This note is concerned with the global well-posedness of nonlinear Schrödinger equations in the continuum with spatially homogeneous random initial data.
Citation: Mitia Duerinckx. On nonlinear Schrödinger equations with random initial data[J]. Mathematics in Engineering, 2022, 4(4): 1-14. doi: 10.3934/mine.2022030
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This note is concerned with the global well-posedness of nonlinear Schrödinger equations in the continuum with spatially homogeneous random initial data.
Motivated by weak turbulence theory, e.g., [10], we consider nonlinear Schrödinger equations with spatially homogeneous statistical ensembles of initial data. As a prototypical example, we study the defocusing cubic equation in Rd,
i∂tu=−△u+|u|2u,u|t=0=u∘. | (1.1) |
A statistical ensemble of initial data amounts to considering an initial condition that is a realization u∘(⋅,ω) of a random field u∘:Rd×Ω→C on some probability space (Ω,P). The spatial homogeneity condition then implies that initial mass and energy diverge:*
*Under an additional ergodicity assumption, it further entails ∫Rd|u∘(⋅,ω)|2=∫Rd|∇u∘(⋅,ω)|2+12|u∘(⋅,ω)|4=∞ almost surely.
E[∫Rd|u∘|2]=∞,E[∫Rd|∇u∘|2+12|u∘|4]=∞. |
This divergence is a key aspect at the very core of weak turbulence: Strichartz' estimates are not applicable in this infinite-energy setting, which is thus in sharp contrast with the finite-energy phenomenology and scattering results [7].
The present note is concerned with the global well-posedness of (1.1) in this infinite-energy setting. The main difficulty is related to the lack of a uniform bound on the propagation speed: mass that is initially spread out might move together and blow up. This contrasts with the case of the nonlinear wave equation, as well as of the discrete nonlinear Schrödinger equation, for which there is an (approximate) finite propagation speed and global well-posedness follows, see [3,Propositions 1-3]. As explained in Examples 2.3 below, periodic and quasi-periodic initial data can in fact be viewed as particular instances of the spatially homogeneous random setting. While the periodic case is well understood [2], the almost periodic case remains largely open and we refer to recent work by Oh [11,12] on the topic. In the general random setting, the problem seems to have only been considered very recently by Dodson, Soffer, and Spencer [3], who established local well-posedness in the real analytic category. If the nonlinearity |u|2u in (1.1) is replaced by a regularized version |ϕ∗u|2(ϕ∗u) for some smooth decaying kernel ϕ, then the problem is strongly reduced and global well-posedness is obtained in [3] in the Ck category.
Our main result in this note states the global well-posedness of the nonlinear Schrödinger equation (1.1) in the spatially homogeneous energy space provided that a tiny dissipation is added. This tiny dissipation is physically relevant in the context of weak turbulence, e.g., [10], and the constructed solution is controlled uniformly with respect to this dissipation. The definition of a meaningful vanishing-dissipation limit remains an open problem (beyond the local-in-time real analytic framework of [3]). Precise definitions of spatial homogeneity and of the functional space X below are postponed to the next section.
Theorem 1. Let 1≤d<4. Given a probability space (Ω,P), let X be the Banach space of spatially homogeneous jointly measurable random fields v:Rd×Ω→C with
‖v‖X:=E[|∇v|2]12+E[|v|4]14<∞. |
For all ε>0 and u∘∈X, there exists a unique global weak solution uε∈L∞(R+;X) to the equation
(−ε+i)∂tuε=−△uε+|uε|2uε,uε|t=0=u∘, | (1.2) |
in the sense that Duhamel's formula holds almost everywhere,
utε=etε−i△u∘−1ε−i∫t0et−sε−i△(|usε|2usε)ds,t≥0. |
In addition, it satisfies the following dissipation estimates: for all t≥0,
E[|utε|2]≤E[|utε|2]+2ε1+ε2∫t0E[|∇usε|2+|usε|4]ds=E[|u∘|2],E[|∇utε|2+12|utε|4]≤E[|∇u∘|2+12|u∘|4]. |
Notation.
● We denote by C≥1 any constant that only depends on the space dimension d. We use the notation ≲ (resp. ≳) for ≤C× (resp. ≥1C×) up to such a multiplicative constant C. We write ≃ when both ≲ and ≳ hold. We add subscripts to C,≲,≳,≃ to indicate dependence on other parameters.
● The ball centered at x and of radius r in Rd is denoted by Br(x), and we write for abbreviation B(x):=B1(x) and Br:=Br(0).
Given a reference probability space (Ω,P), we recall the notion of statistical spatial homogeneity for random fields.
Definition 2.1. A random field on Rd is a map v:Rd×Ω→C such that for all x∈Rd the function v(x,⋅):Ω→C is measurable. It is said to be (statistically) spatially homogeneous if its finite-dimensional law is shift-invariant, that is, if for any finite set E⊂Rd the law of {v(x+y,⋅)}x∈E does not depend on the shift y∈Rd. In addition, it is said to be jointly measurable if the map v:Rd×Ω→C is jointly measurable. We denote by L0hom(Rd×Ω) the set of spatially homogeneous jointly measurable random fields.
Note that the joint measurability condition ensures that realizations v(⋅,ω) are almost surely measurable functions on Rd and can thus be taken as meaningful initial data in (1.1) or (1.2). The following result by von Neumann [14] gives an alternative characterization of joint measurability in this context, which can be viewed as a stochastic version of Lusin's theorem (see also [8,Section 7.1]).
Lemma 2.2 (Joint measurability; [8,14]). A spatially homogeneous random field v is jointly measurable if and only if it is stochastically continuous, that is, if for all δ>0 it satisfies
P[|v(x,⋅)−v(y,⋅)|>δ]→0as|x−y|→0. |
Examples 2.3. Important examples of spatially homogeneous random fields are found among Gaussian fields, and we also explain how periodic and almost periodic settings can be viewed as particular instances of this random framework (see also [13,p.846]).
(a) Gaussian fields: A gauge-invariant Gaussian random field v is a family {v(x,⋅)}x∈Rd of complex-valued Gaussian random variables such that v and eiθv have the same finite-dimensional law for all θ∈[0,2π). Equivalently, this means for all x,y,
E[v(x,⋅)]=0,Cov[v(x,⋅);v(y,⋅)]=0, |
and we denote by c(x,y):=Cov[¯v(x,⋅);v(y,⋅)] the covariance function. This random field v is spatially homogeneous if and only if c is of the form c(x,y)=c0(x−y) for some function c0:Rd→C. Note that c0 is necessarily a positive definite bounded function. In addition, the field v is stochastically continuous, hence jointly measurable by Lemma 2.2, if and only if c0 is continuous at the origin.
(b) Periodic setting: Given a 1-periodic measurable function vper:Rd→C, we choose the probability space (Ω,P) as the periodic cell [0,1)d endowed with Lebesgue's measure, and we define an associated random field v:Rd×Ω→C by v(x,ω):=vper(x+ω). The latter is clearly spatially homogeneous and jointly measurable, and for ω=0 we recover v(x,0)=vper(x).
(c) Almost periodic setting: Denote by B(Rd) the Bohr compactification of the additive group (Rd,+) and let b:Rd→B(Rd) the associated continuous homomorphism, see e.g., [9]. By definition, given an almost periodic function vap:Rd→C, there exists a continuous function Vap:B(Rd)→C such that vap=Vap∘b. We choose the probability space (Ω,P) as B(Rd) endowed with its normalized Haar measure, and we define a random field v:Rd×Ω→C by v(x,ω):=Vap(b(x)+ω), where we use the notation '+' for the group law on B(Rd). This random field is clearly spatially homogeneous and jointly measurable, and for ω=0 we recover v(x,0)=vap(x).
In this section, we define more carefully the functional space X used in the statement of Theorem 1. For 1≤q<∞, we denote by Lqhom(Rd×Ω) the Banach space of spatially homogeneous jointly measurable random fields v∈L0hom(Rd×Ω) such that the following norm is finite,
‖v‖Lqhom(Rd×Ω):=‖v‖Lq([0,1)d×Ω). |
By spatial homogeneity, see Definition 2.1, this is in fact equivalent to
‖v‖Lqhom(Rd×Ω)=‖v(x,⋅)‖Lq(Ω)for any x∈Rd. |
As Lqhom(Rd×Ω) is invariant under spatial translations and as its elements are almost surely locally Lq-integrable, the spatial gradient ∇ can be defined on Lqhom(Rd×Ω) and its domain is denoted by W1,qhom(Rd×Ω). More generally, for all s≥0, we define Ws,qhom(Rd×Ω) as the Banach space of random fields v∈Lqhom(Rd×Ω) such that the following norm is finite,
‖v‖Ws,qhom(Rd×Ω):=‖(1−△)s2v‖Lq([0,1)d×Ω), |
and for q=2 we use the usual notation Hshom(Rd×Ω):=Ws,2hom(Rd×Ω). In these terms, the space X in Theorem 1 coincides with H1hom∩L4hom(Rd×Ω).
Next, we give an alternative description of these spaces and we explain how equations (1.1) or (1.2) in this spatially homogeneous random setting are equivalent to abstract equations on the probability space; this construction is standard for corrector equations in stochastic homogenization theory, see e.g., [13,Section 2] and [8,Section 7.1]. Let u∘∈L0hom(Rd×Ω) be a reference random field. Since we consider equations with realizations of u∘ as initial data, we can henceforth assume that the probability space (Ω,P) is endowed with the σ-algebra σ(u∘) generated by u∘.† Translations u∘(⋅,ω)↦u∘(⋅+x,ω) then induce a unique multiplicative linear action T={Tx}x∈Rd of the additive group (Rd,+) on the algebra of random variables.
† That is, the σ-algebra generated by all sets of the form {ω∈Ω:u∘(x1,ω)∈A1,…,u∘(xn,ω)∈An} with n≥1, x1,…,xn∈Rd, and Borel subsets A1,…An⊂C.
Lemma 2.4 (Properties of T).
(i) For 1≤q≤∞, the maps Tx's are isometries on Lq(Ω).
(ii) For 1≤q<∞, the action T is a C0-group of isometries on Lq(Ω).
Proof. Item (i) follows from the fact that u∘ is spatially homogeneous. We turn to the proof of (ii). Given q<∞, it remains to check that ‖TxX−X‖Lq(Ω)→0 as |x|→0 for all X∈Lq(Ω). By a truncation argument, it suffices to argue for X∈L∞(Ω). By the joint measurability of u∘, the map (x,ω)↦(TxX)(ω) is also jointly measurable, hence stochastically continuous by Lemma 2.2. Writing for any δ>0,
‖TxX−X‖qLq(Ω)≤δq+(2‖X‖L∞(Ω))qP[|TxX−X|>δ], |
the conclusion follows from stochastic continuity.
In terms of T, we can define the extension X♯∈Lqhom(Rd×Ω) of any random variable X∈Lq(Ω), and the restriction v♭∈Lq(Ω) of any random field v∈Lqhom(Rd×Ω), via
X♯(x,ω):=(TxX)(ω),v♭(ω)=v(0,ω), |
and we note that (X♯)♭=X and (v♭)♯=v, thus yielding a canonical isomorphism
Lq(Ω)≅Lqhom(Rd×Ω). | (2.1) |
For 1≤q<∞, as T is a C0-group of isometries on Lq(Ω), cf. Lemma 2.4(ii), we can define the T-gradient ∇♭ as the generator of this group. It is a densely defined operator on Lq(Ω), its domain is denoted by W1,q(Ω), and it is skew-adjoint on L2(Ω). Alternatively, this operator ∇♭ can be reinterpreted via the isomorphism (2.1):
W1,q(Ω)≅W1,qhom(Rd×Ω),(∇♭X)♯=∇X♯for all X∈W1,q(Ω). |
We also define the corresponding T-Laplacian −△♭:=−∇♭⋅∇♭ on Lq(Ω), which is nonnegative and essentially self-adjoint on L2(Ω). For all s≥0, we denote by Ws,q(Ω) the Banach space of random variables X∈Lq(Ω) such that X♯∈Ws,qhom(Rd×Ω).
The above construction entails that spatially homogeneous solutions of (1.2) are equivalent to solutions of a corresponding abstract equation on the probability space. Note that expressions like etε−i△v in (2.2) below make sense almost surely for v∈Lqhom(Rd×Ω) since the kernel of etε−i△ has Gaussian decay while realizations of v have subexponential growth almost surely, cf. (3.13). In the periodic case, in view of Example 1(b), ∇♭ is the periodic gradient and this result amounts to reducing (1.2) to the corresponding equation on the periodic cell Ω=[0,1)d.
Lemma 2.5. Given T>0, the following two properties are equivalent:
● A random field uε∈L∞([0,T];H1hom∩L4hom(Rd×Ω)) is a weak solution of (1.2) in the sense that Duhamel's formula holds almost everywhere,
utε=etε−i△u∘−1ε−i∫t0et−sε−i△(|usε|2usε)ds,0≤t≤T. | (2.2) |
● We have utε=(Utε)♯ where Uε∈L∞([0,T];H1∩L4(Ω)) is a weak solution of the following abstract equation on the probability space,
(−ε+i)∂tUε=−△♭Uε+|Uε|2Uε,Uε|t=0=(u∘)♭, | (2.3) |
in the sense that Duhamel's formula holds almost everywhere,
Utε=etε−i△♭(u∘)♭−1ε−i∫t0et−sε−i△♭(|Usε|2Usε)ds,0≤t≤T. |
In contrast with the periodic case, the T-Laplacian −△♭ on L2(Ω) typically has absolutely continuous spectrum and no spectral gap above 0. As shown in [4], the spectrum can actually be arbitrary and depends on the structure of the underlying probability space; we focus here for simplicity on the Gaussian setting.
Lemma 2.6 (Spectrum of T-Laplacian; [4]). Assume that u∘∈L2hom(Rd×Ω) is Gaussian in the sense of Example 1(a) and that its covariance function c0 has an absolutely continuous Fourier transform. Then, the spectrum of −△♭ on L2(Ω) is [0,∞) and is made of a simple eigenvalue at 0 embedded in absolutely continuous spectrum.
In particular, this entails that Poincaré's inequality and compact Rellich embeddings do not hold on H1(Ω). In addition, we show that Sobolev embeddings also fail and that the parabolic semigroup {et△♭}t≥0 yields no improved integrability. Heuristically, this lack of functional tools is related to the fact that the T-gradient ∇♭ only contains information on a finite set of directions, while Ω is typically an infinite product space. This constitutes a key difficulty for the analysis of nonlinear equations such as (1.2) in this setting.
Lemma 2.7 (Lack of functional tools). Assume that u∘∈L2hom(Rd×Ω) is Gaussian in the sense of Example 1(a) and that its covariance function c0 is integrable. Then,
(i) Poincaré's inequality: ‖X−E[X]‖L2(Ω)≤C‖∇♭X‖L2(Ω) does not hold on H1(Ω).
(ii) Compact Rellich embedding: Hm(Ω) is not compactly embedded in L2(Ω) for any m>0.
(iii) Sobolev embedding: ‖X‖Lq(Ω)≤C‖X‖Hm(Ω) does not hold on Hm(Ω) for any m≥0 and q>2.
(iv) Parabolic improvement of integrability: ‖ez△♭X‖Lq(Ω)≤C‖X‖L2(Ω) does not hold on Lq(Ω) for any q>2 and any z∈C with ℜz≥0.
Proof. We start with items (i) and (ii). They can both be viewed as consequences of Lemma 2.6, but we rather provide a quick direct proof. Given a real-valued test function ξ∈C∞c(Rd), consider the Gaussian random variables
Xn(ω):=n−d2∫Rdξ(yn)u∘(y,ω)dy,n≥1. |
A direct computation yields as n↑∞,
‖Xn‖2L2(Ω)→R:=‖ξ‖2L2(Rd)∫Rdc0,n2‖∇♭Xn‖2L2(Ω)→‖∇ξ‖2L2(Rd)∫Rdc0, | (2.4) |
which contradicts the validity of Poincaré's inequality, hence proves (i). Since the sequence (Xn)n is bounded in Hm(Ω) for any m≥0, and since it converges weakly but not strongly to 0 in L2(Ω), item (ii) follows.
We turn to the proof of item (iii). A simple application of Wick's theorem yields for all integers r≥1,
‖Xn‖2rL2r(Ω)=r!‖Xn‖2rL2(Ω). |
Extending this to all r≥1, and combining with (2.4), we find
‖Xn‖2rL2r(Ω)n↑∞→RrΓ(r+1). | (2.5) |
Also note that (2.4) yields ‖∇mXn‖2rL2r(Ω)→0 for all r,m≥1. Hence, given q≥2 and m≥0, we deduce for all integers p≥1,
‖Xpn‖Lq(Ω)‖Xpn‖Hm(Ω)n↑∞→Γ(qp+1)1/qΓ(2p+1)1/2. |
For q>2, the right-hand side blows up exponentially as p↑∞, thus contradicting the validity of Sobolev's inequality.
It remains to prove item (iv) and we start with the easy case when ℜz=0. The essential self-adjointness of the T-Laplacian −△♭ on L2(Ω) entails that e−z△♭ is unitary on L2(Ω) in that case, hence the inequality ‖ez△♭X‖Lq(Ω)≤C‖X‖L2(Ω) for all X∈L2(Ω) would in fact imply ‖X‖Lq(Ω)≤C‖X‖L2(Ω) for all X∈L2(Ω), a contradiction.
We turn to the proof of (iv) for ℜz>0. As the operator ez△ has kernel Kz(x):=(4πz)−d2e−14z|x|2, and as realizations of the Gaussian random field X♯n have subexponential growth almost surely, we can write in view of (2.1),
ez△♭Xpn=∫RdKz(x)X♯n(x,⋅)pdx. |
Appealing to Wick's theorem as above, using (2.4) in form of
E[X♯n(x,⋅)¯X♯n(y,⋅)]n↑∞→R, |
and noting that ∫RdKz=1, we find for all r≥1 and all integers p≥1,
‖ez△♭Xpn‖2rL2r(Ω)n↑∞→RprΓ(pr+1). |
Hence, combined with (2.5), given q≥2, we deduce for all integers p≥1,
‖ez△♭Xpn‖Lq(Ω)‖Xpn‖L2(Ω)n↑∞→Γ(12qp+1)1/qΓ(p+1)1/2. |
For q>2, the right-hand side blows up exponentially as p↑∞, and the conclusion follows.
We turn to the proof of Theorem 1, where we recall X=H1hom∩L4hom(Rd×Ω) with the above notation. In order to overcome the lack of functional tools to study the nonlinear equation (1.2) in this setting, cf. Lemma 2.7, we rather focus on almost sure realizations in local Sobolev spaces, for which standard tools are available. Dissipation is crucial to compensate for the lack of finite propagation speed and allows to prove well-posedness in local Sobolev spaces with polynomial growth, which is then post-processed into a well-posedness result in the spatially homogeneous random setting.
Proof of Theorem 1. We start by introducing local Sobolev spaces with polynomial growth, which are natural spaces to control almost sure realizations of random fields. Given ℓ≥0, we consider the uniformly localized Lq and H1 norms with ℓ-growth,
‖v‖Lquloc,ℓ(Rd):=supx0∈Rd(⟨x0⟩−qℓ−∫B(x0)|v|q)1q,‖v‖H1uloc,ℓ(Rd):=supx0∈Rd(⟨x0⟩−2ℓ−∫B(x0)(|v|2+|∇v|2))12, |
and we denote by Lquloc,ℓ(Rd) and H1uloc,ℓ(Rd) the corresponding subspaces of L1loc(Rd). We also consider the uniformly localized energy functional with ℓ-growth,
Euloc,ℓ(v):=supx0∈Rd(⟨x0⟩−2ℓ−∫B(x0)(|∇v|2+12|v|4)). |
As shown below, cf. (3.13), given ℓ>dq, realizations of a random field v∈Lqhom(Rd×Ω) belong almost surely to Lquloc,ℓ(Rd). With this in mind, we start by studying the nonlinear Schrödinger equation (1.2) in H1uloc,ℓ(Rd), and next we exploit uniqueness to construct a unique dynamics in H1hom∩L4hom(Rd×Ω). Note that H1uloc,ℓ(Rd) embeds in L4uloc,ℓ(Rd) by the Sobolev embedding (with d≤4), while on the contrary H1hom(Rd×Ω) does in general not embed in L4hom(Rd×Ω), cf. Lemma 2.7(iii). The proof is split into two main steps.
Step 1. Global well-posedness in H1uloc,ℓ(Rd).
For d<4, given ε>0 and ℓ≥0, we show that for all v∘∈H1uloc,ℓ(Rd) the equation
(−ε+i)∂tvε=−△vε+|vε|2vε,vε|t=0=v∘ |
admits a unique global weak (Duhamel) solution vε in L∞loc(R+;H1uloc,ℓ(Rd)). We split the proof into four further substeps.
Substep 1.1. Equivalent definition of Lquloc,ℓ(Rd).
In terms of the exponential cut-off χ(x):=e−|x|, setting for abbreviation χx0:=χ(⋅−x0), we show that for all q,ℓ we have
‖v‖Lquloc,ℓ(Rd)≃q,ℓsupx0∈Rd(⟨x0⟩−qℓ∫Rdχx0|v|q)1q. | (3.1) |
Indeed, as χx0≳1 on B(x0), the left-hand side in (3.1) is clearly bounded above by the right-hand side. The converse inequality follows from the following,
(⟨x0⟩−qℓ∫Rdχx0|v|q)1q≤(⟨x0⟩−qℓ∑z∈Zd∫B(x0+z)χx0|v|q)1q≲(⟨x0⟩−qℓ∑z∈Zde−|z|⟨x0+z⟩qℓ)1q‖v‖Lquloc,ℓ(Rd)≲‖v‖Lquloc,ℓ(Rd). |
Substep 1.2. Localized parabolic estimates.
Given ε>0, ℓ≥0, and 1≤p≤q≤∞ with d2(1p−1q)<1, we show for all g∈Lquloc,ℓ(Rd) and t≥0,
‖etε−i△g‖Lquloc,ℓ(Rd)≲p,q,ℓ,εt−d2(1p−1q)eCεt‖g‖Lpuloc,ℓ(Rd), | (3.2) |
and in addition, provided d2(1p−1q)<12,
‖∇etε−i△g‖Lquloc,ℓ(Rd)≲p,q,ℓ,εt−d2(1p−1q)−12eCεt‖g‖Lpuloc,ℓ(Rd). | (3.3) |
Multiplying the parabolic evolution {etε−i△g}t≥0 with the exponential cut-off χx0, and using Duhamel's formula, we easily find
χx0(etε−i△g)=etε−i△(χx0g)+1ε−i∫t0et−sε−i△((△χx0)(esε−i△g))ds−2ε−i∫t0∇⋅et−sε−i△((∇χx0)(esε−i△g))ds. | (3.4) |
Note that the parabolic semigroup {etε−i△}t≥0 has kernel Ktε(x):=(ε−i4πt)d2e−ε−i4t|x|2, which implies by Young's convolution inequality, for all g∈C∞c(Rd) and 1≤p≤q≤∞, letting r be such that 1p+1r=1q+1,
‖etε−i△g‖Lq(Rd)=‖Ktε∗g‖Lq(Rd)≤‖Ktε‖Lr(Rd)‖g‖Lp(Rd)≲p,q,εt−d2(1p−1q)‖g‖Lp(Rd), | (3.5) |
and similarly,
‖∇etε−i△g‖Lq(Rd)≲p,q,εt−d2(1p−1q)−12‖g‖Lp(Rd). | (3.6) |
For 1≤p≤q≤∞, taking the Lq norm in (3.4) and using these parabolic estimates, we obtain
‖χx0(etε−i△g)‖Lq(Rd)≤Cp,q,εt−d2(1p−1q)‖χx0g‖Lp(Rd)+Cε∫t0(1+(t−s)−12)‖χx0(esε−i△g)‖Lq(Rd)ds, |
and the claim (3.2) easily follows from Grönwall's inequality together with (3.1). Using (3.6) instead of (3.5), the claim (3.3) is obtained in a similar way.
Substep 1.3. Local well-posedness in Lquloc,ℓ(Rd).
Given ε>0, ℓ≥0, and 3≤q≤∞ with q>d, we show that for all v∘∈Lquloc,ℓ(Rd) there exists T>0 such that the equation
(−ε+i)∂tvε=−△vε+|vε|2vε,vε|t=0=v∘ |
admits a unique weak (Duhamel) solution vε in L∞([0,T];Lquloc,ℓ(Rd)). In case q>6dd+2, we further have vε∈L∞([0,T];H1uloc,ℓ(Rd)) provided v∘∈H1uloc,ℓ(Rd).
To prove well-posedness in L∞([0,T];Lquloc,ℓ(Rd)), we argue by a Picard fixed-point argument: for T>0 we define an operator ΦT,ε(⋅;v∘) on L∞([0,T];Lquloc,ℓ(Rd)) by
(ΦT,ε(v;v∘))t:=etε−i△v∘−1ε−i∫t0et−sε−i△(|vs|2vs)ds,0≤t≤T, | (3.7) |
and it suffices to show that for all v,w∈L∞([0,T];Lquloc,ℓ(Rd)),
‖ΦT,ε(v;v∘)‖L∞([0,T];Lquloc,ℓ(Rd))≲q,ℓ,εeCεT(‖v∘‖Lquloc,ℓ(Rd)+‖v‖3L∞([0,T];Lquloc,ℓ(Rd))), | (3.8) |
‖ΦT,ε(v;v∘)−ΦT,ε(w;v∘)‖L∞([0,T];Lquloc,ℓ(Rd))≲q,ℓ,εT1−dqeCεT‖(v,w)‖2L∞([0,T];Lquloc,ℓ(Rd))‖v−w‖L∞([0,T];Lquloc,ℓ(Rd)). | (3.9) |
It remains to prove these two estimates. Applying the localized parabolic estimate (3.2) with exponents q3≤q, we find for all 3≤q≤∞ with q>d,
‖(ΦT,ε(v;v∘))t‖Lquloc,ℓ(Rd)≲q,ℓ,εeCεt(‖v∘‖Lquloc,ℓ(Rd)+∫t0(t−s)−dq‖|vs|2vs‖Lq/3uloc,ℓ(Rd)ds)≲q,ℓ,εeCεt(‖v∘‖Lquloc,ℓ(Rd)+t1−dq‖v‖3L∞([0,t];Lquloc,ℓ(Rd))), |
which proves (3.8). Similarly, we find
‖(ΦT,ε(v;v∘))t−(ΦT,ε(w;v∘))t‖Lquloc,ℓ(Rd)≲q,ℓ,εeCεt∫t0(t−s)−dq‖|vs|2vs−|ws|2ws‖Lq/3uloc,ℓ(Rd)ds, |
and thus, further using Hölder's inequality in form of
‖|vs|2vs−|ws|2ws‖Lq/3uloc,ℓ(Rd)≲‖(vs,ws)‖2Lquloc,ℓ(Rd)‖vs−ws‖Lquloc,ℓ(Rd), |
we deduce
‖(ΦT,ε(v;v∘))t−(ΦT,ε(w;v∘))t‖Lquloc,ℓ(Rd)≲q,ℓ,εt1−dqeCεt‖(v,w)‖2L∞([0,t];Lquloc,ℓ(Rd))‖v−w‖L∞([0,t];Lquloc,ℓ(Rd)), |
which proves (3.9).
It remains to show that for q>6dd+2 this local weak (Duhamel) solution vε=ΦT,ε(vε;v∘) in L∞([0,T];Lquloc,ℓ(Rd)) also belongs to L∞([0,T];H1uloc,ℓ(Rd)) provided v∘∈H1uloc,ℓ(Rd). Without loss of generality we choose q≤6. As the condition on q ensures d2(3q−12)<12, we appeal to the localized parabolic estimate (3.3) with exponents q3≤2, to the effect of
‖∇vtε‖L2uloc,ℓ(Rd)≲q,ℓ,εeCεt(‖∇v∘‖L2uloc,ℓ(Rd)+∫t0(t−s)−d2(3q−12)−12‖vs‖3Lquloc,ℓ(Rd)ds)≲q,ℓ,εeCεt(‖∇v∘‖L2uloc,ℓ(Rd)+‖v‖3L∞(R+;Lquloc,ℓ(Rd))), |
hence vtε∈L∞([0,T];H1uloc,ℓ(Rd)).
Substep 1.4. Conclusion: global well-posedness in H1uloc,ℓ(Rd).
We argue that it suffices to prove the following localized energy estimate: for all T>0 and v∘∈H1uloc,ℓ(Rd), if vε∈L∞([0,T];H1uloc,ℓ(Rd)) is a weak (Duhamel) solution of
(−ε+i)∂tvε=−△vε+|vε|2vε,vε|t=0=v∘, | (3.10) |
then we have for all 0≤t≤T,
Euloc,ℓ(vtε)≲ℓe12εtEuloc,ℓ(v∘). | (3.11) |
Since by definition
Euloc,ℓ(g)≃‖g‖2H1uloc,ℓ(Rd)+‖g‖4L4uloc,ℓ/2(Rd), |
we can naturally combine this energy estimate (3.11) together with the local well-posedness result of Step 3 with q=4. As the restriction q>6dd+2 reduces in that case to d<4, and as H1uloc,ℓ(Rd) embeds in L4uloc,ℓ(Rd) by the Sobolev embedding for d<4, this leads us to the claimed global well-posedness result.
It remains to prove the energy estimate (3.11). If the solution vε of (3.10) was smooth, then, using the standard notation ⟨a,b⟩:=ℜ(ˉab) for the scalar product in C, we could compute by Eq (3.10) and Young's inequality,
∂t∫Rdχx0(|∇vε|2+12|vε|4)=2∫Rdχx0(⟨∇vε,∇∂tvε⟩+⟨|vε|2vε,∂tvε⟩)=2∫Rdχx0⟨−△vε+|vε|2vε,∂tvε⟩−2∫Rd∇χx0⋅⟨∇vε,∂tvε⟩=−2ε∫Rdχx0|∂tvε|2−2∫Rd∇χx0⋅⟨∇vε,∂tvε⟩≤12ε∫Rdχx0|∇vε|2, |
hence, by Grönwall's inequality,
∫Rdχx0(|∇vtε|2+12|vtε|4)≤e12εt∫Rdχx0(|∇v∘|2+12|v∘|4). | (3.12) |
This estimate can be justified in our non-smooth setting by an approximation procedure as e.g., in [5,6], and the claim (3.11) then follows from (3.1).
Step 2. Global well-posedness in H1hom∩L4hom(Rd×Ω).
Let d<4. Given u∘∈H1hom∩L4hom(Rd×Ω), we prove the existence of a unique almost sure global weak (Duhamel) solution uε in L∞(R+;H1hom∩L4hom(Rd×Ω)) of Eq (1.2). We split the proof into four further substeps.
Substep 2.1. Existence and uniqueness for realizations.
Let ℓ>d2 be fixed. The localized energy of a realization u∘(⋅,ω) is trivially bounded by
Euloc,ℓ(u∘(⋅,ω))≲Mu∘(ω):=∑z∈Zd⟨z⟩−2ℓ−∫B(z)(|∇u∘(⋅,ω)|2+12|u∘(⋅,ω)|4). |
As u∘ belongs to H1hom∩L4hom(Rd×Ω), the choice ℓ>d2 ensures E[Mu∘]<∞. This implies that there is a subset Ω0⊂Ω with maximal probability such that
Euloc,ℓ(u∘(⋅,ω))≲Mu∘(ω)<∞for all ω∈Ω0, | (3.13) |
hence u∘(⋅,ω)∈H1uloc,ℓ(Rd). Therefore, in view of Step 1, for all ω∈Ω0, there exists a unique weak (Duhamel) solution uε(⋅,ω)∈L∞loc(R+;H1uloc,ℓ(Rd)) of
(−ε+i)∂tuε(⋅,ω)=−△uε(⋅,ω)+|uε(⋅,ω)|2uε(⋅,ω),uε(⋅,ω)|t=0=u∘(⋅,ω), | (3.14) |
and it satisfies the following energy estimate for all t≥0,
Euloc,ℓ(utε(⋅,ω))≲ℓe12εtEuloc,ℓ(u∘(⋅,ω))≲e12εtMu∘(ω). | (3.15) |
Substep 2.2. Measurability.
For all T we show that the above-defined map Ω0→L∞([0,T];H1uloc,ℓ(Rd)):ω↦uε(⋅,ω) is Bochner measurable.
On the one hand, by Fubini's theorem, the joint measurability of u∘ on Ω×Rd ensures that the map Ω0→H1uloc,ℓ(Rd):ω↦u∘(⋅,ω) is weakly measurable, hence also Bochner measurable by Pettis' theorem [1,Lemma 11.37] as H1uloc,ℓ(Rd) is a separable Banach space. On the other hand, arguing again as in Substep 1.3, we note that the solution operator H1uloc,ℓ(Rd)→L∞([0,T];H1uloc,ℓ(Rd)):u∘(⋅,ω)↦uε(⋅,ω) is locally Lipschitz continuous. The Bochner measurability of uε follows by composition.
Substep 2.3. Spatial homogeneity.
For all T, we show that uε belongs to the space L∞([0,T];H1hom∩L4hom(Rd×Ω)) up to modification on null sets.
For all ω∈Ω0 and x∈Rd, since by definition (Txu∘)(⋅,ω)=u∘(⋅+x,ω), cf. Section 2.2, the uniqueness of a weak (Duhamel) solution entails for almost all t,y,
(Txutε)(y,ω)=utε(y+x,ω). | (3.16) |
In other words, uε satisfies an "almost everywhere'' version of spatial homogeneity, and it remains to modify it on null sets to make it spatially homogeneous in the sense of Definition 2.1. By the measurability statement of Substep 2.2 and by the bound (3.15) with Mu∘∈L1(Ω), we have uε∈L2(Ω0;L∞([0,T];H1uloc,ℓ(Rd))). Up to modification on null sets, we deduce uε∈L∞([0,T];L2(Ω;H1uloc,ℓ(Rd))), and we then define for all δ>0,
Utε,δ(ω):=−∫Bδutε(⋅,ω). |
By definition and by (3.15), the family (Uε,δ)δ>0 is bounded in L∞([0,T];H1∩L4(Ω)). Up to an extraction, Uε,δ converges weakly to some Uε in that space as δ↓0. For all x, this implies that TxUε,δ converges weakly to TxUε. Now by (3.16), we find TxUtε,δ(ω)=−∫Bδ(x)utε(⋅,ω) for almost all t,x,ω. Passing to the limit yields TxUtε(ω)=utε(x,ω) for almost all t,x,ω, and the claim follows.
Substep 2.4. Conclusion.
It remains to check dissipation estimates. We start with mass dissipation. Let χR(x):=χ(1Rx). If the solution uε was smooth, Eq (1.2) would allow to compute, almost surely,
∂t∫RdχR|uε(⋅,ω)|2=2∫RdχR⟨uε(⋅,ω),∂tuε(⋅,ω)⟩=−2∫RdχR⟨uε(⋅,ω),1ε−i(−△uε(⋅,ω)+|uε(⋅,ω)|2uε(⋅,ω))⟩=−2εε2+1∫RdχR(|∇uε(⋅,ω)|2+|uε(⋅,ω)|4)−2∫Rd∇χR⋅⟨uε(⋅,ω),1ε−i∇uε(⋅,ω)⟩, |
hence, by integration, with |∇χR|≤1RχR,
|∫RdχR|utε(⋅,ω)|2−∫RdχR|u∘(⋅,ω)|2+2εε2+1∫t0∫RdχR(|∇uε(⋅,ω)|2+|uε(⋅,ω)|4)|≤1R∫t0∫RdχR(|uε(⋅,ω)|2+|∇uε(⋅,ω)|2). |
Up to convolving uε with a smooth kernel and passing to the limit, this estimate is easily justified in our non-smooth setting. Now taking the expectation, using the spatial homogeneity, and letting R↑∞, the claimed mass dissipation identity follows.
We turn to energy dissipation. Repeating the argument for (3.12), but replacing the exponential cut-off χx0 by χR, we get
∫RdχR(|∇utε(⋅,ω)|2+12|utε(⋅,ω)|4)≤e12εR2t∫RdχR(|∇u∘(⋅,ω)|2+12|u∘(⋅,ω)|4). |
Now taking the expectation, using the spatial homogeneity, and letting R↑∞, the claimed energy dissipation estimate follows.
The author thanks Bjoern Bringmann and Laure Saint-Raymond for bringing up this problem and for discussions on the topic, and acknowledges financial support from the CNRS-Momentum program.
The author declares no conflict of interest.
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