Global well-posedness of the energy-critical stochastic nonlinear wave equations

We consider the Cauchy problem for the defocusing energy-critical stochastic nonlinear wave equations (SNLW) with an additive stochastic forcing on $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ with $d \geq 3$. By adapting the probabilistic perturbation argument employed in the context of the random data Cauchy theory by B\'enyi-Oh-Pocovnicu (2015) and Pocovnicu (2017) and in the context of stochastic PDEs by Oh-Okamoto (2020), we prove global well-posedness of the defocusing energy-critical SNLW. In particular, on $\mathbb{T}^d$, we prove global well-posedness with the stochastic forcing below the energy space.


Introduction
We consider the following Cauchy problem for the defocusing energy-critical stochastic nonlinear wave equation (SNLW) on M = R d or T d (with T = R/2πZ) for d ≥ 3: where u is real-valued, ξ is the space-time white noise on R + × M, and φ is a bounded operator on L 2 (M).The aim of this paper is to show global well-posedness of (1.1).
Let us first mention some backgrounds on the energy-critical NLW.Consider the following deterministic defocusing NLW on R d with d ≥ 3: It is well known that the following dilation symmetry for λ > 0 u(t, x) → u λ (t, x) := λ d−2 2 u(λt, λx) maps solutions of NLW (1.2) to solutions of NLW (1.2).A direct computation yields so that the scaling critical Sobolev regularity for NLW (1.2) is s c = 1.Also, the energy defined by is conserved under the flow of (1.2).In view of the Sobolev embedding Ḣ1 (R d ) ֒→ L For this reason, we refer to Ḣ1 (R d ) as the energy space for NLW (1.2).Moreover, we say that NLW (1.2) is energy-critical.On T d , although there is no dilation symmetry, the heuristics provided by the scaling analysis still hold and we say that (1.2) on T d is energy-critical.
Note that for energy-subcritical NLW, after proving local well-posedness, one can easily obtain global well-posedness by using the conservation of the energy, which provides an a priori control of the Ḣ1 -norm of the solution.For the energy-critical NLW, however, there is a delicate balance between the linear and nonlinear parts of the equation.In the energy-critical setting, only the energy conservation itself is not enough to obtain global wellposedness, which makes the problem quite intricate.Still, after substantial efforts of many mathematicians, we now know that the energy-critical defocusing NLW (1.2) is globally wellposed in Ḣ1 (R d ) and all solutions in the energy space scatter.See [55,20,21,52,53,25,2,1,31,32,56].On the other hand, in the periodic setting, the global well-posedness results for (1.2) in Ḣ1 (R d ) immediately implies corresponding global well-posedness for (1.2) in H 1 (T d ), thanks to the finite speed of propagation.We also point out that these well-posedness results for (1.2) in the energy space are sharp in the sense that ill-posedness for (1.2) on R d occurs below the energy space; see [11,18,38].
(1.6) See (2.2) in Subsection 2.1 for a precise definition.On R d , we obtain the following global well-posedness result.
Here, the assumption on φ is chosen in such a way that the stochastic convolution Ψ lies in the energy space Ḣ1 (R d ).See Lemma 2.6 below.
For the proof of Theorem 1.1, we first establish local well-posedness of (1.1) in Section 3 and then prove global well-posedness of (1.1) in Subsection 4.1 below.To prove wellposedness of (1.1), we use the first order expansion u = Ψ + v and consider the following equation for v: ( By viewing (1.7) as an energy-critical NLW for v with a perturbation term, we adapt the probabilistic perturbation theory developed in [4,51] in the context of random data Cauchy theory.See Lemma 2.5 below.The perturbation theory has also been previously used to prove global well-posedness for other equations in deterministic settings, stochastic settings, or random initial data settings.See [58,13,28,29,4,34,33].
In order to apply the perturbation argument for the equation (1.7), we need to make sure that the stochastic convolution Ψ is small on short time intervals.This can be done since Ψ can be bounded in Strichartz spaces (i.e.L q in time and L r in space), as in [33].Nevertheless, due to the complicated nature of the wave equations, establishing the space-time regularity of Ψ is non-trivial.See Lemma 2.6 for more details.
Another important ingredient in carrying out the perturbation argument in our setting is an a priori energy bound.This is achieved by a rigorously justified application of Ito's lemma.See also [17,33,8,9].
We now switch our attention to the periodic setting.Here, we assume that φ is a a Fourier multiplier operator on L 2 (T d ).Namely, for any n ∈ Z d , Then, the defocusing energy-critical SNLW (1.1) is globally well-posed in H 1 (T d ) in the sense that the following statement holds true almost surely; given (u 0 , u 1 ) ∈ H 1 (T d ), there exists a global-in-time solution u to (1.1) with (u, ∂ t u)| t=0 = (u 0 , u 1 ).
Note that compared to the R d case, there is an improvement in the regularity assumption of the noise term.This is mainly because of the better space-time regularity of the stochastic convolution Ψ on a bounded domain.See Lemma 2.8 for more details.Moreover, our global well-posedness result for d ≥ 5 is optimal in the sense that the regularity of φ cannot be further lowered.See Remark 1.4 below.
Thanks to the finite speed of propagation, the proof of Theorem 1.2 follows from the same strategies for the R d case.However, due to the lower regularity assumption of the noise, the stochastic convolution does not belong to the energy space H 1 (T d ).Thus, for the a priori energy bound, instead of using Ito's lemma, we use a Gronwall-type argument developed by [7,39] in the context of random data Cauchy theory.
We conclude this introduction by stating several remarks.Remark 1.3.(i) In [39,51], the authors studied the defocusing energy-critical NLW (1.2) on R d with initial data below the energy space.In particular, using the Wiener randomization of the initial data, they proved global well-posedness of (1.2) by establishing an energy bound via a Gronwall-type argument.
However, at this point, we do not know whether one can prove any global well-posedness results for the defocusing energy-critial SNLW (1.1) with initial data below the energy space.The main obstacle is that, even with randomized initial data, the Gronwall-type argument as in [7,39,51] is not directly applicable due the lack of space-time regularity of the stochastic convolution Ψ.
(ii) Compared to the R d case, the situation is better on T d and we can prove almost sure global well-posedness of the stochastic energy-critical defocusing NLW (1.1) below the energy space.Specifically, we consider the following equation for d ≥ 3: where φ satisfies the same condition as in Theorem 1.2 and (u ω 0 , u ω 1 ) is a randomization of (u 0 , u 1 ) defined by where {g n,j } n∈Z d ,j∈{0,1} is a sequence of independent mean zero complex-valued random variables conditioned such that g −n,j = g n,j for all n ∈ Z d .Moreover, we assume that there exists a constant c > 0 such that on the probability distributions µ n,j of g n,j , we have ˆeγ•x dµ n,j (x) ≤ e c|γ| 2 , j = 0, 1 for all γ ∈ R 2 when n ∈ Z d \ {0} and all γ ∈ R when n = 0.Then, due to better integrability of V (t)(u ω 0 , u ω 1 ) compared to V (t)(u 0 , u 1 ) for any t ≥ 0 (see, for example, [40,  .Note that when s < −1, the stochastic convolution Ψ is merely a distribution, and hence a proper renormalization is needed to make sense of the power-type nonlinearity.For this purpose, the nonlinearity must be an integer power of the form u k , which is only true for d ≤ 4.This means that our global well-posedness result of (1.1) on T d for d ≥ 5 is the best that we can achieve.
In the case d = 4, it is natural to ask whether it is possible to extend the well-posedness theory of (1.1) on T 4 with the stochastic convolution Ψ being merely a distribution.The answer is no.Indeed, in a recent preprint [35], the authors showed an ill-posedness result for the following (renormalized) stochastic NLW on T d : where k ≥ 2 is an integer and φ is a Fourier multiplier operator with φ ∈ HS(L 2 (T d ), H s (T d )), s < −1.Hence, our global well-posedness result for the defocusing energy-critical SNLW (1.1) on T 4 is sharp up to the endpoint.It would be of interest to show global well-posedness of (1.1) in the endpoint case s = −1 for d = 4, which seems to require a more intricate analysis.See also Remark 4.5 (ii) below.

Preliminary results and lemmas
In this section, we recall some notations, definitions, useful lemmas, and previous results.For two positive numbers A and B, we use A B to denote A ≤ CB for some constant C > 0. Also, we use shorthand notations for space-time function spaces, such as We recall that if H 1 , H 2 are Hilbert spaces, then for a linear operator φ from H 1 to H 2 , we denote as the Hilbert-Schmidt operator norm of φ, where {e n } n∈N is an orthonormal basis of H 1 .
2.1.Preliminary lemmas on Sobolev spaces.In this subsection, we recall Sobolev spaces on R d and T d and also some useful estimates.
For s ∈ R, we denote by Ḣs (R d ) the homogeneous L 2 -based Sobolev space with the norm , where f is the Fourier transform of f .We denote by Ḣs (R d ) the inhomogeneous L 2 -based Sobolev space with the norm For 1 < p ≤ ∞, we denote by W s,p (R d ) the L p -based Sobolev space with the norm where F −1 denotes the inverse Fourier transform.
On T d , for s ∈ R, we denote by H s (T d ) the inhomogeneous L 2 -based Sobolev space with the norm We also define We now recall some useful estimates for Sobolev spaces, starting with the following fractional chain rule.For a proof, see [12]. 1   Lemma 2.1.Let d ≥ 1, s ∈ (0, 1), and r > 2. Let 1 < p, p 1 < ∞ and 1 < p 2 ≤ ∞ satisfying . Then, we have We also need the following Gagliardo-Nirenberg interpolation inequality.The proof of this inequality follows directly from Sobolev's inequality and interpolation.Lemma 2.2.Let d ≥ 1, 1 < p 1 , p 2 < ∞, and s 1 , s 2 > 0. Let p > 1 and θ ∈ (0, 1) satisfying Then, we have 2.2.Previous results on wave equations.In this subsection, we record some results on wave equations.Let us first recall the Strichartz estimate for linear wave equations.Let γ ∈ R and d ≥ 2. We say that a pair (q, r) is Ḣγ (R d )-wave admissible if q ≥ 2 and 2 ≤ r < ∞ satisfy The following Strichartz estimates for wave equations are well-studied.See, for example, [19,30,27].
Lemma 2.3.Let γ > 0, d ≥ 2, (q, r) be Ḣγ (R d )-wave admissible, and ( q, r) be Ḣ1−γ (R d )wave admissible.Let u be the solution of where q ′ and r ′ are Hölder conjugates of q and r, respectively. 1Here, we use the version in [54,Theorem 3.3.1].As pointed out in [54], the proof in [12] needs a small correction, which yields the fractional chain rule in a less general context.See also [26,54,57].
In the energy-critical setting, we will frequently use the following Strichartz space: where one can easily check that the pair ) is Ḣ1 -wave admissible.Another frequently used pair is (∞, 2), which is Ḣ0 -wave admissible.
We now recall the following global space-time bound on the solution to the deterministic defocusing energy-critical NLW.
For a proof of Lemma 2.4, see [51,Lemma 4.6] and also the references therein, where the steps can be easily extended to d ≥ 3.
Lastly, we recall the following long-time perturbation lemma from [51,Lemma 4.5].The lemma in [51] was stated for d = 4, 5, but it can be easily extended to d ≥ 3.
Lemma 2.5.Let d ≥ 3, (v 0 , v 1 ) ∈ Ḣ1 (R d ), and M > 0. Let I ⊂ R + be a compact time interval and t 0 ∈ I. Let v be a solution on I × R d of the following perturbed equation Let (w 0 , w 1 ) ∈ Ḣ1 (R d ) and let w be the solution of the defocusing energy-critical NLW: Then, there exists ε(M ) > 0 sufficiently small such that if x ≤ ε for some 0 < ε < ε(M ), then the following holds: for all Ḣ1 -wave admissible pairs (q, r), where C(•) is a non-decreasing function.

Regularity of stochastic convolutions.
In this subsection, we study regularity properties of several stochastic object.
We first consider the stochastic convolution Ψ as defined in (1.6), which we now provide a more precise definition.By fixing an orthonormal basis {e k } k∈N of L 2 (R d ), we define W as the following cylindrical Wiener process on L 2 (R d ): Here, β k is defined by , where •, • t,x denotes the duality pairing on R + × R d .As a result, {β k } k∈N is a family of mutually independent real-valued Brownian motions.The stochastic convolution Ψ is then given by where S is the Fourier multiplier as defined in (1.5).
We now show the following lemma, which establishes the regularity property of Ψ.
almost surely.Moreover, for any finite p ≥ 1, we have for some constant C = C(T, p) > 0.
Remark 2.7.One can use integration by parts to write almost surely.This allows us to compute that almost surely.As in Lemma 2.6 (i), given d ≥ 1, T > 0, and φ ∈ HS(L 2 , H s ) for some s ∈ R, we can show that for any finite p ≥ 1.
We now consider the stochastic convolution Ψ in (1.6) on T d , which we denote by Φ := Ψ to avoid confusion.We recall that in the T d setting, we assume that φ is a Fourier multiplier operator on L 2 (T d ): for all n ∈ Z d , we have for some φ n ∈ C. Thus, Φ is given by Here, the operator S T d has the same form as S in (1.5) but on the periodic domain T d and Given R ≥ 1, we define the following stochastic convolution on R d : Also, in view of Remark 2.7, we also also take a time derivative and obtain We now state the following lemma regarding the regularity of the above stochastic objects.
For the estimate (2.11), we can apply the Sobolev embedding L ∞ (R d ) ֒→ W ε,r 1 (R d ) with 0 < ε ≪ 1 and 2 ≤ r 1 < ∞ satisfying εr 1 > d and then use the above steps to obtain the desired result.The fact that Φ R is continuous in time follows from adapting the above modifications to the proof of [22, Proposition 2.1] using Kolmogorov's continuity criterion.The steps for establishing the estimate (2.12) are similar.

Local well-posedness of the energy-critical stochastic NLW
In this section, we briefly go over local well-posedness of the defocusing energy-critical SNLW (1.1) on R d .We first show the following local well-posedness result in a slightly more general setting.Recall that the operator V (t) is as defined in (1.5) and the X(I)-norm is as defined in (2.1).Proposition 3.1.Let d ≥ 3 and (u 0 , u 1 ) ∈ Ḣ1 (R d ).Then, there exists 0 < η ≪ 1 such that if for some time interval I = [t 0 , t 1 ] ⊂ R, then the Cauchy problem Here, the uniqueness of v holds in the set {v ∈ X(I) : v X(I) ≤ 3η}.
Proof.By writing (3.2) in the Duhamel formulation, we have where S(t) is as defined in (1.5).We would like to run the contraction argument on the ball where η > 0 is to be chosen later.Suppose that (u 0 , u 1 ) Ḣ ≤ A for some A > 0.Then, by the Strichartz estimate (Lemma 2.3) and (3.1), for any v ∈ B a,I , we have for some constant C 1 > 0, where in the last inequality we choose 0 < η ≪ 1 in such a way that Also, by the Strichartz estimate (Lemma 2.3) and the fundamental theorem of calculus, for any v 1 , v 2 ∈ B η,I , we have for some constant C 2 > 0, where in the last inequality we further shrink 0 < η ≪ 1 if possible so that 2C 2 (3η) Thus, Γ is a contraction from B η,I to itself, so that the desired local well-posedness of the equation As a consequence of local well-posedness in Proposition 3.1, we have the following blowup alternative.Lemma 3.2.Let d ≥ 3 and (u 0 , u 1 ) ∈ Ḣ1 (R d ).Let f be a function such that f X([0,T ]) < ∞ for any T > 0. Let T * = T * (u 0 , u 1 , f ) > 0 be the forward maximal time of existence of the following Cauchy problem for v: Then, we have either We now go back to the defocusing energy-critical SNLW (1.1) on R d .By writing u = v + Ψ with Ψ being the stochastic convolution as defined in (2.2), we see that v satisfies We also note that by Lemma 2.6, given φ ∈ HS(L almost surely for any t 0 ≥ 0, τ > 0, and some θ > 0. This norm can be made arbitrarily small if we make τ > 0 to be small.Thus, the local well-posedness result (Proposition 3.1) and the blowup alternative (Lemma 3.2) apply to the equation (3.4) by letting f = Ψ.

Proof of global well-posedness
In this section, we show the proofs of the two global well-posedness results: Theorem 1.1 and Theorem 1.2.

4.1.
Global well-posedness on R d .In this subsection, we prove Theorem 1.1, global wellposedness of the defocusing energy-critical SNLW (1.1) on R d .As mentioned in Section 1, the proof is based on the perturbation lemma (Lemma 2.5).However, to carry out the perturbation argument, we first need to show an a priori energy bound using Ito's lemma.In order to justify the use of Ito's lemma, we need to perform an approximation procedure for the defocusing energy-critical SNLW (1.1) as below.
Given N ∈ N, we denote P N to be a smooth frequency projection onto {|ξ| ≤ N }.We consider the following truncated defocusing energy-critical SNLW: where We define the truncated stochastic convolution Ψ N by (2.2) with φ replaced by P N φ.Due to the boundedness of P N , we can easily see from Lemma 2.6 that almost surely for any t 0 ≥ 0, T > 0, and some θ > 0. Thus, again due to the boundedness of P N , we see that the local well-posedness result (Proposition 3.1) and the blowup alternative (Lemma 3.2) apply to the following equation for v This shows that the truncated defocusing energy-critical SNLW (4.1) is locally well-posed.
Let us now show the following lemma regarding the convergence of u N to the solution u to SNLW (1.1).Lemma 4.1.Let d ≥ 3, φ ∈ HS(L 2 , L 2 ), and (u 0 , u 1 ) ∈ H 1 .Then, the following holds true almost surely.Assume that u is a solution to the defocusing energy-critical SNLW (1.1) on [0, T ] for some T > 0, and assume that u N is a solution to the truncated defocusing energycritical SNLW (4.1) on [0, T N ] for some T N > 0. Also, let R > 0 be such that u X([0,T ]) ≤ R, where the X([0, T ])-norm is as defined in (2.1).Then, by letting S N = min(T, T N ), we have Remark 4.2.In Lemma 4.1, due to the lack of global well-posedness of the truncated equation (4.1) at this point, we need to assume that the existence time T N of (4.1) depends on N .One can avoid this issue by inserting a H 1 -norm truncation as in [17], so that the nonlinearity becomes Lipschitz and hence one has global well-posedness for the truncated equation.In this paper, we choose to proceed with the existence time T N dependent on N , which turns out to be harmless at a later point (see the proof of Proposition 4.3).
As in the proof of Proposition 3.1, by using the Duhamel formulation (1.4), the Strichartz estimate (Lemma 2.3), and the fundamental theorem of calculus, we obtain for any 0 ≤ τ ≤ min(t 1 , T N ).Then, from (4.4), by the Lebesgue dominated convergence theorem applied to (Id − P N )u 0 , (Id − P N )u 1 , and (Id − P N )N (u) along with (4.2) and (4.3), we have for some absolute constant C > 0, given N ≥ N 0 (ω, ε, u) sufficiently large.By taking η > 0 to be sufficiently small (independent of ε), we can then use a standard continuity argument to get and given N ≥ N 0 (ω, ε, u).By (4.5) and taking η > 0 to be sufficiently small (independent of ε), we then obtain We now repeat the above arguments on I 1 to obtain x ) < 4C 2 ε.By applying the above arguments repetitively, we obtain that for j = 0, 1, . . ., J − 1, Thus, we have Since ε > 0 can be arbitrarily small and J depends only on R > 0 and an absolute constant η > 0, we can conclude the desired convergence results.
We now show the following a priori bound on the energy E as defined in (1.3).
Let u be the solution to the defocusing energy-critical SNLW (1.1) with (u, ∂ t u)| t=0 = (u 0 , u 1 ) and let T * = T * (ω, u 0 , u 1 ) be the forward maximal time of existence.Then, given any T 0 > 0, there exists C = C( (u , u 1 ) Ḣ1 , φ HS(L 2 ,L 2 ) , T 0 ) > 0 such that for any stopping time T with 0 < T < min(T * , T 0 ) almost surely, we have Proof.We write the energy E in (1.3) as A direct computation yields and Given R > 0, we define the stopping time where the X([0, τ ])-norm is as defined in (2.1).We also set T 2 := min(T, T 1 ).Note that we have almost surely in view of the blowup alternative in Lemma 3.2, so that we have T 2 ր T almost surely as R → ∞.Let u N be the solution to the truncated defocusing energy-critical SNLW (4.1) with maximal time of existence T * N = T * N (ω).By Lemma 4.1, we can deduce that there exists an almost surely finite number N 0 (ω) ∈ N such that T * N ≥ T 2 for any N ≥ N 0 (ω).Indeed, suppose not, then there exists Ω 0 ⊂ Ω with positive measure such that for all ω ∈ Ω 0 , there exists a sequence of increasing numbers {N j (ω)} j∈N ⊂ N such that T * N j (ω) < T 2 .By the blowup alternative (Lemma 3.2), we know that for all ω ∈ Ω 0 and j ∈ N, there exists This contradicts with the convergence of u N j (ω) to u in X([0, T N j (ω) ]) as N j (ω) → ∞ in Lemma 4.1, given that we have We can now work with (4.1) on [0, T 2 ].Note that we can write (4.1) in the following Ito formulation: By Ito's lemma (see [16,Theorem 4.32]) along with (4.7) and (4.8), we obtain For the third term on the right-hand-side of (4.9), by Hölder's inequalities, the Lebesgue dominated convergence theorem applied to (Id − P N )N (u), and Lemma 4.1, we have almost surely, as N → ∞.Thus, by applying the Burkholder-Davis-Gundy inequality to (4.9) along with (4.10), Hölder's inequality, and Cauchy's inequality, we obtain for some absolute constant C > 0 and ε > 0 arbitrarily small given N ≥ N 0 (ω, ε, u) sufficiently large.This shows that By Fatou's lemma and (4.11), we have for some constant C = C( (u 0 , u 1 ) Ḣ1 , φ HS(L 2 ,L 2 ) , T 0 ).In view of the almost sure convergence of T 2 to T , we then obtain the desired energy bound using Fatou's lemma.
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1.Let u be a local-in-time solution to the defocusing energy-critical SNLW (1.1) given by Proposition 3.1.We let v = u − Ψ, where v satisfies where Given T > 0, if we suppose that the solution u exists on [0, T ], then by the energy bound in Proposition 4.3, we have Then, by Lemma 2.6 and Remark 2.7, we obtain Given a target time T > 0, we pick any t 0 ∈ [0, T ) and suppose that the solution v to (4.12) has already been constructed on [0, t 0 ].Our goal is to show the existence of a unique solution v to (4.12) on [t 0 , t 0 + τ ] ∩ [0, T ] with τ > 0 independent of t 0 .In this way, we can iterate the argument so that a global solution v to (4.12) can be constructed on [0, T ], which then concludes the proof of Theorem 1.1.

4.2.
Global well-posedness on T d .In this subsection, we focus on global well-posedness of the defocusing energy-critical SNLW (1.1) on the periodic setting and present the proof of Theorem 1.2.We would like to invoke the finite speed of propagation of the wave equation to reduce our global well-posedness problem on T d to global well-posedness on R d , which we have presented in Subsection 4.1.However, instead of using Ito's lemma to establish an energy bound for the solution u to (1.1), we use a Gronwall-type argument to show an a priori energy bound for the solution v to the perturbed NLW (1.7).
Let us first show the a priori energy bound in a slightly more abstract setting.Consider the following perturbed equation for v on R d : where N (u) = |u| 4 d−2 u and z is a space-time function satisfying some regularity assumptions to be specified below. ) Let v be the solution to the equation (4.22) with (v, ∂ t v)| t=0 = (u 0 , u 1 ) and let T * = T * ω (u 0 , u 1 ) be the forward maximal time of existence.Then, there exists a constant C = C T 0 , (u 0 , u 1 ) H 1 , z * > 0 such that for any stopping time T with 0 < T < min(T * , T 0 ) almost surely, we have Here, • * refers to any norm associated with the function spaces in (4.23).
Proof.We consider three cases.In this case, by assumption, we have z ∈ C([0, T ]; W )), where ε ′ > 0 is sufficiently small.By (4.28) and Taylor's theorem, we have for some θ ∈ (0, 1).For I 2 , by the Cauchy-Schwarz inequality, Hölder's inequality, and Cauchy's inequality, we obtain We now consider I 1 .For 0 ≤ t 1 ≤ t 2 ≤ T , by integration by parts, Hölder's inequalities, and Young's inequalities, we obtain ˆt2 for some 0 < δ < 1.By duality, Hölder's inequality, Lemma 2.1, and Lemma 2.2, we have for any 0 ≤ t 1 ≤ t 2 ≤ T .By Gronwall's inequality and Sobolev's embedding, we get E( v(t)) ≤ C( (u 0 , u 1 ) H 1 )e C( z * , (u 0 ,u 1 ) H 1 )t for any 0 < t ≤ T , which implies the desired energy bound.In this case, by assumption, we have z ∈ C([0, T ]; L r 2 (R 4 )) for any 2 ≤ r 2 ≤ ∞.Thus, by (4.22), we have By the fundamental theorem of calculus, we have Thus, by (4.28), (4.29), Hölder's inequality, and the Cauchy-Schwarz inequality, we obtain By Gronwall's inequality, we get for any 0 < t ≤ T , which implies the desired energy bound.In this case, by assumption, we have z ∈ L q ([0, T ]; L r 1 (R d )) for 1 ≤ q < ∞ and 2 ≤ r 1 < ∞.Proceeding as in Case 2 using the fundamental theorem of calculus, we obtain Let 0 ≤ t 1 ≤ t 2 ≤ T .For I 4 , by the Cauchy-Schwarz inequalities, we obtain ˆt2 For I 1 , by Hölder's inequalities, we obtain ˆt2 Thus, combining (4.30), (4.31), (4.32) and using Gronwall's inequality, we obtain E( v(t)) ≤ C( (u 0 , u 1 ) H 1 )e C( z * , (u 0 ,u 1 ) H 1 )t for any 0 < t ≤ T , which implies the desired energy bound.Remark 4.5.(i) To make the computations in the above proof more rigorous, we need to work with smooth solutions (v N , ∂ t v N ) to (4.22) with truncated initial data (P ≤N u 0 , P ≤N u 1 ) and truncated perturbation term P ≤N z, where P ≤N denotes the sharp frequency projection onto {|n| ≤ N }.Using similar arguments as in Lemma 4.1, we can show that (v N , ∂ t v N ) converges to (v, ∂ t v) in C([0, T ]; Ḣ1 (R d )).After establishing an upper bound for E(v N , ∂ t v N ) that is independent of N , we can take N → ∞ to obtain the desired energy bound for E( v).(ii) In the case when d = 4 and σ = 0, the argument in Case 3 in the above proof breaks down.Due to the lack of an L ∞ -bound in the endpoint case σ = 0, we have to rely on the fact that 8 d−2 < 2d d−2 when dealing with I 3 term in the above proof.However, when d = 4, the inequality 8 d−2 < 2d d−2 does not hold and we do not know how to overcome this issue at this point.It seems that a more intricate analysis is needed to establish an energy bound for the endpoint case σ = 0 for d = 4.
We are now ready to show Theorem 1.2 by reducing it to the R d case.To achieve this, we adjust the use of the finite speed of propagation as in [40, Appendix A] to our stochastic setting.
Proof of Theorem 1.2.Let T > 0 be a target time and N ∈ N. Given initial data (u 0 , u 1 ) ∈ H 1 (T d ), we define u j,T (x) := η T (x) We recall the definition of Φ T in (2.9) and define Φ N,T as Φ T with the summation restricted to |n| ≤ N .By Lemma 2.8, it is not hard to see that Φ N,T converges to Φ T in C([0, T ]; W s+1−ε,∞ (R d )) given φ ∈ HS(L 2 , H s ).Note that Φ N,T and Φ T satisfy ∂ 2 t Φ N,T − ∆Φ N,T = η T P ≤N (φξ) and ∂ 2 t Φ T − ∆Φ T = η T (φξ), respectively, both with zero initial data.We also recall that Φ is the stochastic convolution as defined in (2.7) and we let Φ N be the truncation of Φ onto frequencies {|n| ≤ N }.By the finite speed of propagation for the linear solutions, we have It is easy to see that (u 0,T , u 1,T ) ∈ H 1 (R d ) given (u 0 , u 1 ) ∈ H 1 (T d ).In addition, given the range of s provided in the statement of Theorem 1.2, we obtain by Lemma 2.8 that where 1 ≤ q < ∞, 2 ≤ r 1 < ∞, 2 ≤ r 2 ≤ ∞, ε > 0, and σ ∈ R satisfying where the inclusion follows from Lemma 2.8.
n∈Z d u j (n)e in•x , x ∈ R d u j,N,T (x) := η T (x) |n|≤N u j (n)e in•x , x ∈ R dfor j = 0, 1, where η T is a smooth cutoff function on [−2πT, 2πT ] as defined in(2.8).We consider the following equation for u N on R d :∂ 2 t u N,T − ∆u N,T + N (u N,T ) = η T P ≤N (φξ) (u N,T , ∂ t u N,T )| t=0 = (u 0,N,T , u 1,N,T ), (4.33)where P ≤N denotes the sharp frequency cutoff onto {|n| ≤ N } and N (u) = |u| 4 d−2 u.Since the initial data and the noise are smooth, there exists a unique (smooth) global solution u N,T to (4.33) on R d .By the finite speed of propagation, we have that u N := u N | [0,T ]×T d is a solution to the following equation on T d : ∂ 2 t u N − ∆u N + N (u N ) = P ≤N (φξ) (u N , ∂ t u N )| t=0 = (P ≤N u 0 , P ≤N u 1 ).