On the 3D Navier–Stokes equations with a linear multiplicative noise and prescribed energy

For a prescribed deterministic kinetic energy, we use convex integration to construct analytically weak and probabilistically strong solutions to the 3D incompressible Navier–Stokes equations driven by a linear multiplicative stochastic forcing. These solutions are defined up to an arbitrarily large stopping time and have deterministic initial values, which are part of the construction. Moreover, by a suitable choice of different kinetic energies which coincide on an interval close to time 0, we obtain non-uniqueness.


Introduction 1.1 Motivation and Previous Works
Proving existence and smoothness of strong solutions to the incompressible Navier-Stokes equations is a longstanding open problem in the field of fluid dynamics.It is the subject of one of the Millennium Prize Problems and caused, especially in the recent years, worldwide much attention.
In 2009 De Lellis and Székelyhidi developed the method of convex integration which permits them to construct infinitely many weak solutions to the incompressible Euler equations which dissipate the total kinetic energy and satisfy the global and local energy inequality [DLS09], [DLS10], [DLS13].Together with Isett, Buckmaster and Vicol, they used this scheme to prove Onsager's conjecture in 2016 and 2017, respectively [Is16], [BDLSV17].Their work was a groundbreaking success regarding the theory of weak solutions and was seminal for many further results.
Buckmaster and Vicol applied the technique of convex integration in 2019 to establish existence and non-uniqueness of weak solutions to the incompressible Navier-Stokes equations with finite kinetic energy [BV19a], [BV19b].One year later, in July 2020, a similar result for power law flows in the deterministic setting follows by Burczak, Modena and Székelyhidi [BMS21].
Meanwhile the methods of convex integration even found their way in the theory of stochastic partial differential equations as well.A possible advantage by adding suitable stochastic perturbations into partial differential equations consists in regularizing deterministically ill-posed problems.Considering in particular a linear multiplicative noise provides additionally a certain stabilization effect on the three dimensional Navier-Stokes equations see e.g.Röckner, Zhu and Zhu [RZZ13].However this phenomena does not help when it comes to the question of uniqueness of probabilistically strong solutions to the Navier-Stokes equations, as also shown in the present paper.
A stochastic counterpart to [BV19a], [BV19b] was obtained by Hofmanová, Zhu and Zhu, who were able to show existence and non-uniqueness of analytically weak and probabilistically strong solutions to the incompressible Navier-Stokes equations also with a prescribed energy but additionally driven by a stochastic additive noise [HZZ21a].Chen, Dong, Hofmanová, Zhu and Zhu developed further stochastic versions of convex integration to prove dissipative martingale solutions to 3D stochastic Euler equations, global existence and non-uniqueness for 3D stochastic Navier-Stokes equations with space time white noise, non-unique ergodic solutions for 3D Navier-Stokes equations and Euler equations as well as sharp non-uniqueness of solutions to stochastic Navier-Stokes equations [HZZ21b], [HZZ22a], [HZZ21c], [CDZ10].Very recently first results regarding the 3D Euler equations with transport noise, power-law equations with additive noise and surface quasi-geostrophic equations with irregular spatial perturbations could already be achieved [HLP22], [LZ22], [HZZ22b].
In their previous work [HZZ19] Hofmanová, Zhu and Zhu were concerned with the incompressible Navier-Stokes equations perturbed by three different stochastic forces: one with a multiplicative, one with an additive and one with a non-linear noise.In all these cases they proved that the law of analytically weak and probabilistically strong solutions is not unique and that the solutions violate the corresponding energy inequality.
A series of further results regarding non-uniqueness in law for several stochastic partial differential equations, such as the transport-diffusion equation, the 3D magnetohydrodynamics system or also the 3D Navier-Stokes equations, was attained by Yamazaki and Rehmeier and Schenke [Ya21a], [Ya21b], [Ya22], [RS22].All these equations are perturbed by different kinds of random noise but without any statement concerning a prescribed energy.
In the present paper we follow the ideas of [HZZ19] and [HZZ21a] to prove the existence and non-uniqueness of solutions to the 3D incompressible Navier-Stokes equations driven by a multiplicative noise up to an arbitrarily large stopping time.Opposed to [HZZ19] we are now able to deduce that the kinetic energy of the constructed solution equals to a prescribed energy profile.
To this end we make use of the transformation and convex integration technique therein, but had to reformulate the main iteration.For a detailed introduction into the Fourier analysis of convex integration we refer to [Be23].
For some pressure P : [0, T ] × T 3 × Ω → R the system governs the time evolution of the velocity u : [0, T ] × T 3 × Ω → R 3 of an incompressible fluid with viscosity ν.Throughout the paper the viscosity is, for the sake of simplicity, assumed to be 1, which physically corresponds to water of 20 • C (cf. [AT74], p.1238, Table 3) and moreover we will often deal with the T 3 -periodic extensions of u and P on R 3 , which can be identified with functions on the three dimensional flat torus R 3 \ (2πZ) 3 (cf.Lemma A.1).In this paper we are concerned with finding solutions in the following sense: Definition 1.1.An (F t ) t∈[0,T ] -adapted solution u to (1.1) is said to be probabilistically strong and analytically weak if i) it belongs for some γ ∈ (0, 1) to C [0, T ]; Note that working with divergence free test functions in the definition above allows to eliminate the pressure term, which can be reconstructed after a weak solution has been found.We can now formulate our main result: for some constants 4 < e ≤ ē and e > 0 a probabilistically strong and analytically weak solution u to (1.1), depending explicitly on the given energy e, can be constructed up to a P-a.s.strictly positive stopping time . This solution has deterministic initial value u 0 and belongs to C [0, τ ]; H γ T 3 a.s.for some γ ∈ (0, 1).It obeys and its kinetic energy is given by e, i.e.

u(t) 2
L 2 = e(t), Moreover the following consistency result holds: If two energies with the same bounds e, ē, e coincide for some t ∈ [0, L] everywhere on [0, t], then so do the corresponding solutions on [0, t ∧ τ ].
The proof of Theorem 1.2 is based on a convex integration scheme.That is, after transforming (1.1) to a random PDE, we develop an iteration procedure and apply it to the just received equation.
More precisely, if u solves (1.1), the function v := e −B u is by Itô's formula a solution to the ensuing system where Θ is the stochastic process given by and the converse is also true.In fact, applying Itô's formula to the smooth function dB t and by Itô's product rule and (1.1) we conclude (1.2).

Organization of the Paper
We organize the present paper as follows: Chapter 2 is devoted to the collection of basic notations used throughout this paper.In Chapter 3 we outline the convex integration technique to prove Proposition 4.1.This is the core of the proof of our main Theorem 1.2, which is established in Chapter 4. The Appendix A covers some lemmata used in the previous chapters.

Preliminaries
In order to define several function spaces and operators we need the Fourier transform and the inverse Fourier transform of a function u on T 3 given by for any n ∈ Z 3 and x ∈ T 3 , respectively, where the series shall be understood as the limit of partial sums with square-cut off Moreover, for d ∈ N we will often deal with the spaces of symmetric or traceless d×d-matrices A, designated by R d×d sym and Rd×d , respectively.As usual we consider the Frobenius norm on them and in order to elucidate that the matrix itself is traceless we will frequently write Å instead of A.

Function Spaces
For some N ∈ N 0 ∪ {∞}, T ∈ [0, ∞) and a Banach space (Y, • Y ) we denote the space of all N -times continuously differentiable functions from X ∈ {T 3 , R 3 } to Y by C N XY and from X = (−∞, T ] to Y we shorten C N T Y .Endowed with the natural norms respectively, it is well known that they become Banach spaces.Sometimes we will omit writing X or Y if it is clear from the context which domain or codomain is considered and moreover we will frequently write C XY instead of C 0 XY .In particular u C is the usual supremum norm, which will also be used for more general normed spaces X.There should be no misunderstanding when we talk about C N c -and C N b -functions, meaning the N -times continuous differentiable ones with compact support and the ones that are either bounded from above, bounded from below or even both.Besides it is customary to introduce the space of test functions and we moreover label by C N T,x the space of all N -times continuously differentiable functions on (−∞, T ] × T 3 , equipped with the corresponding norm We will speak of Hölder-continuous functions u on X ∈ {T 3 , R 3 } of exponent N + ι with N ∈ N 0 and ι ∈ (0, 1], whenever is the ι th -Hölder seminorm.For functions on (−∞, T ] we define the space in the same manner.The space of Bochner-integrable functions L p (X; Y ) consists of the equivalence classes of all functions u : X → Y , which coincide almost everywhere and for which the usual L p -norm should be the usual Sobolev space for all k ∈ N 0 and 1 ≤ p ≤ ∞.Moreover we denote by W k,p 0 (X) the closure of D (X) in W k,p X , • W k,p and by W −k,q (X) the dual space of W k,p 0 (X) with 1 p + 1 q = 1.The Bessel potential spaces are for any p, q ∈ (1, ∞), 1 p + 1 q = 1 on T 3 defined in the spirit of [Tr83], [Tr92]: We set for s ≥ 0, whereas we define < ∞ whenever s > 0, which can be identified with the dual space of W s,q T 3 (see also [DNPV11]).Endowed with the canonical norms , respectively, these spaces become Banach spaces and furthermore we stipulate we denote by p (X) the usual space of sequences for which the corresponding norm • p is finite.We will often deal with functions of zero mean.So for convenience we set for any function space X, where P =0 is the projection onto functions with non-zero frequencies, which will be introduced in Definition 2.2 below.

Operators
We will deal with the extended Leray projection P = Id −∇∆ −1 div on the Bessel potential spaces W s,p =0 T 3 for every real s ≥ 0 and p ∈ (1, ∞), which enjoys the following useful property.Lemma 2.1.The Leray projection commutes almost everywhere with any partial derivative on W 1,p =0 T 3 and if we work with time depended functions u, satisfying for some g ∈ L 1 T 3 and all x ∈ T 3 , even on C 1 (−∞, T ]; C =0 T 3 with T > 0. In other words holds for all x ∈ T 3 , each t ∈ R and j = 1, 2, 3.
Moreover, it is very common and practical to introduce the Fourier multiplier operators, which project a function onto its null mean frequencies and onto its frequencies ≤ κ in absolute value.
Definition 2.2.For any u ∈ L p T 3 with 1 < p ≤ ∞ we denote by the projection onto its non-zero frequencies.Furthermore for all real κ ≥ 1 we define the operators P ≤κ and P ≥κ as and respectively, where we set χ κ := χ • κ for the smooth compactly supported function χ ∈ C ∞ c R 3 given by Lemma 2.3.The above operators P =0 , P ≤κ and P ≥κ are for each real κ ≥ 1 and 1 < p ≤ ∞ continuous on L p T 3 , where the implicit constants do not depend on κ.
Lemma 2.4.For all real κ ≥ 1 it holds Lemma 2.5.For all T L 3 -periodic functions u ∈ L p T 3 with L ∈ N and 1 < p ≤ ∞ the operators P ≤κ and P ≥κ can be written as and respectively.If additionally L > κ, we have that

Outline of the Convex Integration Scheme
The convex integration scheme is an iterative procedure giving rise to solutions to several deterministic and stochastic PDE's.Also in the present paper we construct, based on a suitable starting point, a solution v q to (1.2) on (−∞, τ ] perturbed by an error term Rq , called Reynolds stress, on the level q ∈ N 0 .While the iterations v q approach the desired velocity v, solving (1.2) on [0, τ ], the stress tensor Rq becomes step by step infinitesimally small.The convex integration technique provides typically a way to construct even an infinite number of such solutions, as it is also the case in the present paper.
For the construction of our sequence (v q , Rq ) q∈N0 we previously have to fix some parameters, done in the following section, Section 3.1.Section 3.2 is concerned with the key bounds that each pair (v q , Rq ) has to fulfill.In Section 3.3 we recall the definitions of intermittent jets used to give an explicit expression of v q+1 .Section 3.4 is then concerned with the verification of the key bounds of the next iteration step v q+1 , culminating in the proof of convergence of the sequence (v q ) q∈N0 .In Section 3.5 we decompose the Reynolds stress and in Section 3.6 we verify the key bound for this tensor on the level q + 1.We close this chapter in Section 3.7 by proving that our constructed sequence is adapted and deterministic at any time t ≤ 0.
Note that the assumption 33(2π Choosing furthermore a, b sufficiently large and α, β sufficiently small enough, permits to suppose q 0 = 1.

Start of the Iteration
In view of (1.2) we are concerned with an adapted velocity field v q and a symmetric traceless matrix Rq , solving the transformed Navier-Stokes-Reynolds system reads and that obey for q ∈ N 0 , any t ∈ (−∞, τ ] and some universal constant M 0 ≥ 1.Here we have to include also negative times in order to avoid several problems by decomposing the Reynolds stress in Section 3.5.For this purpose the energy e and the Brownian motion B are continuously extended to functions on (−∞, τ ] by taking them equal to the value at t = 0. We furthermore set 0 r=1 := 0 and point out that q r=1 δ 1/2 r is bounded by 2. Indeed, thanks to the assumed a βb ≥ 2 it holds (3.4) at any time t up to the stopping time τ .In other words, the given energy e will be gradually approximated by the kinetic energies of the iterations e B v q .

Construction of v q+1
In order to obtain more regularity we refrain in defining the next iteration v q+1 in mere dependence of v q .Instead we intend to define v q+1 in terms of the mollified velocity field v and a perturbation w q+1 , pointed out in the two subsequent sections.For short, the new velocity field will be given as

Mollification
Let us start with the mollification of the velocity v q .For this end we consider the standard mollifier on R 3 and the shifted mollifier on R, where C space , C time > 0 are chosen, such that ´R3 φ(x) dx = 1 and ´R ϕ(t) dt = 1, as usual.Thus the convolution of v q with the rescaled mollifiers φ := 1 3 φ( • ) and ϕ := 1 ϕ( • ) yield a smooth function in space and time, and of Θ with ϕ just in time.
For short, we consider where R remains traceless, because the mollification acts only componentwise.It is worth noting that the adaptedness of v , R and Θ follows from the fact that the support of ϕ lives in R + .In fact, let Π be the set that contains all partitions of [0, ] of the form {0 = s 0 < s 1 < . . .< s n = } for some n ∈ N. Then for any t ∈ (−∞, τ ] and x ∈ T 3 the function Taking into account that the convolution in space does not influence the behavior of (v q * t ϕ ) in time, v inherits the (F t ) t∈R -adaptedness and so do R and Θ .It is easy to see that v is close to v q w.r.t.• L 2 at any time up to the stopping time τ and fulfills (3.3) as well.More precisely, the subsequent Lemma holds.
Lemma 3.1.The mollification v , defined above, enjoys the following bounds The proof is rather straightforward, so that we do not pursue this here.

Perturbation
Let us now have a closer look at the perturbation w q+1 .We will decompose it into three parts: the principle part w q+1 .Each of them will be defined in terms of the amplitude functions and the intermittent jets, introduced and worked out in [BV19b] and [BV19a], respectively.In what follows we will give a short review of the necessary facts.First, we recall the essential geometric lemma from [BV19a].

Lemma 3.2 (Geometric Lemma).
There exists a family of smooth real-valued functions (γ ξ ) ξ∈Λ , where Λ is a set of finite directions, contained in S 2 ∩ Q 3 , so that each symmetric 3 × 3 matrix R, satisfying R − Id F ≤ 1 2 , admits the representation Second, based on this lemma, we define for all N ∈ N the constant

Amplitude Functions
The function γ ξ in the geometric Lemma 3.2 is used to define the amplitude functions as for any t ∈ (−∞, τ ], x ∈ R 3 , ω ∈ Ω, where η denotes the mollification in time of η q .Since we start our iteration procedure with R0 = 0, we include a small perturbation in the definition of ρ to avoid its degeneracy, whereas the function η q should pump energy into the system.This enables us to confirm the key bounds (3.3) at level q + 1 in Section 3.4.2and 3.6.1,respectively.Moreover, we point out that (3.5) and our choice of parameters a βb ≥ 2, b ≥ 7 (cf.Section 3.1) ensure 2 in the geometric Lemma 3.2, so that the amplitude functions are actually well-defined.Now we would already like to sum up some properties of these functions here.
The proofs of this Lemma and of Lemma 3.6 are collected in Section 3.4.1.In view of [HZZ19] one might heuristically think that it makes sense to define the amplitude func- In this case the first statement of Theorem 1.2 remains true but there would appear several problems in order to deduce the energy equality later on.The factor Θ −2 in the definition of η q is thereby essential to make use of (3.5), so that we can derive practical bounds for amplitude functions later on (cf.e.g.(3.21)), whereas Θ in front of η in the definition of ρ is needed to get a suitable cancellation in the first term of (3.30).

Intermittent Jets
Let us now proceed with the construction of the intermittent jets.To this end consider two smooth functions with support in a ball of radius 1 and center 0, where Φ should solve the Poisson equation φ = −∆Φ.We require φ and ψ to admit the normalizations and ψ to have zero mean.Note that Green's identity implies ´T2 φ dx = ´T2 −∆Φ dx = 0 as well.Moreover, we define q+1 , so that we find by our choice of parameters (c.f.Section 3.1) Since the rescaled cut-off functions remain compactly supported, we will henceforth identify them with their T 2 , T 2 and T-periodic versions.
The vectors ξ ∈ Λ in the geometric Lemma 3.2 are used to construct the building blocks (3.10) of our intermittent jets.Strictly speaking, let us select (α ξ ) ξ∈Λ ⊆ R 3 in such a way that for each ξ, ξ ∈ Λ and z 1 , z 2 ∈ Z, which forces the families φ (ξ) ξ∈Λ and Φ (ξ) ξ∈Λ given by to have mutually disjoint support.Here we consider ψ (ξ) at any time t ∈ R. The vector A ξ ∈ S 2 ∩ Q 3 should be orthogonal to ξ, so that {ξ, A ξ , ξ × A ξ } ⊆ S 2 ∩ Q 3 forms an orthonormal basis for R 3 and n * ∈ N denotes the least common multiple of the denominators of the rational numbers ξ i , (A ξ ) i and (ξ With these preparations in hand, we introduce the intermittent jet and its incompressibility corrector where (ξ) becomes divergence free.Their spatial support is then contained in some cylinders of radius r +r ⊥ n * r ⊥ λ and axis being the line passing through (−µt, at each time t ∈ R.So by possibly shifting α ξ in direction of A ξ , ξ × A ξ , Lemma A.2 guarantees that their supports are still disjoint for all distinct ξ, ξ ∈ Λ.As a consequence of the orthogonal directions of oscillations for the functions defined in (3.10), we may deduce Lemma 3.4.The building blocks ψ (ξ) and φ (ξ) obey for each n, m, N ∈ N 0 , p ∈ [1, ∞) and all multi-indices α, β ∈ N 3 0 .as well as the following fundamental bounds Lemma 3.5.For any N, M ∈ N 0 and p where the implicit constants merely depend on M, N and p.
In the special case N = M = 0, p = 2 Lemma 3.4 and the normalizations in (3.9) even entail Based on this, we are now able to define the principle part the temporal corrector which will provide a better handling of the oscillation error later on, and the incompressibility corrector whose purpose is to ensure that w q+1 is divergence free and to have zero mean.In fact, the expression can be easily verified by a direct computation, so that w q+1 is obviously divergence free and since a (ξ) V (ξ) is a smooth function with periodic boundary conditions, it has additionally zero mean.Notably, these properties carry over to the total perturbation and we may bound each part of it as follows.
Lemma 3.6.At any time t ∈ (−∞, τ ], each component of the perturbation w q+1 can be estimated a) in C t L p for any p ∈ (1, ∞) as In the specific case p = 2 the principle part admits the stronger bound The proof is postponed to Section 3.4.1.
It is worth mentioning that the factor Θ −1/2 in the principle part w q+1 of the perturbation is needed to establish (3.31), which in turn is essential to deduce a handy expression of the oscillation error (3.43).We also define the incompressibility corrector w

Inductive Estimates for v q+1
So far we have developed a sequence (v q ) q∈N0 , solving (3.2) on the level q ∈ N 0 and with the corresponding Reynolds stress constructed below, also on the level q + 1.As we will see in Section 3.4.4this sequence converges in C (−∞, τ ]; L 2 T 3 , so that the limit function will be our desired weak solution to (1.2) on [0, τ ].However, let us take one step after the other.We start by aiming (v q ) q∈N0 to admit the bounds (3.3a) and (3.3b) on the level q + 1.

Preparations
For this purpose we need to control the amplitude functions and intermittent jets, so that we go back to Lemma 3.3, and Lemma 3.6.The key ingredient of the proof of Lemma 3.3 is the ensuing result from [BDLIS15], p.163, where we set Proof of Lemma 3.3.
Let us proceed with a bound for ρ 1/2 . We will verify For this purpose we need 1.Claim: R Proof.Owing to the embedding W 4,1 ⊆ L ∞ (cf.[Ev10], p.284, Theorem 6), Fubini's theorem and (3.3c) we may deduce (3.21) Proof.Combining (3.18), (3.20) and η q ≥ 0 with (3.5), results in For k > 0 we introduce the smooth function Moreover, we use Leibniz's rule together with the fact η q ≥ 0 and (3.5) to calculate As a result of these three bounds In order to find a bound for ρ 1/2 C k t,x we intend to make use of Proposition 3.7 again.This time, however, applied to the function holds, provided k > 0. As a consequence and accordingly (3.21) Let us now have a closer look at . Our aim is to verify (3.23) The case k = N is trivial, whereas Proposition 3.7 and (3.7) again imply and we assert 3.Claim: R .
Applying Proposition 3.7 to the functions and ρ yields according to (3.21) and (3.1b) The penultimate step additionally follows from (3.20) and ρ ≥ , whereas the last step also holds due to (3.1b).
As a result Altogether we therefore bound (3.8b) as provided N > 0. However, the final bound is due to (3.7) and (3.19) even valid for N = 0.The proof of Lemma 3.3 is therefore complete.
Proof of Lemma 3.6.

Verifying the Key Bounds on the Level q + 1
With these preparations in hands, we are now able to justify (3.3a) and (3.3b) on the level q + 1.
As pointed out in Section 3.1, we increase a and b in a fashion that q+1 ≤ 1 2 as well as Kλ 5 q λ −5 q+1 ≤ 1 2 .That means (3.3b) stays true on the level q + 1.

Control of the Energy
It remains to affirm (3.5) at level q + 1, which is equivalent in showing or expressed in terms of the function η q As we will see it is not enough to require only boundedness of the energy e here, rather it is necessary to ask for a uniform bound of its derivative e .From a physical point of view it means that the change of kinetic energy and therefore the acceleration of a fluid can not become arbitrary large.For example, if we consider a river that flows uphill, the gravity will influence its flow rate, so that the gradient can only attend limited values.Anyway, lets come back to the mathematical computations: and we proceed with a bound for I.
Moreover, we already know about the existence of some K > 0, such that Choosing the parameters in such a way that K Finally, plugging all the above estimates from I through to V into (3.30)proves (3.5) on the level q + 1.

Convergence of the Sequence
Moreover, we claim (v q ) q∈N0 to be a Cauchy sequences in C (−∞, τ ]; L 2 T 3 .Thanks to (3.27) and (3.6a) we obtain for any t ∈ (−∞, τ ] and q ∈ N 0 , which results together with a βb ≥ 2 in As a consequence the sequence converges and is therefore in particular Cauchy in R, i.e. for each ε > 0 there exists some N ∈ N 0 , such that

Decomposition of the Reynolds Stress Rq+1
In order to find an expression for the Reynolds error at level q + 1 we plug v q+1 into (3.2) and exploit the formula div A = div Å + 1 3 ∇ tr A together with tr(a ⊗ b) = a • b, which holds for arbitrary quadratic matrices A and vectors a, b, respectively, to derive div( Rq+1 ) − ∇p q+1 =:pcor where we have set p := p q * t ϕ * x φ .Keeping in mind that div a 2 (ξ) P =0 (W (ξ) ⊗ W (ξ) ) has zero mean, since a 2 (ξ) P =0 (W (ξ) ⊗ W (ξ) ) is a smooth function with periodic boundary conditions, we invoke (3.31) to rewrite the oscillation error as div( Rosc ) + ∇p osc = ∂ t w where Equipped with this knowledge it make sense to define Rlin := R∂ t (w with the corresponding pressure terms The Reynolds stress at level q + 1 then becomes Rq+1 = Rlin + Rcor + Rosc + Rcom and remains by construction symmetric and traceless.

Inductive Estimates for the Reynolds Stress Rq+1
3.6.1 Verifying the Key Bound on the Level q + 1 Next, we aim at verifying (3.3c) on the level q + 1.To do so, we need r 2/p−2 ⊥ r 1/p−1 ≤ λ α q+1 , which can be achieved by taking p ∈ (1, 16  16−7α ].We start with a bound for the Linear error.

Linear Error
Remembering that R∆ and R curl are according to Lemma 3.9 bounded operators on C ∞ T 3 ; R 3 , we employ Hölder's inequality, (3.14) and (3.24) to obtain .
II A bound for the second term can be deduced in the same manner: .
Under the assumptions 161α < 1 7 and α > 4βb 2 , we therefore get by Hölder's inequality for some Ŝ > 0. To absorb this implicit universal constant, we impose Ŝ M

Commutator Error
We will estimate each term of Rcom separately.
I To find a bound for Θ − Θ Ct we proceed in a similar way as in (3.36) by using Itô's formula.More precisely we find Together with (3.44) it furnishes II & IV Keeping Cauchy-Schwarz's inequality in mind, we combine (3.44) with (3.6a) and (3.6b) to find Note that the final bound is also valid for IV.
III Moreover, we invoke (3.49), (3.44) and (3.6b) to get V Thanks to the normalizations of mollifiers we have , where V 1 boils owing to (3.44) and Cauchy-Schwarz's inequality down to Here (8) follows similarly to (3.36), this time however with e(s) replaced by v q (s, x) and ē, e by v q C 1 t,x .One more time we employ (3.44) and stick to standard mollification estimates in order to control V 2 as follows Plugging the above bounds into (3.48) and taking (3.3b), (3.3a), (3.4) and (3.1a) into account, leads to Finally, combining 33(2π) 3/2 M 0 m 9/4 ēλ So altogether this proves (3.3c) at level q + 1.

Convergence of the Sequence
Furthermore note that it follows instantly from (3.3c) that Rq q∈N0 is a zero sequence in C (−∞, τ ]; L 1 T 3 .

Adapted
Now, we aim at proving the adaptedness of the next iteration step (v q+1 , Rq+1 ).As already mentioned in Section 3.3.1 we emphasize again that the mollified velocity field v as well as R and Θ remain (F t ) t∈(−∞,τ ] -adapted.Since the energy e is deterministic, the adaptedness of η follows in the same way as for v .These facts in turn yield the adaptedness of ρ and thus also of the amplitude functions a (ξ) as composition of them.Furthermore note, that the building blocks φ, Φ and ψ of the intermittent jets do not depend on ω, so consequently V (ξ) , W (ξ) and the intermittent jets themselves are deterministic as well.Together with the fact that any partial derivative of a (ξ) in space remains (F t ) t∈(−∞,τ ] -adapted, establishes the adaptedness of w (p) q+1 and w (c) q+1 .Moreover, the Leray projection P and obviously also the projection P =0 onto functions with zero mean preserve adaptedness, which confirms the adaptedness of the temporal corrector w (t) q+1 .To summarize our above considerations: we have proven that v and the total perturbation q+1 are, at any time t ∈ (−∞, τ ], F t -measurable, which finally verifies the adaptedness of v q+1 = v + w q+1 and Rq+1 .

Deterministic
We will use an induction argument to verify, that the constructed sequence v q (t, x), Rq (t, x) q∈N0 is deterministic for each t ≤ 0 and x ∈ T 3 .Obviously the starting point v 0 , R0 = (0, 0) does not depend on ω at any time.So assuming v q (t, x) for all t ≤ 0 and x ∈ T 3 to be deterministic, it can be easily seen that v (t, x, ω) = ˆ 0 ˆ|y|≤ v q (t − s, x − y, ω)ϕ (s)φ (y) dy ds, R (t, x) and Θ (t) are deterministic as well.Combined with the fact, that η q (t, •) is deterministic, ρ(t, •) and hence a (ξ) (t, •) does not depend on ω either.Based on this we are able to conclude that each part of the total perturbation w q+1 (t, •), is deterministic, that is to say w On the one hand, this together with the independence of v (t, •) of ω results in the deterministic behavior of v q+1 at time t ≤ 0. On the other hand it also follows that Rlin (t, •), Rcor (t, •), Rosc (t, •), Rcom (t, •) and thereby Rq+1 (t, •) are deterministic as well, yielding the assertion.

End of the Proof of Theorem 1.1
Summarizing our previous results, we formulate the following Proposition.
Proof of Theorem 1.2.Existence: According to Proposition 4.1 there exists a sequence v q , Rq q∈N0 , so that v q (t), Rq (t) is for every q ∈ N 0 and t ≤ 0 deterministic and so that v q , Rq solves (3.2) on (−∞, τ ] in the weak sense.That means for any divergence free test function ϕ ∈ C ∞ T 3 ; R 3 the pair v q , Rq satisfies ˆT3 v q (t, x) − v q (0, x) • ϕ(x) dx Furthermore we may assume the existence of a limit lim q→∞ v q =: v ∈ C (−∞, τ ]; L 2 T 3 , which causes that each term on the left hand side of (4.1) will converge for q → ∞ pointwise.In fact integrating by parts again and Cauchy-Schwartz's inequality furnish So passing to the limit on both sides of (4.1) shows that v is an analytically weak solution to (1.2) with deterministic initial condition.Therefore u := e B v solves (1.1) on [0, τ ] in the probabilistically strong and analytically weak sense; of course also with a deterministic initial condition u 0 .
Regularity: It remains to verify that the convergence of (v q ) q∈N0 to v even takes place in C (−∞, τ ]; H γ T 3 .For this end we combine Hölder's inequality with exponents 1 γ and 1 1−γ with Plancherel's theorem to accomplish So if we impose γ ∈ 0, β 5+β , the above series will converge as it can be bounded by a geometric series.Thus the sequence is as a convergent sequence particularly Cauchy in R, which means that for all ε > 0 there exists some N ∈ N 0 so that holds for every n ≥ k ≥ N .In other words (v q ) q∈N0 is also a Cauchy sequence in the Banach space C (−∞, τ ]; H γ T 3 for γ ∈ 0, β 5+β , furnishing the existence of the limit in C (−∞, τ ]; H γ T 3 , which equals, by virtue of the uniqueness of the limit, to v.
for any ε > 0, sufficiently large q ≥ 0 and every t ∈ [0, τ ] and ω ∈ Ω.Note that the constant on the right hand side neither depends on t nor on ω.

Figure 3 . 1 :
Figure 3.1: Implication scheme of the adaptedness of each process up to the desired iteration step vq+1.
q+1 do not depend on ω at each time up to time 0. The functions φ, Φ and ψ and hence the intermittent jets W (ξ) as well as its incompressibility corrector W (c) (ξ) and V (ξ) are by definition deterministic.