Well-posedness for a stochastic 2D Euler equation with transport noise

We prove the existence of a unique global strong solution for a stochastic two-dimensional Euler vorticity equation for incompressible flows with noise of transport type. In particular, we show that the initial smoothness of the solution is preserved. The arguments are based on approximating the solution of the Euler equation with a family of viscous solutions which is proved to be relatively compact using a tightness criterion by Kurtz.


Introduction
Consider the two-dimensional Euler equation modelling an incompressible flow perturbed by transport type stochasticity with initial condition ω 0 , where (ξ i ) i are time-independent divergence-free vector fields, the operator £ i is given by £ i ω t = ξ i · ∇ω t , and (W i ) i∈N is a sequence of independent Brownian motions. Classically, u t stands for the velocity of an incompressible fluid and ω t = curl u t is the corresponding fluid vorticity. The stochastic part considered here follows the Stochastic Advection by Lie Transport (SALT) theory (see [15,18,27,33]) and corresponds to a stochastic integral of Stratonovich type. The Euler equation is used to model the motion of an incompressible inviscid fluid. A representative aspect in this context is the study of the fluid vortex dynamics modelled by the vorticity equation. There is a vast literature on well-posedness in the deterministic setting, see e.g. [4,10,24,35,38,39,44,51], and references therein.
The introduction of stochasticity into ideal fluid dynamics has received special attention over the past two decades. On one hand, comprehensive physical models can be obtained when the stochastic term accounts for physical uncertainties [15,16,18,33], whilst, in some cases, the regularity properties of the deterministic solution can be improved when the right type of stochasticity is added [21,26,27,31]. Global existence of smooth solutions for the stochastic Euler equation with multiplicative noise in both 2D and 3D has been obtained in [31]. In [8], a weak solution of the Euler equation with additive noise is constructed as an inviscid limit of the stochastic damped 2D Navier-Stokes equations. A martingale solution constructed also as a limit of Navier-Stokes equations but with cylindrical noise can be found in [13]. Existence and uniqueness results with different variations in terms of stochastic forcing and approximations can be found in [6,19,42,45,46] and references therein. An overview of results on this topic is provided in [7].
The analysis of nonlinear stochastic partial differential equations with noise of transport type has recently expanded substantially, see e.g., [2,3,5,15,16,18,30,33]. Existence of a solution for the two-dimensional stochastic Euler equation with noise of transport type has been considered in [12]. While in [12] the authors prove the existence and pathwise uniqueness of a distributional solution in L ∞ (T 2 ), in this paper we are concerned with the existence of a strong solution and give conditions under which the solution enjoys smoothness properties. 1 In [25], a random point vortices system is used to construct a so-called ρ-white noise solution. Local well-posedness and a Beale-Kato-Majda blow-up criterion for the three-dimensional case in the space W 2,2 (T 3 ) has been obtained in [18]. Full well-posedness for a point vortex dynamics system based on this equation has been proven in [28]. The linear case has been considered in [26] and in [29].
In the sequel, T 2 is the two-dimensional torus, k ≥ 2 is a fixed positive integer, and W k,2 (T 2 ) is the usual Sobolev space (see Sect. 2). The main result of this paper is the following: Theorem: Under certain conditions on the vector fields (ξ i ) i the two-dimensional stochastic Euler vorticity equation admits a unique global (in time) solution which belongs to the space W k,2 (T 2 ). Moreover, ω t is a continuous function of the initial condition.

Remark 1
As stated above, the stochastic terms in (1) are stochastic integrals of Stratonovich type. We interpret Eq. (1) in its corresponding Itô form, that is The assumptions on the vector fields (ξ i ) i are described in Sect. 2. In short, they are assumed to be sufficiently smooth, their corresponding norms to decay sufficiently fast as i increases, so that the infinite sums in (1), respectively, in (2) make sense in the right spaces (see condition (4) below). Importantly, we do not require the additional assumption 2 used in [12]. As a result, in the Itô version (2) of the SPDE, the term 1 2 ∞ i=1 ξ i · ∇(ξ i · ∇ω t ) does not reduce to c ω t . This would simplify the analysis as, in this case, the Laplacian commutes with higher order derivatives. Morever, it commutes with the operation of convolution with the Biot-Savart kernel, an essential ingredient used in [12]. The general term 1 2 ∞ i=1 ξ i ·∇(ξ i ·∇ω t ) makes the analysis harder. We succeed in controlling it by considering it in tandem with the term coming from the quadratic variation of the stochastic integrals (see Lemma 25i.) appearing in the evolution equation for the process t → ω t 2 . A similar technical difficulty appears when trying to control the high-order derivatives of the vorticity. Nonetheless, this is achieved through a set of inequalities (see Lemma 25) that have first been introduced in the literature by Krylov and Rozovskii [32,36] and recently used by Crisan et al. [18]. Again, we emphasize that these rather surprising inequalities hold true without imposing assumptions on the driving vectors (ξ i ) i other than on their smoothness and summability. This finding is particularly important when using this model for the purpose of uncertainty quantification and data assimilation: for example, in [15][16][17], the driving vectors (ξ i ) i are estimated from data and not a priori chosen. The methodology used in these papers does not naturally lead to driving vector fields that satisfy Assumption (3) so removing it is essential for our research programme.
We emphasize that the appearance of the second order differential operator ω → 1 2 ∞ i=1 ξ i · ∇(ξ i · ∇ω) in the Itô version of the Euler equation does not give the equation a parabolic character, even if one assumes the restriction (3) with c chosen strictly positive. Equation (1) is truly a transport type equation and one cannot expect the initial condition to be smoothed out. The best scenario is to prove that the initial level of smoothness of the solution is preserved. This is indeed the main finding of our research. Moreover, we show that our result can be extended to cover also L ∞solutions in the Yudovich sense.
The paper is organised as follows: In Sect. 2 we introduce the main assumptions, key notations, and some preliminary results. In Sect. 3 we present our main results: in Sect. 3.1 we show that the solution is almost surely pathwise unique, while in Sect. 3.2 we prove existence of a strong solution (in the sense of Definition 3). In Sect. 4 we proceed with an extensive analysis of a truncated form of the Euler equation: uniqueness (Sect. 4.1) and existence -based on a new approximating sequence introduced in Sect. 4.2. At the end of this section we show continuity with respect to initial conditions for the original equation. In Sect. 5 we show existence, uniqueness, and continuity for the approximating sequence of solutions constructed in Sect. 4.2. In Sect. 6 we show that the family of approximating solutions is relatively compact. In Sect. 7 we present an extension of the main results to the Yudovich setting. The paper is concluded with an "Appendix" that incorporates a number of proofs of the technical lemmas and statements of some classical results.

Preliminaries
We summarise the notation used throughout the paper. Let ( , F, (F t ) t , P) be a filtered probability space, with the sequence W i i∈N of independent Brownian motions defined on it. Let X be a generic Banach space. Throughout the paper C is a generic notation for constants, whose values can change from line to line.
• L p (T 2 ; X ) 3 is the class of all measurable p-integrable functions f defined on the two-dimensional torus, with values in X ( p is a positive real number). The space is endowed with its canonical norm f p = ally, for p = ∞ we denote by L ∞ the space of essentially bounded measurable functions. • For a, b ∈ L 2 T 2 , we denote by ·, · the scalar product , with m a positive integer and 1 ≤ p < ∞. A detailed presentation of Sobolev spaces can be found in [1]. • C m (T 2 ; X ) is the (vector) space of all X -valued functions f which are continuous on T 2 with continuous partial derivatives D α f of orders |α| ≤ m, for m ≥ 0. C ∞ (T 2 ; X ) is regarded as the intersection of all spaces C m (T 2 ; X ). Note that on the torus all continuous functions are bounded.
with the uniform convergence norm over compact subintervals of [0, ∞). • L p (0, T ; X ) is the space of measurable functions from [0, T ] to X such that the norm which are right-continuous and have limits to the left, endowed with the Skorokhod topology. This topology is a natural choice in this case because its underlying metric transforms D([0, ∞); X ) into a complete separable metric space. For further details see [23] Chapter 3, Section 5, pp. 117-118. • Given a : T 2 → R 2 , we define the differential operator £ a by £ a b := a · ∇b for any map b : T 2 → R such that the scalar product a · ∇b makes sense. In line with this, we use the notation Denote the dual of £ a by £ a that is £ a b, c = b, £ a c . • For any vector u ∈ R 2 we denote the gradient of u by ∇u = (∂ 1 u, ∂ 2 u) and the corresponding orthogonal by ∇ ⊥ u = (∂ 2 u, −∂ 1 u).
Assumptions on the vector fields (ξ i ) i . The vector fields ξ i : T 2 → R 2 are chosen to be time-independent and divergence-free and for numerical purposes they need to be specified from the underlying physics. For the analytical aspects, we assume that Condition (4) ensures that for any f ∈ W 2,2 (T 2 ), Provided ω ∈ L 2 (0, T ; W 2,2 (T 2 , R)), condition (5a) ensures that the infinite sum of stochastic integrals is well defined and belongs to L 2 (0, T ; L 2 (T 2 , R)). Similarly, condition (5b) ensures that the process s → £ 2 i ω s is well-defined and belongs to L 2 (0, T ; L 2 (T 2 , R)) provided the solution of the stochastic partial differential equation (1) belongs to a suitably chosen space (see Definition 3 below). In particular, the Itô correction in (2) is welldefined. The conditions above are needed also for proving a number of required a priori estimates (see Lemma 25 in the "Appendix").
holds P-almost surely for all ϕ ∈ C ∞ (T 2 , R). 4 Here and everywhere else u is implicitly defined as the velocity field whose vorticity is ω, in other words ω = curl u = ∂ 2 u 1 − ∂ 1 u 2 . See further details in Remarks 4 and 21. 5 Equation (7) is interpreted as an identity between elements in L 2 (T 2 ; R). The same applies to the identity Brownian motions and the identity

Remark 4
The velocity field u is not uniquely identified through the equation ω = curl u. Indeed, any two velocity fields that differ by a constant will lead to the same vorticity map ω. Instead we identify u through the "explicit" formula u = ∇ ⊥ −1 ω, see details in Remark 21 in the "Appendix". In particular, since u and ω are defined in terms of partial derivatives of other functions, they must have zero average on the torus: This is due to the fact that ω = curl u = ∂ 2 u 1 −∂ 1 u 2 therefore ∂ 1 u 2 dx = 0 as we have periodic boundary conditions. Similarly, u = ∇ ⊥ ψ where ψ is the streamfunction (see the "Appendix") and therefore T 2 udx = 0. Note that if ω 0 has zero average, then ω t will have zero average, as it is immediate that all the terms appearing in the Euler equation (either (1) or (2)) have zero average.

Remark 6
Naturally, if ω t is a strong solution in the sense of Definition 3, then it is also a weak/distributional solution. In this sense, our result enhances the solution properties presented in [12] at the expense of stronger assumptions on the initial condition of the stochastic partial differential equation, but without the need to impose the additional constraint (3). Note also that if ω t is a weak/distributional solution with paths in L 2 (0, T ; W k,2 (T 2 )) then, by integration by parts, the equation has a strong solution.

Main results
We restate the existence and uniqueness result, this time with complete details: , then the two-dimensional stochastic Euler vorticity Eq. (1) The proof of Theorem 7 is contained in Sects. 3.1 and 3.2 . We state next a result that shows the continuity with respect to initial conditions:

Theorem 8
Let ω,ω be two strong solutions of Eq. (1). Define A as the process ( ω s k,2 + 1)ds, for any t ≥ 0. Then there exists a positive constant C independent of the two solutions, such that The proof of Theorem 8 is incorporated in Sect. 4.3.

Remark 9
Insofar, Theorems 7 and 8 are less general than, for example, the corresponding results in [12], since the initial condition of Eq. (1) is assumed to be in W k,2 (T 2 ). The relaxation to initial conditions in L ∞ (T 2 ) is done in Sect. 7 where the well-posedness is achieved without the additional constraint (3). However, the importance of these results lies in their application to modelling and to the numerical analysis of Eq. (1). In particular, Theorem 7 states that the initial smoothness of the solution is carried over for all times. From a modelling perspective, this is quite important: should the vorticity of a fluid be modelled by Eq. (1), it is essential to have it uniquely defined everywhere. This is not the case if vorticity is only known to be in L ∞ (T 2 ).
As an immediate consequence of a Sobolev embedding theorem, this is achieved, for example, if ω ∈ W 2,2 (T 2 ). Separately, if Eq. (1) is used to model the evolution of the state in a data assimilation problem (see, e.g. [17] for further details) and the observable is, say, the fluid velocity at a given set of points, then that observable should be well defined at those chosen points. Again, just by assuming ω ∈ L ∞ (T) does not ensure this. Finally, Theorems 7 and 8 are also important when one is interested in the numerical approximation of Eq. (1). For example, when a finite element numerical approximation is used, the smoothness of the solution influences the choice of the basis and governs the rate of convergence of the numerical approximation (higher rates of convergence require higher smootheness), see [11]. Moreover, Theorem 8 is useful to transfer the convergence from local to global numerical approximations, in the required Sobolev norm.

Remark 10
At the expense of additional technical arguments, Theorems 7 and 8 can be extended to cover solutions in W k, p (T 2 ) with p ≥ 2. More precisely, the existence of solutions in W k, p (T 2 ) of the sequence of linearised truncated equations in Sect. 4.2 can be obtained by an extension (albeit non-trivial) of the results in [32] from deterministic to random coefficients. In addition, the results in Lemmas 24 and 25 would need to be extended to cover the necessary a priori bounds in W k, p (T 2 ). 7

Pathwise uniqueness of the solution of the Euler equation
The uniqueness of the solution of Eq. (1) is an immediate corollary of inequality (9) with k = 0. However, the proof of (9) requires the existence of an approximating sequence which is constructed as part of the existence results. We present below a direct proof which avoids this, given the fact that pathwise uniqueness is required for the proof of existence of a strong (probabilistic) solution. Suppose that Eq. (1) admits two global (F t ) t -adapted solutions ω 1 and ω 2 with values in the space C [0, ∞); W k,2 (T 2 ) and letω := ω 1 − ω 2 . Consider the corresponding velocities u 1 and u 2 such that curl u 1 = ω 1 , curl u 2 = ω 2 andū := u 1 −u 2 . Since both ω 1 and ω 2 satisfy (2), their difference satisfies By an application of the Itô formula one obtains Note that the first and the last terms in the above identity are null (see Lemma 25) 8 and that 7 These two extensions of Theorems 7 and 8 have been suggested to us by an anonymous referee. We thank the referee for this. 8 The application of the Lemma requires that the two solutions ω 1 and ω 2 belong to W k,2 (T 2 ) with k ≥ 2.
To deduce (9) we need a similar control (albeit not an identity) for higher order derivatives. This is done by using the approximating sequence constructed in Sect. 4 and then taking the limit. This is the reason why we cannot prove directly (9).
This is true since by the Sobolev embedding theorem (see [1] Theorem 4.12 case A) one has ∇ω 1 t 4 ≤ C ω 1 t k,2 and using also the Biot-Savart law one has ū t 4 ≤ Since we only have a priori bounds for the expected value of the process t → ω 1 t k,2 and not for its pathwise values, the uniqueness cannot be deduced through a Gronwall type argument. Instead, we proceed as follows: let A be the process defined as A t := t 0 C ω 1 s k,2 ds, for any t ≥ 0. This is an increasing process that stays finite P-almost surely for all t ≥ 0 as the paths of ω 1 This leads to We conclude that e −A t ω t 2 2 = 0, and since e −A t cannot be null due to the (pathwise) finiteness of A t , we deduce that ω t 2 2 = 0 almost surely, which gives the claim. The above argument uses the fact that the terms ω t , £ 2 iω t and £ iωt , £ iωt are well defined. In other words, even though we only wish to control the L 2 -norm of the vorticity, we have to resort to higher order derivatives. This is permitted as we assumed that ω ∈ W k,2 (T 2 ) for k ≥ 2. By applying a similar argument, to control the W k,2 (T 2 )norm of the vorticity we would need to control terms of the form ∂ kω t , ∂ k £ 2 iω t and ∂ k £ iωt , ∂ k £ iωt . This is no longer allowed because we do not have enough smoothness in the system. To overcome this difficulty we will make use of a smooth approximating sequence for the vorticity equation, see Sect. 4.3.

Remark 11
The above uniqueness result is somewhat stronger than the uniqueness deduced from inequality (9). It shows that a solution of (1) will be unique in the larger space L 2 (T 2 ) rather than in the space W k,2 (T 2 ). Nevertheless, inequality (9) shows the Lipschitz property of the solution with respect to the initial condition in the weaker W k−1,2 (T 2 )-norm. It is not possible to show the same result in the stronger W k,2 (T 2 )-norm.
Remark 12 We note that, in contrast to the deterministic version of the Euler equation, the minimal k that ensures the existence of a strong solution is k = 2. This is because of the occurrence of the term £ 2 i ω in the Itô version of Eq. (1). Moreover, if we insist on the Stratonovich representation of Eq. (1), then we need to use the evolution equation of £ i ω to deduce the covariation between £ i ω and W i required for the rigorous definition of the Stratonovich integral. This, in turn, requires k ≥ 3 as the term £ 3 i ω appears in this evolution equation.
Nonetheless, the methodology from this paper can be used to cover initial conditions ω 0 ∈ W k,2 (T 2 ) with k < 2. In this case we have to content ourselves with weak/distributional solutions. Whilst this is not the subject of this paper, such a solution can be shown to exist as long as the product ωL u ϕ makes sense in a suitably chosen sense. We need to interpret the nonlinear term in a weak form as a generalised function and replace it with t 0 ω s , u s · ∇ϕ ds. Then the same methodology can be applied as long as we can control uω in a suitably chosen norm. In Sect. 7 we do this for the case ω 0 ∈ L ∞ (T 2 )(which is the so-called Yudovich setting).

Existence of the solution of the Euler equation
The existence of the solution of Eq. (1) is proved by first showing that a truncated version of it has a solution, and then removing the truncation. In particular we will truncate the non-linear term in (1) by using a smooth function f R equal to 1 on [0, R], equal to 0 on [R + 1, ∞), and decreasing on [R, R + 1], for arbitrary R > 0. We then have the following: , then the following equation . In particular, if k ≥ 4, then the solution is classical.

Remark 14
The truncation in terms of the norm ω R t k−1,2 and not ω R t k,2 is not incidental as it suffices to control the norm u R t k,2 (see Proposition 22).
We prove Proposition 13 in Sect. 4. For now let us proceed with the proof of global existence for the solution of the Euler equation (1). (1) is global.

Proposition 15 The solution of the stochastic 2D Euler equation
, and therefore, on [0, τ R ] the solution of the truncated equation (11) is in fact a solution of (1) with all required properties. It therefore makes sense to define the . This definition is consistent as, following the uniqueness property of the solution of the truncated equation (see The process ω defined this way is a solution of the Euler equation (1) on the interval [0, sup R τ R ). To obtain a global solution we need to prove that sup R>0 τ R = ∞ P-almost surely.
In order to finish the proof of global existence we use Lemma 24 and the fact that It follows that and therefore P(A ) = 0. This concludes the global existence for the solution of the Eq. (1).

Uniqueness of solution for the truncated equation
We use a similar strategy as the one used to prove the uniqueness of the solution of the (un-truncated) Euler Eq. (1). Suppose that Eq. (11) admits two global (F t ) t -adapted solutions ω 1,R and ω 2,R with values in the space C [0, ∞); W k,2 (T 2 ) . We prove that ω 1,R and ω 2,R must coincide. In the following, we will formally drop the dependence on R of the two solutions. As above, letω := ω 1 − ω 2 and consider the corresponding velocities u 1 and u 2 such that curl u 1 = ω 1 , curl u 2 = ω 2 andū := u 1 − u 2 . Since both ω 1 and ω 2 satisfy (11), their difference satisfies . By an application of the Itô formula and after eliminating the null terms (see Lemma 25, Remark 28, and use the fact that u 2 t is divergence-free), one obtains One can show that (see [18] for a proof) there exists a constant C = C(R) such that and to finally deduce that It follows that (note that the stochastic term is null) Similar arguments are used to control ∂ αω where α is a multi-index with |α| ≤ k − 1 and to deduce that there exists a constant C such that where we use the control (see Lemma 25) Some care is required for the case when |α| = k − 1 as ∂ α £ 2 iω t is no longer welldefined. In this case, by using the weak form of the Eq. (11) iω t we can proceed as above by using that The above is true for functions in W k+1,2 (T 2 ) and, by the continuity of both sides in the above inequality, it is also true for functions in the larger space . Using the above one can deduce that there exists ( ω 1 s k,2 + 1)ds for any t ≥ 0, hence the uniqueness of the solution of the truncated equation holds.

Existence of solution for the truncated equation
The strategy of proving that the truncated Eq. (11) has a solution is to construct an approximating sequence of processes that will converge in distribution to a solution of (11). This justifies the existence of a weak solution. Together with the pathwise uniqueness of the solution of this equation, we then deduce that strong uniqueness holds. Recall where ν n = 1 n is the viscous parameter (n > 0) and u where Theorem 16 If ω ν n ,R,n 0 ∈ C ∞ (T 2 ) is a function with null spatial mean, then the twodimensional stochastic vorticity equation (13) admits a unique global (F t ) t -adapted solution ω ν n ,R,n = {ω ν n ,R,n The proof of this theorem is provided in Sect. 5.

Proposition 17
The laws of the family of solutions (ω ν n ,R,n ) ν n ∈[0,1] are relatively compact in the space of probability measures over D([0, T ], L 2 (T 2 )) for any T ≥ 0.
The proof of Proposition 17 is left for Sect. 6.

Proof of existence of the solution of Eq. (11)
Using a diagonal subsequence argument we can deduce from Proposition 17 and the fact that lim n→∞ ω ν n ,R,n 0 = ω 0 the existence of a subsequence (ω ν n j ) j with lim j→∞ ν n j = 0, which is convergent in distribution over D([0, ∞), L 2 (T 2 )). We show that the limit of the corresponding distributions is the distribution of a stochastic 9 The operator curl −1 is the convolution with the Biot-Savart kernel, see Remark 21 for details. 10 The stochastic Itô integral is understood here in the usual sense, see [20] .
process that solves (11). This justifies the existence of a weak (probabilistic) solution. By using the Skorokhod representation theorem (see [9] Section 6, pp. 70), there exists a space (˜ ,F,P) and a sequence of processes (ω ν n ,R,n ,ũ ν n ,R,n , ( W i,n ) i , n = 1, ∞) which has the same distribution as that of the original converging subsequence and which converges (when n → ∞) almost surely to a triplet (ω R ,ũ R , Note that ω ν n ,R,n andω ν n ,R,n have the same distribution, so that for any test function ϕ ∈ C ∞ (T 2 ) we have Note that there exists a constant C = C(T ) such that whereẼ is the expectation with respect toP. We prove this in Lemma 25 for the original sequence, but sinceω ν n ,R,n satisfies the same SPDE, the same a priori estimates hold forω ν n ,R,n . Since the space of continuous functions is a subspace of the space of càdlàg functions and the Skorokhod topology restricted to the space of continuous functions coincides with the uniform topology, it follows that the sequence (ω ν n ,R,n ,ũ ν n ,R,n , ( W i,n ) i , n = 1, ∞) converges (when n → ∞)P-almost surely to (ω R ,ũ R , ( W i ) i ) also in the uniform norm. It also holds that also the limit of the right hand side of (17) converges to 0 (we use here the control ∞ i=1 £ i ϕ 2 2 ≤ C ϕ 2 1,2 assumed in (5b)). The convergence of the stochastic integrals in (15) is obtained by means of Theorem 32. To obtain this we use a countable dense set (ϕ j ) j ∈ W 2,2 (T 2 ) and show the convergence of (15) for this particular sequence. The convergence of (15) for an arbitrary ϕ ∈ W 2,2 (T 2 ) follows via a density argument. Let us identify precisely the sequence of processes to which we apply Theorem 32. Let X n := ω ν n ,R,n ,ũ ν n ,R,n , ω ν n ,R,n , £ i ϕ j ; i, j ∈ N W n := 0, 0,W i, j,n ; i, j ∈ N , whereW i, j,n =W i,n for all i, j, n ∈ N. Note that From the above, the sequence (X n , W n ) n converges in distribution to (X , W ) where By a similar application of the Skorokhod representation theorem, we can also assume that, on (˜ ,F,P), the above convergence holdsP-almost surely (as well as in L 2 (P)). Let us prove the convergence of the remaining terms in (15): •ω ν n ,R,n convergesP-almost surely toω R in D([0, ∞), L 2 (T 2 )). Since ϕ is bounded, it follows that ω ν n ,R,n t , ϕ − −−→ n→∞ ω R t , ϕ and ω ν n ,R,n 0 , ϕ − −−→ n→∞ ω R 0 , ϕ P -almost surely (as well as in L 2 (P)), for any ϕ ∈ C ∞ (T 2 ). • The second term on the right hand side of (15) converges to 0 when n → ∞ because the integral is uniformly bounded in L 2 (P) (again, because of (16)) and ν n → 0 when n → ∞. • Using the fact thatũ ν n−1 ,R,n−1 is the convolution betweenω ν n−1 ,R,n−1 and the Biot-Savart kernel, we obtain thatũ ν n ,R,n converges toũ RP -almost surely (as well as in L 2 (P)). • In order to show the convergence of the nonlinear term, one can write For the first term we have The term on the right hand side converges to 0 in L 2 (P) and all other terms are controlled uniformly in n. For the second term we have For the third term, using Hölder's inequality and an interpolation argument, we obtain: • Lastly, the integrals coming from the Itô correction term are treated in a similar fashion using that: We have shown so far that there exists a weak/distributional solution in the sense of Definition 3. part b. on the space (˜ ,F,P). However, sinceω R belongs to the space W k,2 (T 2 ) → C k−m (T 2 ) the solution is also strong, again, as a solution on (˜ ,F ,P) (and not on the original space). It follows that (ω,ũ, ( W i ) i ) is a martingale solution of the truncated Euler equation (11) in the sense of Definition 3 part c. Together with the pathwise uniqueness proved in Sect. 4.2 and using the Yamada-Watanabe theorem for the infinite-dimensional setting (see, for instance, [48]) we conclude the existence of a strong solution of the truncated Euler equation.
From the above, it follows that the limiting process is continuous in the L 2 -norm and, by interpolation, in the W k−ε,2 -norm for any ε > 0. To show continuity in the W k,2 -norm we follow the standard argument: we observe first that the sequence is weakly continuous in this norm, so it suffices to show that the W k,2 -norm of the process is continuous. For this we apply the Kolmogorov-Čentsov criterion. By Fatou's lemma, we have that and the right hand side of the above inequality can be controlled by C(t − s) 2 using an argument similar to the one from Sect. 4.3, where C is a constant independent of s, t and n. Therefore ω R ∈ C([0, T ], W k,2 (T 2 )) 11 and the solution exists in the sense of Definition 3, part a. Now using the embedding W k,2 (T 2 ) → C k−m (T 2 ) with 2 ≤ m ≤ k and k ≥ 4 we conclude that the solution is classical when k ≥ 4.

Proof of Theorem 8
We are finally ready to show continuity with respect to initial conditions. As stated in the theorem, let ω,ω be two C [0, ∞); W k,2 (T 2 ) -solutions of Eq. (1) and define A as the process A t := t 0 ( ω s k,2 + 1)ds, for any t ≥ 0. Let ω R ,ω R be their corresponding truncated versions and also let (ω ν n ,R,n t ) n≥0 and (ω ν n ,R,n t ) n≥0 be, respectively, the corresponding sequences constructed as in Sect. 4.2 on the same space after the application of the Skorokhod representation theorem. By Fatou's lemma, applied twice, we deduce that where A n is the process defined by A n t := t 0 ( ω ν n ,R,n s k,2 + 1)ds, for any t ≥ 0. Following a similar proof with that of the uniqueness of the Euler equation, one then deduces that there exists a positive constant C independent of the two solutions and independent of R and n such that which gives the result. We emphasize that we use here the fact that the processes (ω ν n ,R,n t ) n≥0 and (ω ν n ,R,n t ) n≥0 take values in W k+2,2 (T 2 ) as an essential ingredient, a property that was not true for either the solution of the Euler equation or its truncated version.

Existence and uniqueness of the approximating sequence
We show that the sequence (ω ν n ,R,n t ) n given by formula (13), that is,  [49]. All assumptions required by Theorem 1 and Theorem 2 in [49], Chapter 4, are fulfilled. Therefore there exists a unique solution ω ν n ,R,n t which belongs to the class L 2 (0, T ; W k,2 (T 2 )) ∩ C([0, T ], W k−1,2 (T 2 )) and satisfies Eq. (13) for all t ∈ [0, T ] and for all ω in ⊂ with P( ) = 1. Furthermore, since the conditions are fulfilled for all k ∈ N, using Corollary 3 from p. 141 in [49], we obtain that ω ν n ,R,n for any n ≥ 1, using the Biot-Savart law and an inductive argument. One has u ν n−1 ,R,n−1 t = K ω ν n−1 ,R,n−1 t with K being the Biot-Savart kernel defined in the "Appendix". The convolution between K and ω ν n−1 ,R,n−1 t is commutative, so we have Since ω ν n−1 t is in C ∞ (T 2 ) by Corollary 3 (at step n − 1), and using the fact that K ∈ L 1 (T 2 ), we conclude that u ν n−1 ,R,n−1 t ∈ C ∞ (T 2 ). This, together with the initial Assumption (4), ensures that all the coefficients of Eq. (13) are infinitely differentiable. The uniform boundedness is ensured by the truncation K R (ω ν n−1 ,R,n−1 t ), as proven in Lemma 25 from the "Appendix".

Proposition 18 There exists a constant C = C(T ) independent of n and R such that
In particular, by the Kolmogorov-Čentsov criterion (see [37]), the processes ω ν n ,R,n have continuous trajectories in L 2 (T 2 ).  [s,t] ω ν n ,R,n

Proof Consider s ≤ t. Then
Next we have The penultimate inequality is true given that and the last term is finite in expectation due to the a priori estimates proved in Lemma 25 vi. Similarly, we can prove that which, together with (19) gives a control on the second term of (18). For the last term we use the Burkholder-Davis-Gundy inequality and obtain due to the initial Assumption (4) and the a priori estimates (25). The conclusion now follows by a direct application of the Kolmogorov-Čentsov criterion.

Relative compactness of the approximating sequence of solutions
In this section we prove that the approximating sequence of solutions constructed in Sect. 4.2 is relatively compact in the space D([0, T ], L 2 (T 2 )). 12

Proof of Proposition 17
In order to prove relative compactness we use Kurtz' criterion for relative compactness. For completeness we state the result in "Appendix", see Theorem 29. To do so we need to show that, for every η > 0 there exists a compact set K η,t ⊂ L 2 (T 2 ) such that sup n P ω ν n ,R,n t / ∈ K η,t ≤ η. The compact we use is K η,t := ω ∈ W k,2 (T 2 )| ω k,2 < C η 1 4 where C is the constant appearing in the a priori estimates (25). By a Sobolev compact embedding theorem, K η,t is a compact set in L 2 (T 2 ) and sup n P ω ν n ,R,n To prove relative compactness, we need to justify part b) of Kurtz' criterion, as per Theorem 29. For this we will show that there exists a family (γ n δ ) 0<δ<1 of nonnegative random variables such that The filtration (F n t ) t corresponds here to the natural filtration (F ω νn ,R,n t ) t . We will use the mild form of Eq. (13), that is Aiming to construct the family (γ n δ ) 0<δ<1 of nonnegative random variables, we still need to control the two stochastic terms. The first one is more delicate and in order to obtain a suitable control on it we will use the so-called factorisation formula (see e.g. [20] Section 5.3.1). More precisely, we use the fact that where C(α) is a constant which depends on α > 0 only. Using also the semigroup property S n (t − s) = S n (t − r )S n (r − s) for s < r < t, one can write We choose α ∈ (0, 1/2) such that all integrals are well-defined. Now the fourth term in (20) can be estimated as follows: and therefore For the fifth term in (20) we have We can now define ] ≤ c 2 . The integrands in the integrals above converge pointwise to 0 when l → 0 due to the strong continuity of the semigroup S n . At the same time, they are bounded by integrable functions, therefore the convergence is uniform in space by the dominated convergence theorem. Then the requirement lim δ→0 sup n E γ n δ = 0 is met. In conclusion all the conditions required by Kurtz' criterion are fulfilled and therefore (ω ν n ,R,n t ) ν n is relatively compact.

Recovering the solution of the Euler equation in the Yudovich setting
As an application of Theorem 7, we prove existence of the solution of the stochastic Euler equation under the relaxed assumption that ω 0 ∈ L ∞ (T 2 ). This is the so-called Yudovich setting, see e.g. [52] or [44] Section 8.2. By doing so, we duplicate the result in [12] without the need to impose Assumption (3). The result is as expected: we show existence of a weak solution ω t ∈ L ∞ (T 2 ) of Eq. (1 ) re-cast in its weak form (8).
Note that whilst the definition of a weak solution only requires ω t ∈ L 2 (T 2 ), we are showing here that Eq. (1) has a solution in the (smaller) space L ∞ (T 2 ). The strategy is quite similar with that employed for proving Theorem 7. We briefly explain here the main steps without going into details. For an arbitrary ω 0 ∈ L ∞ (T 2 ), let (ω n 0 ) n ∈ W k,2 (T 2 ) be a uniformly bounded sequence such that ω n 0 converges to ω 0 almost surely. Consider next the sequence of strong solutions (ω n ) n ∈ W k,2 (T 2 ) of Eq. (1), with the corresponding velocities (u n ) n ∈ W k+1,2 (T 2 ). Then (ω n ) n is tight as a sequence with paths in C([0, T ]; W −2,2 (T 2 )) (see Lemma 19 below). We can therefore extract a subsequence (we re-index it if necessary) (ω n ) n which converges in distribution to ω over the space C([0, T ]; W −2,2 (T 2 )). By appealing to Theorem 32 (via another reduction to a subsequence, see Sect. 4.2) we can deduce the convergence of the stochastic integrals. Consider a countable dense set (ϕ j ) j ∈ W 4,2 (T 2 ). We show the convergence using this sequence, and then the convergence for an arbitrary ϕ ∈ W 4,2 (T 2 ) follows via a density argument. Let X n := ω n , u n , ω n , £ i ϕ j ; i, j ∈ N W n := 0, 0, W i, j,n ; i, j ∈ N , where W i, j,n = W i,n for all i, j, n ∈ N. The sequence (X n , W n ) n converges in distribution to (X , W ), where and W i, j = W i for all i, j ∈ N. Then Note that the stochastic integrals that appear above are interpreted as W −2,2 (T 2 )processes. Then, using a Skorokhod representation argument similar to the one in Sect To complete the argument we need to prove that the sequence (ω n ) n is tight as a sequence with paths in C([0, T ]; W −2,2 (T 2 )). The main ingredient in the argument is to show that, just as in the deterministic case, the vorticity ω n is propagated by inviscid flows, and therefore its L p -norm is conserved for any p ∈ (0, ∞]. For this we characterize the trajectories of the Lagrangian fluid particles as the solutions of the following stochastic flow Recall that, following from Theorem 7, (ω n ) n has paths in W k,2 (T 2 ) with the corresponding velocities (u n ) n having paths in W k+1,2 (T 2 ). By Theorem 4.6.5 in [40] the mapping x → X n t (x) is a C k−1 -diffeomorphism, for all n ∈ N. 13 The stochastic process X n t (x) models the evolution of the Lagrangian particle path corresponding to a fluid parcel starting from an arbitrary value x ∈ T 2 . Each Lagrangian path evolves according to a mean drift flow perturbed by a random flow which aims to model the rapid oscillations around the mean. By using the Itô-Wentzel formula, see e.g. [37], Let B Z (0, R) ⊂ Z be the closed ball of radius R in the Z-norm. Then B Z (0, R) is a compact set in C [0, T ]; W −2,2 T 2 . This follows from the generalized Arzela-Ascoli theorem see Lemma 1 in [50] (use the fact that L 2 T 2 is compactly embedded in W −2,2 T 2 ) combined with Lemma 5 in [50]). The tightness then follows as This is true as sup n E[ ω n 4 Z ] < ∞. To show the latter claim use the fact that sup n ω n 2 < ∞ due to the existence of the stochastic flow (see above) and that there exists a constant C = C (T ) independent of n such that Lemma 20 Let (ω n ) n be the sequence constructed above via the Skorokhod representation theorem, and ϕ ∈ W 1,2 (T 2 ). Then Proof From (23), the sequence (ω n ) n is uniformly bounded and converges to ω almost surely in C [0, T ]; W −2,2 T 2 . Let Also sup t∈[0,T ] ||ω t || L 2 (T 2 ) ≤ M. We first deduce that (ω n t ) n converges to ω t in L 2 w T 2 almost surely. To see this, choose an arbitrary ϕ and ε > 0. Let ϕ ε ∈ W 2,2 T 2 such that ||ϕ − ϕ ε || L 2 (T 2 ) < ε/4M and N such that ω n t , ϕ ε − (ω t , ϕ ε ) < ε/2 for all n ≥ N . Then, for all n ≥ N ,we have hence the claim.
Next observe that, since u n = K ω n , where K is the Biot-Savart kernel defined in (25), we can deduce that (u n ) n is uniformly bounded and converges to u := K ω almost surely. Moreover it is uniformly continuous (see Corollary 2.18 in [12]). By Arzela-Ascoli theorem we deduce that u is continuous and u n converges uniformly to u almost surely. More precisely we have, almost surely, Then The first term converges to 0 as u n converges uniformly to u. The second term converges to 0 as (ω n ) n is uniformly bounded and converges to ω in L 2 w T 2 . The limit in (24) then follows by the bounded convergence theorem.
ψ-and therefore the reconstruction of u from ω is ensured by the incompressibility condition div u = 0 [44]. A periodic, distributional solution of ψ = −ω is given by ( [28]) where G is the Green function of the operator − on T 2 , G(x) = k∈Z 2 \{0} e ik·x k 2 . Then the vector field u = ∇ ⊥ ψ is uniquely derived from ω as follows: (25) where K is the so-called Biot-Savart kernel It is known that G is smooth everywhere except at x = 0, and that K ∈ L 1 (T 2 ).
For the following result we recall a few elementary results of Fourier analysis. We embed L 2 T 2 into L 2 T 2 ; C and consider the basis of functions e 2πiξ ·x ; ξ ∈ Z 2 . Then every f ∈ L 2 T 2 ; C can be expressed as where f (ξ ) = T 3 e −2πiξ ·x f (x) dx, ξ ∈ Z 2 are the corresponding Fourier coefficients. We have the classical Parseval identity (see e.g. [44]) If v ∈ L 2 T 2 ; R 2 is a vector field with components v i , i = 1, 2, we write v (ξ ) = T 2 e 2πiξ ·x v (x) dx and we have, in a similar way, that Since u and ω are partial derivatives of other functions on the torus, they must have zero average: Hence u 1 (0, 0) = u 2 (0, 0) = ω (0, 0) = 0 and the term corresponding to ξ = (0, 0) does not appear in the Fourier expansion for u 1 , u 2 and, respectively ω.
For every s ≥ 0, let W s,2 T 2 ; C be the fractional Sobolev space of all functions f ∈ L 2 T 2 ; C such that It is a simple exercise to show that there exist a constant C > 1 such that if s ∈ N the It follows that this definition coincides with the definition given in Sect. 2 for integer s ∈ N. Therefore we can extend the norm f s,2 defined for s ∈ N to arbitrary s > 0 to be given by 2 We denote by W s,2 σ T 2 , R 2 the space of all zero mean divergence free (divergence in the sense of distribution) vector fields v ∈ L 2 T 2 ; R 2 such that all components v i , i = 1, 2 belong to W s,2 T 2 ; R . For a vector field v ∈ W s,2 σ T 2 , R 2 the norm v s,2 is defined by the identity v 2 where v i 2 s,2 is defined above. We thus have again v 2 s,2 := ξ ∈Z 2 \{0} |ξ | 2s (| v 1 (ξ )| + | v 1 (ξ )|) 2 . For f ∈ W s,2 T 2 ; C , we denote by (− ) s/2 f the function of L 2 T 3 ; C with Fourier coefficients |ξ | s f (ξ ). For even integers s ∈ N , this definition coincides with the classical definition. Similarly, we write − −1 f for the function having Fourier coefficients |ξ | −2 f (ξ ). We use the same notations for vector fields, meaning that the operations are made componentwise.
The Biot-Savart operator is the reconstruction of a zero mean divergence free vector field u from a divergence free vector field ω such that curl u = ω. As stated in Remark 21 on the 2D torus it is given by u = − curl −1 ω. It follows that the Fourier coefficients of u are given by In the next proposition we highlight the smoothing properties of the Biot-Savart kernel K .

Remark 23
The norm · m,2 is equivalent to the norm defined as ||| f ||| := f 2 + D m f 2 , therefore it is enough to show that all properties hold for the L 2 norm of f and for the L 2 norm of the maximal derivative D m f (see [14] pp. 217).
Let ω be the solution of the Euler Eq. (1) and ω ν n ,R,n the solution of the linear approximating Eq. (12). In the following lemmas we collect a number of identities and a priori estimates. Lemma 24 in particular in proving that the solution of the Euler equation is global.
Proof By the Biot-Savart law We use the truncation Then given that ∇ω 4 ≤ C ∇ω 1,2 ≤ C ω 2,2 ≤ C u 3,2 by the Sobolev embedding W 1,2 → L 4 and the Biot-Savart law. So For the second integral we use the fact that That is If u 3,2 ≤ 1 we choose = 1, otherwise we choose = 1 u 3/2 3,2 . We have ∇u ∞ ≤ C 1 + 1 + 3 2 ln u 3,2 ω ∞ Using the inequality 1 + ln + x ≤ C ln(e + x) for sufficiently large C, we have ∇u t ∞ ≤ C 1 + 1 + ln + u t 3,2 ω t ∞ ≤ C 1 + ln(e + u t 3,2 ) ω t ∞ and using the transport property and Biot-Savart u t 3,2 ≤ ω t 2,2 ≤ ω t 2,2 we have By the Itô formula We have By the Burkholder-Davis-Gundy inequality We control the quadratic variation of the stochastic integral using the fact that We write In consequence, the Burkholder-Davis-Gundy inequality gives Using this in 28 we obtain By the Fatou lemma this holds also for the original solution ω Using the transport property (see relation (22) in Sect. 7), ω t ∞ = ω 0 ∞ , we conclude that there exists a constant C( ω 0 k,2 , T ) which is independent of the truncation radius R such that E sup s∈ [0,t] ln(e + ω s 2 k,2 ) ≤ C( ω 0 k,2 , T ).

Lemma 25
The following properties hold: i. For any f ∈ W 2,2 (T 2 ) we have ii. If ω t , ω ν n ,R,n t ∈ W k,2 (T 2 ) then the following L 2 -transport formulae 16 hold Palmost surely: ω t L 2 = ω 0 L 2 and ω ν n ,R,n iii. If ω ∈ W k,2 (T 2 ), then (P n−1,n ω ν n ,R,n t , £ i ω ν n ,R,n t = 0 since the vector fields (ξ i ) i are assumed to be divergencefree. Therefore We want to estimate the other terms independently of ν n . All terms are estimated above. Let Obviously B t is a local martingale.  The following lemma is instrumental in showing that the limit of the approximating sequence satisfies the Euler equation in W k,2 (T 2 ) although the relative compactness property holds in D [0, T ], L 2 (T 2 ) . It is also essential when proving a priori estimates for the truncated solution ω R (see Lemma 27).
Lemma 26 i. Assume that (a n ) n is a sequence of functions such that lim n →∞ a n = a in L 2 (T 2 ) and sup n>1 a n s,2 < ∞ for s ≥ 0. Then a ∈ W s,2 (T 2 ) and a s,2 < sup n>1 a n s,2 . Moreover, lim n →∞ a n = a in W s ,2 (T 2 ) for any s < s. ii. Assume that a n : → W s,2 (T 2 ) is a sequence of measurable maps such that, lim n →∞ a n = a in L 2 (T 2 ), P-almost surely or lim n →∞ a n = a in distribution. Further assume that sup n>1 E[ a n p s,2 ] < ∞. Then, P-almost surely, a ∈ W s,2 (T 2 ) and E[ a p s,2 ] ≤ sup n>1 E[ a n p s,2 ], for any p > 0.
For the second part we can write a n − a s ,2 = where the first inequality is true due to the fact that |λ| ≥ M. Now we can choose M such that the last term is strictly smaller than 2 . Likewise, n can be chosen such that λ∈Z 2 ,|λ|≤M (1 + |λ| 2s )|(â n −â)λ| < 2 hence a n converges to a in W s ,2 . ii. From above it follows that

Lemma 27
Let ω R t be the solution of the truncated Euler equation (11). There exists a constantC(T ) independent of R such that Theorem 32 (Theorem 4.2 in [41] or Theorem 1 in [34]) Let (F n t ) t be a filtration and (X n ) n a sequence of (F n t ) t -adapted processes with càdlàg trajectories with values in a separable Hilbert space H . Let (W n ) n be a sequence of cylindrical Brownian motions on H (cf. Example 3.18 in [41]). If (X n , W n ) n converges in distribution to (X , W ), in the Skorokhod topology, then (X n , W n , X n dW n ) converges in distribution to (X , W , XdW ) in the Skorokhod topology. If the first convergence holds in probability, then the convergence of the stochastic integrals holds also in probability.