{'fraction_non_alphanumeric': 0.0696112002080534, 'fraction_numerical': 0.030081054137228554, 'mean_word_length': 3.4983427568726846, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 23, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Space-time approximation of local strong solutions to the 3D stochastic Navier-Stokes equations

We consider the 3D stochastic Navier-Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates in for the energy error with respect to convergence in probability, that is convergence of order 1 in space and of order (up to) 1/2 in time. The result holds up to the possible blow-up of the (time-discrete) solution. Our approach is based on discrete stopping times for the (time-discrete) solution.


Introduction
We are concerned with the numerical approximation of the 3D stochastic Navier-Stokes equations which read as

1)
P-a.s. in Q T := (0, T ) × T 3 , where T > 0, µ > 0 is the viscosity and u 0 is a given initial datum. The momentum equation is driven by a cylindrical Wiener process W and the diffusion coefficient Φ(u) takes values in the space of Hilbert-Schmidt operators, see Section 2.1 for details. The analysis of (1.1) has a long history starting with the paper [2], where a semi-deterministic approach is applied. A further milestone is the existence of martingale solutions to (1.1) shown in [16]. These solutions are weak in the analytical sense (derivatives exist only in the sense of distributions and singularities may occur) and weak in the probabilistic sense (the probability space is not a priori given but is an integral part of the solution). By now most results from the deterministic case have found their stochastic counterpart, an overview is given in [15] and [25]. Since the well-posedness of (1.1) (and its deterministic version) is a big open problem, the existence of weak solutions is the best one can hope unless one is satisfied with a local-intime result. There exists various results concerning local strong pathwise solutions to (1.1), cf. [1,12,19,23,24]. These solutions are defined on a given stochastic basis and are regular with respect to the spatial variable but only exist up to a stopping time (a precise formulation is given in Definition 2.1). The only information about the latter we have is that it is P-a.s. strictly positive. It is yet unclear if the presence of noise changes the well-posedness for (1.1). On the one hand, there are results based on the method of convex integration showing that stochastic perturbations do not render the ill-posedness of problems in fluid mechanics, cf. [8,21,22]. On the other hand, it was recently proved in [17] that a carefully chosen transport noise in (1.1) can delay the blow-up of the vorticity.
There is less known about the numerical approximation of (1.1), though there was recently some progress on the 2D case. In particular, it is shown in [7] and [13] that for any ξ > 0 as h, τ → 0 (where α < 1 2 and β < 1 are arbitrary); see also [3,4] for related results. Here u is the solution to (1.1) and u h,m the approximation of u(t m ) with discretisation parameters τ = T /M (in time) and h (in space). The relation (1.2) tells us that the convergence with respect to convergence in probability is of order (almost) 1/2 in time and 1 in space. A similar convergence result for the pathwise error is not expect due to non-Lipschitz nonlinearity in (1.1). A result such as (1.2) heavily relies on the spatial regularity of the solution and can consequently not be expected for the 3D problem we are interested here. The only reachable outcome is the approximation of martingale solution, which converges in law to the solution up to taking a subsequence. A corresponding result has been proved in [11].
In this paper we are taking a different perspective and study the approximation of local strong pathwise solutions. We prove a counterpart of (1.2) which holds locally in time, that is, up to a discrete stopping which replaces M in maximum and sum. We obtain a result for the temporal discretisation in Theorem 4.10 as well as for the error between time-and space-time discretisation, see Theorem 5.3. Both combined give the convergence rate for the spatiotemporal discretisation just announced, see Theorem 5.1. The analysis of the temporal error in Section 4.2 is reminiscent of the estimates for the space-time error for the 2D Dirichlet problem from [9]. They rely on a discrete version of the stopping time for the continuous solution and estimates for the latter. The analysis of the error between time-and spacetime discretisation in Section 5 is more delicate and has not been performed in [9]. Building up on ideas from [10] we introduce a discrete stopping time for the time-discrete solution which announces the blow-up, similar to the stopping time for the continuous solution. In Lemma 4.4 we prove that for τ → 0 we can perform a given number of time steps with a high probability before the blow-up. These replaces the strict positivity of the stopping time in the continuous set-up and justifies the subsequent analysis.
We believe that our approach will be applicable to a wide range of stochastic PDEs which are well-posed locally in time, in particular stochastic Euler equations and stochastic compressible Navier-Stokes equations. Surprisingly, results on the numerical approximation of local solutions to stochastic PDEs do not seem to exists so far.

Mathematical framework
2.1. Probability setup. Let (Ω, F, (F t ) t≥0 , P) be a stochastic basis with a complete, rightcontinuous filtration. The process W is a cylindrical U-valued Wiener process, that is, W (t) = j≥1 β j (t)e j with (β j ) j≥1 being mutually independent real-valued standard Wiener processes relative to (F t ) t≥0 , and (e j ) j≥1 a complete orthonormal system in a separable Hilbert space U. Let us now give the precise definition of the diffusion coefficient Φ(u) taking values in the set of Hilbert-Schmidt operators L 2 (U; H), where H can take the role of various Hilbert spaces. We assume that Φ(u) ∈ L 2 (U; L 2 (T 3 )) for u ∈ L 2 (T 3 ), and Φ(u) ∈ L 2 (U; W 1,2 (T 3 )) for u ∈ W 1,2 (T 3 ), together with If we are interested in higher regularity some further assumptions are in place and we require additionally that Φ(u) ∈ L 2 (U; W 2,2 (T 3 )) for u ∈ W 2,2 (T 3 ), together with Assumption (2.1) allows us to define stochastic integrals. Given an (F t )-adapted process u ∈ L 2 (Ω; C([0, T ]; L 2 (T 3 ))), the stochastic integral is a well-defined process taking values in L 2 (T 3 ) (see [14] for a detailed construction). Moreover, we can multiply by test functions to obtain Similarly, we can define stochastic integrals with values in W 1,2 (T 3 ) and W 2,2 (T 3 ) respectively if u belongs to the corresponding class.
2.2. The concept of solutions. We give the definition of a strong pathwise solution to (1.1) which exists up to a stopping time t. The velocity field belongs P-a.s. to C([0, t]; W 2,2 div (T 3 )). Definition 2.1. Let (Ω, F, (F t ) t≥0 , P) be a given stochastic basis with a complete right-continuous filtration and an (F t )-cylindrical Wiener process W . Let u 0 be an F 0 -measurable random variable with values in W 2,2 div (T 3 ). The tuple (u, t) is called a local strong pathwise solution to (1.1) with the initial condition u 0 provided (a) t is a P-a.s. strictly positive (F t )-stopping time; (b) the velocity field u is (F t )-adapted and u(· ∧ t) ∈ C([0, T ]; W 2,2 div (T 3 )) ∩ L 2 (0, T ; W 3,2 div (T 3 )) P-a.s., (c) the momentum equation holds P-a.s. for all ϕ ∈ C ∞ div (T 3 ) and all t ≥ 0. We finally define a maximal strong pathwise solution.

Definition 2.2 (Maximal strong pathwise solution). Fix a stochastic basis with a cylindrical Wiener process and an initial condition as in Definition 2.1. A triplet
is a maximal strong pathwise solution to system (1.1) provided (a) t is a P-a.s. strictly positive (F t )-stopping time; (b) (t R ) R∈N is an increasing sequence of (F t )-stopping times such that t R < t on the set [t < ∞], lim R→∞ t R = t P-a.s., and with the convention that t R = ∞ if the set above is empty; (c) each triplet (u, t R ), R ∈ N, is a local strong pathwise solution in the sense of Definition 2.1.
The following result can be proved along the lines of [23], where the stochastic Navier-Stokes equations on the whole space R 3 are considered with fractional differentiability σ ∈ (3/2, 2). As mentioned on [23, page 2] the case of differentiability σ = 2 we are interested in is even easier. Theorem 2.3. Suppose that (2.1)-(2.5) hold, and that u 0 ∈ L 2 (Ω, W 1,2 div (T 3 )). Then there is a unique maximal global strong pathwise solution to (1.1) in the sense of Definition 2.2.
2.3. Finite elements. We work with a standard finite element set-up for incompressible fluid mechanics, see e.g. [18]. We denote by T h a quasi-uniform subdivision of T 3 into simplices of maximal diameter h > 0. For K ⊂ R 3 and ℓ ∈ N 0 we denote by P ℓ (K) the polynomials on K of degree less than or equal to ℓ. Let us characterize the finite element spaces V h (T 3 ) and P h (T 3 ) as We will assume that i, j ∈ N to get (2.9) below. In order to guarantee stability of our approximations we relate V h (T 3 ) and P h (T 3 ) by the discrete inf-sup condition, that is we assume that where C > 0 does not depend on h. This gives a relation between i and j (for instance the choice (i, j) = (1, 0) is excluded whereas (i, j) = (2, 0) is allowed). Finally, we define the space of discretely solenoidal finite element functions by The following results concerning the approximability of Π h are well-known (see, for instance [20]). There is c > 0 independent of h such that we have for all p ∈ W 1,2 (T 3 ), and for all p ∈ W 2,2 (T 3 ). Note that (2.11) requires the assumption j ≥ 1 in the definition of P h (T 3 ), whereas (2.10) also holds for j = 0.

Regularity of solutions
3.1. Estimates for the continuous solution. In this section we derive the crucial estimates for the continuous solution, which hold up to the stopping time t R for some R ≫ 1.
Lemma 3.1. Let (Ω, F, (F t ) t≥0 , P) be a given stochastic basis with a complete rightcontinuous filtration and an (F t )-cylindrical Wiener process W . Suppose that u 0 is an F 0 -measurable random variable with values in W 2,2 div (T 3 ). Let (u, (t R ) R∈N , t) be the maximal strong pathwise solution to (1.1), cf. Definition 2.2. (a) Assume that u 0 ∈ L r (Ω, L 2 div (T 3 )) for some r ≥ 2 and that Φ satisfies (2.1). Then we have Here c = c(r, T ) is independent of R.
Proof. Let us suppose that u 0 ∈ L ∞ (Ω, W 2,2 (T 3 )). This assumption can be removed eventually by truncating u 0 . Similar to [24] we consider the solution to a truncated problem. For R > 1 and ζ ∈ C ∞ c ([0, 2)) with 0 ≤ ζ ≤ 1 and ζ = 1 in [0, 1] we set ζ R := ζ(R −1 ·) . Let u R be an (F t )-adapted process with holds P-a.s. for all ϕ ∈ W 1,2 div (T 3 ) and all t ∈ [0, T ]. It can be shown by means of a Glarking approximation (and highger order energy estimates which hold thanks to the periodic boundary conditions) that a unique strong pathwise solution to (3.4) exists. We finally note that u R (· ∧ t R ) = u(· ∧ t R ) such that is sufficient to prove the claimed estimates for u R instead of u.
Estimate (3.1) is the standard a priori estimate which can be proved by applying Itô's formula to the functional t → u R 2 L 2 x and using the cancellation of the convective term. As far as (3.2) is concerned we can apply Itô's formula to t → ∇u R 2 L 2 x and use (3.4), which yields We only show how to estimate the convective term II, which significantly differs from the 2D case. We refer to [9, Lemma 3.1], where it is shown how to control the remaining integrals independently of R. We have by definition of ζ R and the embedding For (c) is we argue similarly to (b) 1 and apply Itô's formula to the mapping t → T 3 |∆u R (t)| 2 dx which shows It holds using the embedding x ds for any δ > 0. The expectation of the first term can be controlled by the estimates from (b). As for the stochastic terms we obtain The proof is complete.
3.2. Stochastic pressure decomposition. Since we will be working with discretely divergencefree function spaces in the finite-element analysis, it is inevitable to introduce the pressure function. For ϕ ∈ C ∞ (T 3 ) we can insert Here ∆ −1 is the solution operator to the Laplace equation with respect to periodic boundary conditions on T 3 . Note that ∆ −1 satisfies This corresponds to the stochastic pressure decomposition introduced in [5] (see also [6,Chap. 3] for a slightly different presentation). In the following we will analyse how the regularity of u transfers to π det and Φ π . Arguing as in [7, Corollary 2.5] and using (3.5)-(3.7) we obtain . Consequently, Lemma 3.1 implies the following.
Here c = c(r, T ) is independent of R.
As in [7, Corollary 2.6] we can combine Lemmas 3.1 and 3.2 to conclude the following result concerning the time regularity of u.
(a) Let the assumptions of Lemma 3.1 (b) be satisfied for some r > 2.

Time discretisation
We now consider a temporal approximation of (1.1) on an equidistant partition of [0, T ] with mesh size τ = T /M , and set t m = mτ . Let u 0 be an . For given u m−1 and ∆ m W , to verify the existence o af unique u m solving (4.1) is straightforward since the problem is linear in u m . Lemma 4.1. Assume that u 0 ∈ L 2 q (Ω, L 2 div (T 3 ; R 2 )) for some q ∈ N and that (2.1) holds. Then the iterates (u m ) M m=1 given by (4.1) satisfy the following estimate uniformly in M : Proof. The proof of (4.2) is identical to [11,Lemma 3.1]. Note that, different from [11], we consider a semi-implicit algorithm, which does not impact the proof since the convective term still cancels when testing with u m .

4.1.
Estimates for the time-discrete solution. In order to obtain ''local-in-time" estimates we consider a trunacted variant of (4.1) similarly to (3.4). Let u R m ∈ W 1,2 div (T 3 ) be such for every ϕ ∈ W 1,2 div (T 3 ) it holds true P-a.s.
with u R 0 = u 0 . Arguing as for Lemma 4.1 and noticing that the convective term in (4.3) still cancels when testing with u R m once can show for where c = c(q, T, Φ, u 0 ) > 0 is independent of R. We are now going to prove R-dependent estimates for (u R m ) M m=1 which we transfer eventually to (u m ) M m=1 by introducing a suitable discrete stopping time.
Proof. The proof is reminiscent of [10, Lemma 2.8] but differs in the estimates for the convective term and the nonlinear diffusion coefficient. We proceed formally, a rigorous proof can be obtained using a Galerkin approximation. We test (4.3) by ∆u m obtaining For δ > 0 we have by definition of ζ R and the embedding W 2,2 (T 3 ) ֒→ L ∞ (T 3 ) (4.6) Summing up and choosing δ sufficiently small shows By the discrete Gronwall lemma we have P-a.s.
Proof. We argue similarly to the proof of Lemma 4.2 testing this time (4.3) by ∆ 2 u m obtaining For δ > 0 we have by definition of ζ R and the embeddings x . Summing up, choosing δ sufficiently small and using continuity of ∇ 2 ∆ −1 on L 2 (T 3 ) shows By the discrete Gronwall lemma we have P-a.s.
For R > 0, we define the (discrete) (F tm )-stopping time The crucial point is now to show a counterpart of the strict positivity of the stopping time from the continuous solution, cf. Definition 2.1. This is the content of the next lemma, which states that s d R ≥ τ with high probability. Proof. Arguing as in the proof of Lemma 4.3 we have where the expectation of the last term can be estimated by cℓτ e cR 2 as a consequence of Lemma 4.3. For the stochastic terms we have again Absorbing the δ-terms we conclude by Markov's inequality. By assumption the right-hand sides vanishes as τ → 0.  (a) Suppose that u 0 ∈ L 2 q (Ω, W 1,2 div (T 3 )) and that additionally (2.2) and (2.3) hold. Then we have (b) Suppose that u 0 ∈ L 2 q (Ω, W 2,2 div (T 3 )) and that additionally (2.2)-(2.5) hold. Then we have Here c = c(q, T, Φ, u 0 ) > 0 is independent of R.
Remark 4.6. In the estimates from Lemmas 4.2 and 4.3 it is also possible to control higher moments provided the corresponding moments are bounded for the initial datum. This transfers also to Corollary 4.5 and, in particular, implies ). Now we are going to introduce the pressure function for (4.1). For ϕ ∈ W 1,2 (T 3 ) we can insert ϕ − ∇∆ −1 div ϕ ∈ W 1,2 div (T 3 ) in (4.1) and obtain . Similar to [7,Lemma 3 & 4] we give some estimates for π det m and Φ π m−1 .
Lemma 4.7. Assume that u 0 ∈ L 4 (Ω, W 1,2 div (T 3 )) and that Φ satisfies (2.1)-(2.3). For all m ∈ {1, ..., j R } the random variable π det m is F tm -measureable, has values in W 1,2 (T 3 ) and we have uniformly in τ Proof. The F tm -measurability of π det m follows directly from the one of u m By continuity of the operator ∇∆ −1 div on L 2 (T 3 ) we have x P-a.s. using also Sobolev's embedding in three dimensions. Now, summing with respect to m, applying expectations and using Corollary 4.5 (a) with q = 2 yields the claim. Proof. As for Lemma 4.7 the proof mainly relies on the continuity of ∇∆ −1 div on L 2 (T 3 ). Here, we have by ( Summing over m, applying expectations and using Lemma 4.1 finishes the proof.

4.2.
Temporal error analysis. For every m ≥ 1 introduce the discrete stopping time which is obviously F tm -measurable. Furthermore, we define m R as the unique index in {1, 2, . . . , M } such that t R M = t mR . Our main effort in this section is devoted to the proof of the following theorem.
where (u m ) M m=1 is the solution to (4.1). The constant c in (4.13) is independent of τ and R. Our main result on the temporal error is now a direct consequence of Theorem 4.13: Supposing that R = R(τ ) ≤ c −1/2 √ −2ε log τ as τ → 0 (note that this includes, in particular, any choice of fixed R ∈ N), where ε > 0 is arbitrary, and relabeling α we have proved the following result.
Eventually, we will take the maximum with respect to m ∈ {1, . . . , m R } and apply expectations. Let us explain how to deal with E max m I 1 (m)], . . . , E[max m I 5 (m)] independently. We have , where the expectation of the last term can be controlled for m ≤ m R by τ 2α+1 R 12 using Corollary 3.3 and t R m ≤ t R . We proceed by for m ≤ m R using the embedding W 2,2 (T 3 ) ֒→ L ∞ (T 3 ) and t R m ≤ t R . We rewrite I 3 (m) as and obtain for any δ > 0 and m ≤ m R x . The last term will be dealt with by Gronwall's lemma leading to a constant of the form ce cR 2 .
In order to estimate the stochastic term I 5 we write is an (F t )-martingale we gain by the Burgholder-Davis-Gundy inequality (using that t R M ≤ t R by definition) Here, we also used (2.1) as well as Young's inequality for δ > 0 arbitrary. Finally, we can control the last term by . Applying (2.9) as well as Lemma 3.1 (b) and Corollary 3.
As far as M 2 m,1 is concerned we argue similarly. Using Cauchy-Schwartz inequality, Young's inequality, Itô-isometry and (2.1) we have as a consequence Lemma 3.1 (b) (using also (2.8)) of Corollary 3.3 (b). Collecting all estimates, choosing δ small enough and applying Gronwall's lemma yields the claim.

Space-time discretisation
Now we consider a fully practical scheme combining the implicite Euler scheme in time (as in the last section) with a finite element approximation in space. For a given h > 0 let u h,0 be an F 0 -mesureable random variable with values in V h div (T 3 ) (for instance Π h u 0 ). We aim at constructing iteratively a sequence of random variables u h,m with values in V h div (T 3 ) such that for every ϕ ∈ V h div (T 3 ) it holds true P-a.s.
uniformly in h and τ . It is also shown there that the sequence convergences in law to a martingale solution to (1.1). We strengthen this result in short-time (where the stopping times j R and m R are introduced below (4.8) and (4.12), respectively) by proving an optimal convergence rate with respect to convergence in probability in the following theorem. Here we suppose that R = R(τ, h) ≤ c −1/4 4 −ε min{τ, h 2 } as τ, h → 0, which inlcudes, in particular, any choice of fixed R ∈ N.
be the unique maximal global strong solution to (1.1) from Theorem 2.3. Then we have for any ξ > 0, α < 1 and β < 1, is the solution to (5.1). [9,Section 4] in the 2D case. The advantage of such an approach is that the stopping time j R is not needed. We believe, however, that the plainly temporal error as well as the error between the temporal and spatio-temporal discretisation given in Theorem 5.3 below are of independent interest.

Remark 5.2. It is possible to obtain Theorem 5.1 by a direct comparison between the spacetime discretisation and the exact solution (avoiding the time discretisation is an intermediate step) as in done
Theorem 5.1 follows from combining Theorem 4.10 with the following result concerning the error between the temporal and spatio-temporal discretisation, the proof of which is the main aim of this section. Here we suppose that R = R(h) ≤ c −1/4 4 √ −εh 2 as h → 0.
Theorem 5.3. Let u 0 ∈ L 8 (Ω, W 2,2 div (T 3 )) be F 0 -measurable and assume that Φ satisfies (2.1)-(2.5). Let (u m ) M m=1 be the solution to (4.1). Then we have for any ξ > 0, α < 1 2 β < 1, is the solution to (5.1). Proof. Define the error e h,m = u m − u h,m for m ∈ {0, 1, . . . , j R }. Subtracting (4.11) and (5.1) we obtain for δ > 0 arbitrary. The δ-terms can be absorbed, while the (sum from m = 1, . . . , j R of the) expectations of the other two terms can be bounded by h 2 e cR 2 using Corollary 4.5 (a) and Lemma 4.7. More care is required for I 2 (m). We argue in the spirit of of [9, bounds for I 4 (m) in the proof of Thm. 4.2] but working with 3D embeddings and the definition of j R . First we write I 2 (m) = I 1 2 (m) + I 2 2 (m) + I 3 2 (m), We obtain for any δ > 0 and m ≤ j R x by the embedding W 2,2 (T 3 ) ֒→ L ∞ (T 3 ) and the approximability of Π h from (2.9). The first term can be absorbed for κ small enough, whereas the second one (in summed form and expectation) is bounded by h 4 e cR 2 due Corollary 4.5 (a). Similarly, we have for any m ≤ j R . The last term in the second line will be dealt with by Gronwall's lemma leading to a constant of the form ce cR 4 . The last term in teh second line (in summed form, for m = 1, . . . , j R , and expectation) can be controlled by (4.10), Lemma 4.5 (c) and 5.2 (each of them with with q = 3). The final line is bounded by c(δ) τ R 4 h 2 u m 2 W 2,2 x using (2.9) and hence can be controlled by Lemma 3.1 (c).
In order to estimate the stochastic term I 4 (m) we write In both estimates, teh first term can be absorbed, the second one can be handled by Gronwall's lemma, while the last one is bounded by h 2 using Lemma 4.1.
In order to estimate I 5 (m) we write Id − Π π h ∆ −1 div Φ(u n−1 ) dW div(Π h e h,n − Π h e h,n−1 ) dx =: N 1 m,2 + N 2 m,2 . Following again [7] we have The first term is bounded by h 4 e cR 2 using Corollary 4.5 (a) (recall that u 0 ∈ L 2 (Ω; W 1,2 (T 3 ))). The second term can be absorbed if δ ≪ 1. Finally, Π h e h,n − Π h e h,n−1 where the first term can be absorbed and the second one is bounded by h 2 on account of Lemma 4.1. We conclude that The claim follows now as R = R(h) ≤ c −1/4 4 √ −εh 2 , where ε > 0 is arbitrary, and applying Markov's inequality.