution v to the perturbed NLS (7.2) with f = S ( t ) ψ : ∂ t E ( v )( t ) . (cid:0) k f ( t ) k L ∞ x (cid:1) E ( v )( t ) + k f ( t ) k L x + k f ( t ) ∇ f ( t ) k L x + k v ( t ) ∇ f ( t ) k L x . (7.4) In Lemma 7.1 and Propositions 7.2 and 7.3, we prove a priori estimates for a smooth solution v withsmooth ψ and hence f . By the standard argument via the local theory, one can show that these a prioriestimates also hold for rough solutions as long as they exist. ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , In particular, given
T > , we have sup t ∈ [0 ,T ] E ( v )( t ) ≤ C (cid:0) T, k f k A s ( T ) (cid:1) (7.5) for any solution v ∈ C ([0 , T ]; H ( R )) to the perturbed NLS (7.2) with f = S ( t ) ψ , wherethe A s ( T ) -norm is defined by k f k A s ( T ) := max (cid:16) kh∇i s − f k L ∞ T,x , k f k L T,x , k f k L T W s, x , k f k L T L x , k ψ k L x , k f k Y sT (cid:17) . Proof.
We first prove (7.4). Since we work for fixed t , we suppress the t -dependence in thefollowing. Noting that ∂ t ( | v | ) = 3 | v | Re( v∂ t v ), we have ∂ t E ( v ) = − Re i ˆ ∆ v ∆ vdx | {z } =0 + Re i ˆ | v + f | ( v + f )∆ vdx + Re i ˆ ∆ v | v | vdx − Re i ˆ | v + f | ( v + f ) | v | vdx = Re i ˆ (cid:8) | v + f | ( v + f ) − | v | v (cid:9) ∆ vdx − Re i ˆ | v + f | ( v + f ) | v | vdx =: I + II . (7.6)By Young’s inequality, we haveII = − Re i ˆ | v + f || v | dx | {z } =0 − Re i ˆ | v + f | · f · | v | vdx . (1 + k f k L ∞ x ) ˆ | v | dx + k f k L x . (1 + k f k L ∞ x ) E ( v ) + k f k L x . (7.7)Integrating by parts, we haveI = − Re i ˆ ∇ (cid:8) | v + f | ( v + f ) − | v | v (cid:9) · ∇ vdx. (7.8)Then, from (4.17), (4.25), and (4.26), we have ∇N ( v + f ) − ∇N ( v )= Re (cid:8) G ( v + f ) ∇ ( v + f ) − G ( v ) ∇ v (cid:9) + H ( v + f ) ∇ ( v + f ) − H ( v ) ∇ v = Re (cid:8) G ( v + f ) ∇ f (cid:9) + Re (cid:8) ( G ( v + f ) − G ( v )) ∇ v (cid:9) + H ( v + f ) ∇ f + ( H ( v + f ) − H ( v )) ∇ v, (7.9)where G ( ζ ) = ζ | ζ | and H ( ζ ) = | ζ | are as in (4.25) and (4.26) (with d = 6), respectively.Let us denote by I j , j = 1 , . . . ,
4, the contribution to I in (7.8) from the j th term on theright-hand side of (7.9).Proceeding as in (4.23), we have G ( v + f ) − G ( v ) = ˆ ∂ ζ G ( v + θf ) · f + ∂ ζ G ( v + θf ) · f dθ,H ( v + f ) − H ( v ) = ˆ ∂ ζ H ( v + θf ) · f + ∂ ζ H ( v + θf ) · f dθ. Then, it follows from (4.25) and (4.26) that k G ( v + f ) − G ( v ) k L ∞ x + k H ( v + f ) − H ( v ) k L ∞ x . k f k L ∞ x . (7.10)Hence, from (7.8), (7.9), and (7.10), we have | I + I | . k f k L ∞ x k∇ v k L x . k f k L ∞ x E ( v ) . (7.11)Note that | G ( ζ ) | = | H ( ζ ) | = | ζ | . Then, integrating by parts (in x ), we have | I + I | . k∇ v k L x + k ( v + f ) ∇ f k L x . E ( v ) + k f ∇ f k L x + k v ∇ f k L x . (7.12)Hence, (7.4) follows from (7.6), (7.7), (7.11), and (7.12).Next, we discuss the second estimate (7.5). By solving the differential inequality (7.4)with v | t =0 = 0 in a crude manner, we obtain E ( v )( τ ) ≤ C ˆ τ e C (1+ k f k L ∞ T,x )( τ − t ) n k f ( t ) k L x + k f ( t ) ∇ f ( t ) k L x + k v ( t ) ∇ f ( t ) k L x o dt ≤ Ce C (1+ k f k L ∞ T,x ) T n k f k L τ,x + k f ∇ f k L τ,x + k v ∇ f k L τ,x o (7.13)for any τ ∈ [0 , T ]. The estimate (7.13) is by no means sharp. It, however, suffices for ourpurpose.We can estimate k f ∇ f k L τ,x as in the proof of Proposition 4.1. Namely, by writing k f ∇ f k L τ,x ≤ X N ,N ∈ N N k P N f P N f k L τ,x , (7.14)we separate the estimate into two cases (i) N & N and (ii) N ≪ N . Then, we canestimate the contribution from (i) by k f k L τ W s, x for s > , while we can apply Lemma 3.6and estimate the contribution from (ii) by k f k Y sτ for s > . Hence, we obtain k f ∇ f k L τ,x . k f k L τ W s, x + k f k Y sτ , (7.15)provided that s > .Next, we consider k v ∇ f k L τ L x . By writing k v ∇ f k L τ,x ≤ X N ,N ∈ N N k P N v P N f k L τ,x , we divide the argument into the following two cases:(i) N & N γ and (ii) N ≪ N γ R d , d = 5 , for some γ > N & N γ .By interpolation and Lemma 7.1, we have X N ,N ∈ N N & N γ N k P N v P N f k L τ,x . X N ,N ∈ N N & N γ N − N − γ +2 k P N v P N f k L τ,x . X N ,N ∈ N N & N γ k P N h∇i − v P N h∇i s − f k L τ,x ≤ C ( T ) n sup t ∈ [0 ,τ ] (cid:0) E ( v )( t ) (cid:1) − k ψ k L x + k ψ k L x o kh∇i s − f k L ∞ τ,x (7.16)for any τ ∈ [0 , T ], provided that s > − γ. (7.17)We now turn our attention to (ii) N ≪ N γ . Recall that ( q, r ) = (2 ,
3) is admissi-ble. Hence, by Lemma 3.6, the Duhamel formula (with v | t =0 = 0), the linear estimate(Lemma 3.8) and the Strichartz estimates (Lemma 3.5), we have k P N v P N f k L τ,x . N − N − +2 k P N v k Y τ k P N f k Y τ . N − N − +2 (cid:13)(cid:13)(cid:13)(cid:13) P N ˆ t S ( t − t ′ ) | v + f | ( v + f )( t ′ ) dt ′ (cid:13)(cid:13)(cid:13)(cid:13) Y τ k P N f k Y τ . N − N − +2 (cid:0) k v k L τ L x + k f k L τ L x (cid:1) k P N f k Y τ (7.18)Fix θ ∈ (0 ,
1) (to be chosen later). We apply (7.18) only to the θ -power of the factor in X N ,N ∈ N N ≪ N γ N k P N v P N f k L τ,x . Then, with (7.18), we have X N ,N ∈ N N ≪ N γ N k P N v P N f k L τ,x . X N ,N ∈ N N ≪ N γ N − θ +2 (cid:0) k v k L τ L x + k f k L τ L x (cid:1) θ k P N f k θY τ k N θ − θ − P N v P N f k − θL τ,x By interpolation, . X N ,N ∈ N N ≪ N γ (cid:0) k v k L τ L x + k f k L τ L x (cid:1) θ k P N f k θY sτ k P N v k − θ + L τ,x × k P N h∇i v k θ − L τ,x k P N h∇i s − f k − θL ∞ τ,x , provided that 1 − θ < s. (7.19) Summing over N and N and applying Lemma 7.1, we obtain k v ∇ f k L τ,x ≤ C ( T, k f k A s ( T ) ) n t ∈ [0 ,τ ] (cid:0) E ( v )( t ) (cid:1) − o (7.20)for any τ ∈ [0 , T ], provided that 43 θ + 52 θ < . (7.21)Optimizing (7.17), (7.19), and (7.21), we obtain s > θ = − and γ = 1 − s +.Finally, putting (7.13), (7.15), (7.16), and (7.20) together with v | t =0 = 0, we obtainsup t ∈ [0 ,τ ] E ( v )( t ) ≤ C ( T, k f k A s ( T ) ) n t ∈ [0 ,τ ] (cid:0) E ( v )( t ) (cid:1) − o for any τ ∈ [0 , T ]. Then, (7.5) follows from the standard continuity argument. (cid:3) We conclude this subsection by establishing an energy estimate when d = 5. As men-tioned in Section 1, we study the growth of the following modified energy: E ( v ) = 12 ˆ |∇ v | dx + 310 ˆ | v + f | dx for a solution v to the perturbed NLS (7.2). Proposition 7.3.
Let d = 5 and s > . Then, the following energy estimate holds: given T > , we have sup t ∈ [0 ,T ] E ( v )( t ) ≤ C (cid:0) T, k f k B s ( T ) (cid:1) (7.22) for any solution v ∈ C ([0 , T ]; H ( R )) to the perturbed NLS (7.2) with f = S ( t ) ψ , wherethe B s ( T ) -norm is defined by k f k B s ( T ) := max p = , , q =2 , (cid:16) kh∇i s − f k L ∞ T,x , kh∇i s f k L pT,x , k f k L T L x , k f k L ∞ T L qx , k f k Y sT (cid:17) . The proof of Proposition 7.3 is similar to that of Proposition 7.2 but is more complicateddue to the (higher) fractional power of the nonlinearity.
Proof.
Proceeding as in (7.6) with ∂ t (cid:0) | v + f | (cid:1) = | v + f | Re (cid:0) ( v + f ) ∂ t ( v + f ) (cid:1) , we have ∂ t E ( v ) = Re i ˆ | v + f | ( v + f )∆ vdx + Re i ˆ (∆ v + ∆ f ) | v + f | ( v + f ) dx − Re i ˆ | v + f | dx | {z } =0 = Re i ˆ ∇ (cid:0) | v + f | ( v + f ) (cid:1) · ∇ f dx ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , With (4.17), = 43 Re i ˆ v + f | v + f | Re (cid:0) ( v + f ) ∇ ( v + f ) (cid:1) · ∇ f dx + Re i ˆ | v + f | ∇ ( v + f ) · ∇ f dx = 53 Re i ˆ | v + f | ∇ v · ∇ f dx + 23 Re i ˆ ( v + f ) | v + f | ∇ v · ∇ f dx + 23 Re i ˆ ( v + f ) | v + f | ∇ f · ∇ f dx . k∇ v k L x + (cid:13)(cid:13) | v + f | ∇ f (cid:13)(cid:13) L x + (cid:13)(cid:13) | v + f | ∇ f · ∇ f (cid:13)(cid:13) L x . E ( v ) + (cid:13)(cid:13) | v + f | ∇ f (cid:13)(cid:13) L x + (cid:13)(cid:13) | v + f | ∇ f · ∇ f (cid:13)(cid:13) L x . (7.23)By solving the differential inequality (7.23) with v | t =0 = 0 in a crude manner, we obtain E ( v )( τ ) . ˆ τ e C ( τ − t ) n(cid:13)(cid:13) | v + f | ∇ f (cid:13)(cid:13) L x + (cid:13)(cid:13) | v + f | ∇ f · ∇ f (cid:13)(cid:13) L x o dt ≤ e CT n(cid:13)(cid:13) | v + f | ∇ f (cid:13)(cid:13) L τ,x + (cid:13)(cid:13) | v + f | ∇ f · ∇ f (cid:13)(cid:13) L τ,x o =: e CT (cid:8) I + II (cid:9) (7.24)for any τ ∈ [0 , T ],We first consider I . By H¨older’s inequality, we have (cid:13)(cid:13) | v + f | ∇ f (cid:13)(cid:13) L τ,x . (cid:13)(cid:13) | f | ∇ f (cid:13)(cid:13) L τ,x + (cid:13)(cid:13) | v | ∇ f (cid:13)(cid:13) L τ,x . k f k L ∞ τ,x k f ∇ f k L τ,x + k v k L ∞ τ L x k v ∇ f k L τ L x . (7.25)Arguing as in (7.14), we have k f ∇ f k L τ,x . k f k L τ W s, x + k f k Y sτ , (7.26)provided that s > . On the other hand, by the dyadic decomposition, we have k v k L ∞ τ L x k v ∇ f k L τ L x . n sup t ∈ [0 ,τ ] (cid:0) E ( v )( t ) (cid:1) + k f k L ∞ τ L x o × X N ,N ∈ N N k P N v P N f k L τ L x . (7.27) Then, by interpolation, we have N k P N v P N f k L τ L x ≤ N k P N v P N f k L τ,x k P N v P N f k L τ L x ≤ N k P N v P N f k L τ,x k P N v k L ∞ τ L x k P N f k L τ L ∞ x . n sup t ∈ [0 ,τ ] (cid:0) E ( v )( t ) (cid:1) + k P N f k L ∞ τ L x o × N k P N v P N f k L τ,x k P N f k L τ L ∞ x . (7.28)We now divide the argument into the following two cases:(i) N & N γ and (ii) N ≪ N γ for some γ ∈ (0 ,
1) (to be chosen later). We first estimate the contribution from (i) N & N γ .By interpolation and Lemma 7.1, we have N k P N v P N f k L τ,x k P N f k L τ L ∞ x . N − N − γ +2 k P N v P N f k L τ,x k P N f k L τ L ∞ x . k P N h∇i − v P N h∇i s − f k L τ,x kh∇i s − f k L τ L ∞ x ≤ C ( T ) n sup t ∈ [0 ,τ ] (cid:0) E ( v )( t ) (cid:1) − k ψ k L x + k ψ k L x o kh∇i s − f k L ∞ τ,x (7.29)for any τ ∈ [0 , T ], provided that s > − γ. (7.30)Next, we consider (ii) N ≪ N γ . Recall that ( q, r ) = (cid:0) , (cid:1) is admissible. Then,proceeding as in (7.18) with Lemma 3.6, the Duhamel formula (with v | t =0 = 0), the linearestimate (Lemma 3.8) and the Strichartz estimates (Lemma 3.5), we have k P N v P N f k L τ,x . N − N − +2 k P N v k Y τ k P N f k Y τ . N − N − +2 (cid:0) k v k L τ L x + k f k L τ L x (cid:1) k P N f k Y τ (7.31)As in the proof of Proposition 7.2, we apply (7.31) only to the θ -power for some θ ∈ (0 , N k P N v P N f k L τ L x k P N f k L τ L ∞ x . N − θ +2 (cid:0) k v k L τ L x + k f k L τ L x (cid:1) θ k P N f k θY τ × k N θ − θ − P N v P N f k (1 − θ ) L τ,x k P N f k L τ L ∞ x In the following, we drop the summation over N and N for conciseness of the presentation. Notethat we can simply sum over N and N at the end by losing an ε -amount of derivative. Similar commentsapply to other dyadic summations. ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , By interpolation and Lemma 7.1, . C ( T ) (cid:0) k v k L τ L x + k f k L τ L x (cid:1) θ k P N f k θY sτ k P N f k − θ L τ,x × k P N h∇i v k θL ∞ τ L x k P N h∇i s − f k − θL ∞ τ,x (7.32)for any τ ∈ [0 , T ], provided that 1 − θ < s. (7.33)Hence, from (7.25), (7.26), (7.27), (7.28), (7.29), and (7.32), we obtainI = (cid:13)(cid:13) | v + f | ∇ f (cid:13)(cid:13) L τ,x ≤ C ( T, k f k B s ( T ) ) sup t ∈ [0 ,τ ] n (cid:0) E ( v )( t ) (cid:1) − o (7.34)for any τ ∈ [0 , T ], provided that 12 + 710 θ + θ < . In particular, by choosing θ = − and γ = θ , it follows from (7.30) and (7.33) that theestimate (7.34) holds for s > ≈ . . (7.35)Next, we estimate II in (7.23). By symmetry, we haveII = (cid:13)(cid:13) | v + f | ∇ f · ∇ f (cid:13)(cid:13) L τ,x . X N ,N ∈ N N ≥ N N N (cid:13)(cid:13) | v + f | P N f · P N f (cid:13)(cid:13) L τ,x . X N ,N ∈ N N ≥ N N N (cid:13)(cid:13) | v | P N f · P N f (cid:13)(cid:13) L τ,x + X N ,N ∈ N N ≥ N N N k f k L τ,x k f P N f k L τ,x k P N f k L τ,x =: II + II . We first estimate II . By the dyadic decomposition, we haveII = X N N ,N ∈ N N ≥ N N N k f k L τ,x k P N f P N f k L τ,x k P N f k L τ,x ≤ X N N ,N ∈ N N ≥ N N − s k f k L τ,x k P N f P N h∇i s f k L τ,x k P N h∇i s f k L τ,x ≤ C ( T, k f k B s ( T ) ) X N N ∈ N N − s +2 k P N f P N h∇i s f k L τ,x for any τ ∈ [0 , T ]. If N & N γ for some γ ∈ (0 , N − s +2 k P N f P N h∇i s f k L τ,x . N − N − s − γs +2 k P N h∇i s f P N h∇i s f k L τ,x . k f k L τ W s, x , (7.36) provided that 2 − s − γs <
0, namely s >
22 + γ . (7.37)If N ≪ N γ , then by applying Lemma 3.6, we have N − s +2 k P N f P N h∇i s f k L τ,x . N − s − N − s +2 k P N f k Y sτ k P N f k Y sτ ≪ N − s + γ (2 − s )+2 k P N f k Y sτ k P N f k Y sτ ≪ k f k Y sτ , (7.38)provided that − s + γ (2 − s ) <
0, namely s > γ γ . (7.39)It follows from (7.36) and (7.38) with (7.37) and (7.39) thatII ≤ C ( T, k f k B s ( T ) ) (7.40)for any τ ∈ [0 , T ], provided that s > ≈ . . (7.41)Finally, we estimate II . By H¨older’s inequality, we haveII . X N ,N ∈ N N ≥ N N − s k v k L τ,x k v P N h∇i s − f k L τ,x k P N h∇i s f k L τ,x . (7.42)In the following, we estimate k v P N h∇i s − f k L τ,x . X N ∈ N k P N v P N h∇i s − f k L τ,x . If N & N γ for some γ ∈ (0 , N − s +2 k v P N h∇i s − f k L τ,x . N − N − s − γ +2 k v P N h∇i s − f k L τ,x . C ( T ) kh∇i v k L ∞ τ L x kh∇i s − f k L ∞ τ,x (7.43)for any τ ∈ [0 , T ], provided that 2 − s < γ < N ≪ N γ , then by applying (7.31) to the θ -power k v P N h∇i s − f k L τ,x as before, wehave N − s +2 k v P N h∇i s − f k L τ,x . N − s − θ +2 (cid:0) k v k L τ L x + k f k L τ L x (cid:1) θ k P N f k θY sτ × k N θ − − θ P N v k − θL τ,x k P N h∇i s − f k − θL ∞ τ,x ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , By interpolation and Lemma 7.1, ≤ C ( T ) N − s − θ +2 (cid:0) k v k L τ L x + k f k L τ L x (cid:1) θ k P N f k θY sτ k f k − θ + L ∞ L x × k P N h∇i v k θ − L τ,x k P N h∇i s − f k − θL ∞ τ,x ≤ C ( T, k f k B s ( T ) ) (cid:0) k v k θL ∞ τ L x (cid:1) kh∇i v k θ − L ∞ τ L x (7.44)for any τ ∈ [0 , T ], provided that s > − θ . (7.45)Putting (7.42), (7.43), and (7.44) together, we obtainII ≤ C ( T, k f k B s ( T ) ) sup t ∈ [0 ,τ ] n (cid:0) E ( v )( t ) (cid:1) − o (7.46)by choosing θ ∈ (0 ,
1) such that + θ + θ < γ = θ . In particular, by choosing θ = − , the regularity restriction (7.45) yields s > ≈ . . (7.47)Therefore, it follows from (7.24), (7.34), (7.40), and (7.46) with (7.35), (7.41), and (7.47)that sup t ∈ [0 ,τ ] E ( v )( t ) ≤ C ( T, k f k B s ( T ) ) n t ∈ [0 ,τ ] (cid:0) E ( v )( t ) (cid:1) − o for any τ ∈ [0 , T ], provided that s > . Therefore, (7.22) follows from the standardcontinuity argument. (cid:3) Long time existence of solutions to the perturbed NLS.
Our main goal in thissubsection is to prove long time existence of solutions to the perturbed NLS (7.2) undersome regularity assumptions on the perturbation f (Proposition 7.5). The main ingredientsare the energy estimates (Propositions 7.2 and 7.3) and the following perturbation lemma. Lemma 7.4 (Perturbation lemma) . Given d = 5 or , let ( q d , r d ) be the admissible pairin (4.1) . Let I be a compact interval with | I | ≤ . Suppose that v ∈ C ( I ; H ( R d )) satisfiesthe following perturbed NLS: i∂ t v + ∆ v = | v | d − v + e, satisfying k v k L qdt ( I ; W ,rdx ( R d )) + k v k L ∞ ( I ; H ( R d )) ≤ R for some R ≥ . Then, there exists ε = ε ( R ) > such that if we have k w − v ( t ) k H ( R d ) + k e k N ( I ) ≤ ε for some w ∈ H ( R d ) , some t ∈ I , and some ε < ε , then there exists a solution w ∈ X ( I ) ∩ C ( I ; H ( R d )) to the defocusing NLS (7.1) with w ( t ) = w such that k w k X ( I ) + k v k X ( I ) ≤ C ( R ) , k w − v k X ( I ) ≤ C ( R ) ε, where C ( R ) is a non-decreasing function of R . See [15, 50, 51] for perturbation and stability results on the usual Strichartz and Lebesguespaces. For perturbation lemmas involving the critical X -norm, see [30, 3]. The proof ofLemma 7.4 follows from a straightforward modification of the proof of Lemma 7.1 in [3]and hence we omit details.We now state a long time existence result for the perturbed NLS (7.2). Fix d = 5 or6 and let s ∈ ( s ∗ , s ∗ is as in Theorem 1.5. Then, let δ = δ ( d, s ) > T >
0, suppose that f ∈ e E M,T for some
M >
0, where e E M,T is asin (6.2). Namely, we have k f k Y s ([0 ,T )) + k f k S s ([0 ,T )) ≤ M. (7.48)Then, Lemma 6.2 guarantees existence of a solution v ∈ C ([0 , τ ]; H ( T d )) ∩ X ([0 , τ ]) tothe perturbed NLS (7.2), at least for a short time τ >
0. Furthermore, assume that thereexists
K > k f k A s ( T ) ≤ K when d = 6 and (ii) k f k B s ( T ) ≤ K when d = 5 , (7.49)where A s ( T ) and B s ( T ) are as in Propositions 7.2 and 7.3. Then, it follows from Lemma 7.1and Propositions 7.2 and 7.3 that there exists R = R ( K, T ) > k v k L ∞ ([0 ,T ]; H ( R d )) ≤ R (7.50)for a solution v to (7.2).Under these assumptions, by iteratively applying Lemma 7.4, we obtain the followinglong time existence result for the perturbed NLS (7.2) on [0 , T ]. Proposition 7.5.
Let d = 5 , and s ∈ ( s ∗ , , where s ∗ is as in Theorem 1.5. Given T > ,assume that the hypotheses (7.48) and (7.49) hold. Then, there exists τ = τ ( R, M, T, s ) > such that, given any t ∈ [0 , T ) , the solution v to (7.2) exists on [ t , t + τ ] ∩ [0 , T ] . Inparticular, the energy estimate (7.50) guarantees existence of v on the entire interval [0 , T ] . Proposition 7.5 follows from a straightforward modification of the proof of Proposition 7.2in [3]. Hence, we omit the details of the proof but we briefly describe the main idea inthe following. Given t ∈ [0 , T ), the main idea is to approximate a solution v to theperturbed NLS (7.2) by the global solution w to the original NLS (7.1) with w | t = t = v ( t )on [ t , t + τ ], where τ = τ ( R, M, T, s ) > t ∈ [0 , T ). We achieve thisgoal by iteratively applying the perturbation lemma (Lemma 7.4) on short time intervals.This is possible thanks to (i) the a priori control (7.48) and (7.50) on f and the H -norm of v ( t ), respectively, on [0 , T ] and (ii) the following space-time control on the global solution w to (7.1) due to Vi¸san [53]: k w k L d +2) d − t,x ( R × R d ) ≤ C ( k v ( t ) k H ) = C ( R ) . See the proof of Proposition 7.2 in [3] for details. In the following, we point out the differencebetween the assumptions in Proposition 7.5 above and those in Proposition 7.2 in [3]. Theassumption in [3] would read as “ k f k S s ( I ) ≤ | I | β for any interval I ⊂ [0 , T ]” in our context.Note that we are making a weaker assumption on the S s -norm in (7.48). This is possiblethanks to the appearance of the factor | I | θ in the nonlinear estimate (6.3) in Corollary 6.1. ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , Namely, in this paper, we already exploited the subcritical nature of the perturbation andcreated the factor | I | θ in (6.3). Compare this with Lemma 6.2 in [3].7.3. Proof of Theorem 1.5.
In this subsection, we present the proof of Theorem 1.5.By Borel-Cantelli lemma, it suffices to prove the following “almost” almost sure globalexistence result. See [16, 3].
Proposition 7.6.
Let d = 5 , and s ∈ ( s ∗ , , where s ∗ is as in Theorem 1.5. Given φ ∈ H s ( R d ) , let φ ω be its Wiener randomization defined in (1.2) . Then, given any T, ε > ,there exists a set e Ω T,ε ⊂ Ω such that (i) P ( e Ω cT,ε ) < ε , (ii) For each ω ∈ e Ω T,ε , there exists a (unique) solution u to (1.1) on [0 , T ] with u | t =0 = φ ω . The proof of Proposition 7.6 is analogous to that of Proposition 8.1 in [3]. The maindifference appears in the definitions of Ω and Ω below, incorporating the energy esti-mate (7.50) and the simplified assumption (7.48). Proof.
Fix
T, ε >
0. Set M = M ( ε, k φ k H s ) by M ∼ k φ k H s (cid:16) log 1 ε (cid:17) . Without loss of generality, we assume that ε > M = M ( ε, k φ k H s ) ≥
1. Defining Ω = Ω ( ε ) byΩ := (cid:8) ω ∈ Ω : k φ ω k H s ≤ M (cid:9) , it follows from Lemma 2.1 that P (Ω c ) < ε . (7.51)Given K >
0, define Ω = Ω ( T, K ) byΩ := (cid:8) ω ∈ Ω : k S ( t ) φ ω k F s ( T ) ≤ K (cid:9) , where F s ( T ) = A s ( T ) when d = 6 and = B s ( T ) when d = 5. Then, by Lemmas 2.1 and 2.3,we can choose K = K ( T, ε, k φ k H s ) ≫ P (Ω c ) < ε . (7.52)Hence, the energy estimate (7.50) holds with some R = R ( K, T ) = R ( T, ε ) > τ = τ ( R, M, T, s ) be as in Proposition 7.5. Let δ = δ ( d, s ) > q = − δ . With I j = [ jτ ∗ , ( j + 1) τ ∗ ] for some τ ∗ ≤ τ (to be chosenlater), we partition the interval [0 , T ] as[0 , T ] = [ Tτ ∗ ] [ j =0 I j ∩ [0 , T ]and define Ω by Ω := n ω ∈ Ω : k S ( t ) φ ω k S s ( I j ) ≤ M, j = 0 , . . . , (cid:2) Tτ ∗ (cid:3)o . Then, by Lemma 2.3 and taking τ ∗ = τ ∗ ( T, ε, k φ k H s ) > P (Ω c ) ≤ [ Tτ ∗ ] X j =0 P (cid:16) k S ( t ) φ ω k S s ( I j ) > M (cid:17) ≤ C Tτ ∗ exp (cid:18) − c M τ q ∗ k φ k H s (cid:19) ≤ C Tτ ∗ · τ ∗ exp (cid:18) − c M τ q ∗ k φ k H s (cid:19) ≤ CT exp (cid:18) − c τ q ∗ k φ k H s (cid:19) < ε . (7.53)Finally, set e Ω T,ε := Ω ∩ Ω ∩ Ω . Then, from (7.51), (7.52), and (7.53), we conclude that P ( e Ω cT,ε ) < ε. Moreover, for ω ∈ e Ω T,ε , we can iteratively apply Proposition 7.5 and construct the solution v = v ω to (1.3) on each [ jτ ∗ , ( j + 1) τ ∗ ], j = 0 , . . . , [ Tτ ∗ ] −
1, and (cid:2) [ Tτ ∗ ] τ ∗ , T (cid:3) . This completesthe proof of Proposition 7.6. (cid:3) Probabilistic construction of finite time blowup solutions below theenergy space
In this section, we present the proof of Theorem 1.7. We first recall the following defini-tion of a weak solution to (1.18). See [29].
Definition 8.1.
We say that v is a weak solution to (1.18) on [0 , T ) if v belongs to L d +2 d − loc ([0 , T ) × R d ) and satisfies ˆ [0 ,T ] × R d v · ( − i∂ t ψ + ∆ ψ ) dxdt = iα ˆ R d v · ψ (0) dx + λ ˆ [0 ,T ] × R d | v + εz | d +2 d − · ψ dxdt (8.1)for any test function ψ ∈ C ∞ c ([0 , T ) × R d ).Fix v ∈ H ( R d ). Then, for any α > ε >
0, Proposition 1.3 establishes almostsure local well-posedness of the following Duhamel formulation: v ( t ) = αS ( t ) v − iλ ˆ t S ( t − t ′ ) | v + εz ω | d +2 d − ( t ′ ) dt ′ . (8.2)The following lemma shows that the solution v to (8.2) is indeed a weak solution to (8.1). Lemma 8.2.
Let d = 5 , and − d < s < . Given φ ∈ H s ( R d ) , let φ ω be its Wienerrandomization defined in (1.2) and let z ω = S ( t ) φ ω . Then, given any v ∈ H ( R d ) , α > , ε > , and T > , any local-in-time solution v ∈ C ([0 , T ); H ( R d )) ∩ X ([0 , T )) tothe Duhamel formulation (8.2) is almost surely a weak solution on [0 , T ) in the sense ofDefinition 8.1. By convention, our test function ψ has compact support but does not have to vanish at t = 0. Thesame comment applies to the test function η = η ( t ) below. ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , We first present the proof of Theorem 1.7, assuming Lemma 8.2. We prove Lemma 8.2at this end of this section. Note that while Proposition 1.3 guarantees the existence of thesolution v to (8.2) at least for some small T ω >
0, Lemma 8.2 assumes its existence on[0 , T ) for some given
T > λ = 1 and assume that v satisfies (1.15).The proof of Theorem 1.7 is based on the so-called test function method [55, 56] andwe closely follow the argument in [27]. We first define two test functions η = η ( t ) ∈ C ∞ c ([0 , ∞ ); [0 , θ = θ ( x ) ∈ C ∞ c ( R d ; [0 , η ( t ) = ( ≤ t < , t ≥ , and θ ( x ) = ( ≤ | x | < , | x | ≥ . We also define the scaled test functions η T and θ T by η T ( t ) := η (cid:0) t T ) and θ T ( x ) := (cid:0) x √ T (cid:1) .Finally, we set ψ T ( t, x ) := η T ( t ) θ T ( x ).Given T ≥
1, let v = v ω ∈ X ([0 , T )) be a solution to the Duhamel formulation (8.2) on[0 , T ). Define I and II byI ( T ) = ˆ [0 ,T ) × B √ T | v + εz ω | p · ψ ℓ T dxdt and II( T ) = Im ˆ B √ T v · θ ℓ T dx, (8.3)where B r denotes the ball of radius r centered at 0 in R d and ℓ ∈ N such that ℓ ≥ p ′ + 1.Here, p ′ = d +24 denotes the H¨older conjugate of p = d +2 d − . By Lemma 8.2 and taking thereal part of the weak formulation (8.1), we obtainI ( T ) − α II( T ) = Im ˆ [0 ,T ] × R d v · ∂ t ψ ℓT dxdt + Re ˆ [0 ,T ] × R d v · ∆ ψ ℓT dxdt =: III ( T ) + III ( T ) . (8.4)By ℓ − ≥ ℓp and the triangle inequality, we haveIII ( T ) . T − ˆ [0 ,T ) × B √ T | v | · η ℓ − T θ ℓ T η ′ (cid:0) tT (cid:1) dxdt . T − ˆ [0 ,T ) × B √ T | v + εz ω | · ψ ℓpT dxdt + εT − ˆ [0 ,T ) × B √ T | z ω | dxdt . T (cid:0) I ( T ) (cid:1) p + εT − k z ω k L t,x ([0 ,T ) × B √ T ) . (8.5)A similar computation with ℓ − ≥ ℓp and the triangle inequality yieldsIII ( T ) . T (cid:0) I ( T ) (cid:1) p + εT − k z ω k L t,x ([0 ,T ) × B √ T ) . (8.6)From (8.4), (8.5), and (8.6) with Young’s inequality, we have − α II( T ) ≤ − I ( T ) + CT (cid:0) I ( T ) (cid:1) p + CεT − k z ω k L t,x ([0 ,T ) × B √ T ) ≤ − I ( T ) + C ′ T p ′ + I ( T ) + CεT − k z ω k L t,x ([0 ,T ) × B √ T ) ≤ C ′ T d +24 + CεT − k z ω k L t,x ([0 ,T ) × B √ T ) . (8.7) On the other hand, from (1.15) and (8.3) with a change of variables, we have − II( T ) ≥ T d − k L ( T ) := T d − k ˆ B √ T | x | − k θ ℓ dx. (8.8)Given R > T ≥
1, and ε >
0, define the set Ω
R,ε byΩ
R,ε := (cid:8) ω ∈ Ω : k φ ω k L x ≤ ε − R (cid:9) . Then, it follows from Lemma 2.1 that P (Ω cR,ε ) ≤ C exp (cid:18) − c R ε k φ k L (cid:19) . In particular, P (Ω cR,ε ) → R → ∞ or ε → T ≥
1, we obtain α ≤ CL − ( T ) n T − d +2 k +24 + εT − d + k − k z ω k L t,x ([0 ,T ) × B √ T ) o ≤ CL − ( T ) n T − d +2 k +24 + εT − d +2 k +24 k z ω k L ∞ t L x ([0 ,T ) × B √ T ) o ≤ CL − ( T ) T − d +2 k +24 (1 + R ) (8.9)for ω ∈ Ω R,ε . In the following, we fix
R > ε >
R,ε . Namely, thefollowing argument holds uniformly in ω ∈ Ω R,ε and we suppress the dependence on ω .Suppose that given α >
0, the maximal existence time T ∗ ( α ) ≥
4. Since k < d , we have L (4) < ∞ . In particular, by setting T = 4 in (8.9), we obtain α . R. This in turn implies that there exists α = α ( R ) > T ∗ ( α ) < α ≥ α .Fix α > α . Then, by noting that L ( T ) defined in (8.8) is decreasing on [0 , ∞ ), weconclude from (8.9) that α ≤ CL − (4) T − d +2 k +24 (1 + R ) ≤ C ( R ) T − d +2 k +24 for any 0 < T ≤ T ∗ ( α ) <
4. Hence, we obtain the following upper bound on the maximaltime of existence: T ∗ ( α ) ≤ C ′ ( R ) α − d +2 k +2 . Lastly, (1.17) follows from the blowup alternative (6.5). This proves Theorem 1.7.We conclude this paper by presenting the proof of Lemma 8.2. While the proof isstandard, we include it for completeness.
Proof of Lemma 8.2.
Write the solution v to (8.2) on [0 , T ) as v ( t ) = αS ( t ) v − iλ I [ N ( v + εz ω )]( t ) , where I is as in (3.2) and N ( u ) = | u | d +2 d − . First, we show that the linear part αS ( t ) v satisfies ˆ [0 ,T ] × R d v · ( − i∂ t ψ + ∆ ψ ) dxdt = iα ˆ R d v · ψ (0) dx (8.10)for any test function ψ ∈ C ∞ c ([0 , T ) × R d ). ROBABILISTIC WELL-POSEDNESS OF NLS ON R d , d = 5 , Let v ,n be smooth functions converging to v in H ( R d ). Then, αS ( t ) v ,n , n ∈ N , solvesthe linear Schr¨odinger equation: i∂ t v + ∆ v = 0 and is smooth on [0 , T ) × R d . Integratingby parts, we have ˆ [0 ,T ] × R d αS ( t ) v ,n · ( − i∂ t ψ + ∆ ψ ) dxdt = iα ˆ R d v ,n · ψ (0) dx. (8.11)By H¨older’s inequality and the unitarity of S ( t ) on L ( R d ), we have (cid:12)(cid:12)(cid:12)(cid:12) ˆ [0 ,T ] × R d α ( S ( t ) v − S ( t ) v ,n )( − i∂ t ψ + ∆ ψ ) dxdt (cid:12)(cid:12)(cid:12)(cid:12) . k v − v ,n k L (cid:0) k ψ k W , T L x + k ψ k L T H x (cid:1) −→ . Similarly, the right-hand side of (8.11) converges to the right-hand side of (8.10) as n → ∞ .Hence, (8.10) holds.Next, we consider the nonlinear part − iλ I ( v + εz ω ). Let v n be smooth functions on[0 , T ) × R d converging to v in X ([0 , T )). Then, by Proposition 4.1 with Lemmas 2.1and 2.3, we have (cid:13)(cid:13) I [ N ( v + εz ω )] − I [ N ( v n + εz ω )] (cid:13)(cid:13) C T H −→ , (8.12)almost surely. Let w n = − iλ I [ N ( v n + εz ω )]. Then, w n is the smooth solution to thefollowing inhomogeneous linear Schr¨odinger equation: ( i∂ t w n + ∆ w n = λ | v n + εz | d +2 d − w n | t =0 = 0 . Then, proceeding as above with (8.12) and integrating by parts, we have ˆ [0 ,T ] × R d − iλ I ( v + εz ω ) · ( − i∂ t ψ + ∆ ψ ) dxdt = lim n →∞ ˆ [0 ,T ] × R d w n · ( − i∂ t ψ + ∆ ψ ) dxdt = lim n →∞ ˆ [0 ,T ] × R d ( i∂ t w n + ∆ w n ) · ψ dxdt = lim n →∞ λ ˆ [0 ,T ] × R d | v n + εz | d +2 d − · ψ dxdt = λ ˆ [0 ,T ] × R d | v + εz | d +2 d − · ψ dxdt (8.13)for any test function ψ ∈ C ∞ c ([0 , T ) × R d ). Hence, the weak formulation (8.1) follows from(8.10) and (8.13). This completes the proof of Lemma 8.2. (cid:3) Acknowledgments.
T.O. was supported by the European Research Council (grantno. 637995 “ProbDynDispEq”). M.O. was supported by JSPS KAKENHI Grant numberJP16K17624.
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E-mail address : [email protected] Mamoru Okamoto, Division of Mathematics and Physics, Faculty of Engineering, ShinshuUniversity, 4-17-1 Wakasato, Nagano City 380-8553, Japan
E-mail address : m [email protected] Oana Pocovnicu, Department of Mathematics, Heriot-Watt University and The MaxwellInstitute for the Mathematical Sciences, Edinburgh, EH14 4AS, United Kingdom
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