Euler-Lagrangian approach to 3D stochastic Euler equations

3D stochastic Euler equations with a special form of multiplicative noise are considered. A Constantin-Iyer type representation in Euler-Lagrangian form is given, based on stochastic characteristics. Local existence and uniqueness of solutions in suitable Hoelder spaces is proved from the Euler-Lagrangian formulation.


Introduction
Recently the vorticity equation (Euler type) with noise with div u t = 0, curl u t = ω t , div σ k = 0, has been studied by Darryl Holm and other authors, see [15,14,5]. It can be written as dω t + L ut ω t dt + k L σ k ω t • dW k t = 0, (1.2) where L ut ω t = [u t , ω t ] = u t · ∇ω t − ω t · ∇u t is the Lie derivative. Equations related to fluid dynamics with multiplicative noise appeared in several other works, see for instance [2,11,7,18,10] and many others. However, the geometric structure in (1.1) has special properties, revealed also by the present work. A first intermediate question we address is finding the noise form when the equation is rewritten in the velocity-pressure variables (u, p), instead of the velocity-vorticity variables (u, ω). The dual operators L * σ k appear. This is an intermediate step in order to investigate the main topic of this work, namely the Euler-Lagrangian formulation, called also Constantin-Iyer representation after [3,4]; among related works, see for instance [20,8]. We prove both the (u, p)-formulation and the Euler-Lagrangian one in Proposition 2.1. The Euler-Lagrangian form is then used to prove a local in time existence and uniqueness result for solutions in suitable Hölder spaces, new for equation (1.1). At the end of the paper we heuristically digress on potential singularities from the viewpoint of this stochastic model and its Euler-Lagrangian formulation, adding some remarks to the discussion of P. Constantin [3,Section 5]. Finally we remark that we restrict ourselves to the three dimensional torus T 3 for simplicity, though the results remain valid in more general settings under suitable conditions.

Equation for the velocity and its representation
Let L * σ k be the adjoint operator of L σ k with respect to the inner product in L 2 (T 3 , R 3 ): Note that if the vector fields u and v are divergence free, then div(L u v) = 0, but this is not necessarily true for L * u v. Consider the following equation of characteristics, associated to (1.1): Denote by X t also the stochastic flow associated to this equation, when defined.
where P is the Leray-Hodge projection and * means the transposition of matrices.
Remark 2.2. The stochastic equation (2.2) is understood as follows: for any smooth divergence free field v on T 3 , Proof of Proposition 2.1. Let us show the first fact. Using curl u t = ω t , we have the vector identities Hence, equation (2.2) can be rewritten as where p t is a new pressure. Taking curl and using the facts we get the equation (1.2). Next we prove the second assertion. Let u t be expressed as in (2.3). For any divergence free vector field v, we have where in the last step we used the change of variable formula and the fact that X t preserves the volume measure. Recall that (∇X −1 t (X t )) v(X t ) = (X −1 t ) * v is the pull-back of v by the flow X t , thus we obtain This equality holds for any divergence free vector field v. Since X t is the flow generated by the SDE in (2.1), by Kunita's formula (see [17, p. 265]), Substituting this expression into (2.5) yields Since the vector fields L σ k v, L us v and L 2 σ k v are all divergence free, we apply (2.5) and get Note that therefore, we obtain (2.4).

Local existence of the representation (2.3)
In this section we aim at proving the local existence of the system (2.3), by following the arguments in [3,16]. First we introduce some notations about Hölder (semi-)norms. For a function or vector field u defined on T 3 and α ∈ (0, 1), l ∈ N, where ∂ m denotes the derivative with respect to the multi-index m ∈ N 3 . Note that · 0 is the usual supremum norm. We denote by C l and C l,α the Hölder spaces with norms u l and u l,α , respectively. We use the idea of [1,19] to solve the SDE in the system (2.3). More precisely, we first solve the equation without drift: where I is the identity diffeomorphism of T 3 . Under the assumption that k σ k 2 l+3,α ′ < ∞ for some l ∈ N and α ′ ∈ (0, 1), the above equation generates a stochastic flow {ϕ t } t≥0 of C l+2,α -diffeomorphisms on T 3 , where α ∈ (0, α ′ ).
In this section we denote by ω a generic random element in a probability space Ω; there will be no confusion with the notation of vorticity, since the latter does not appear in the current section. For a given random vector field u : which is the pull-back of the field u t (ω, ·) by the stochastic flow {ϕ t (ω, ·)} t≥0 . If we denote by K t (ω, x) = (∇ϕ t (ω, x)) −1 , i.e., the inverse of the Jacobi matrix, theñ From this expression we see that if u ∈ C([0, T ], C l+1,α ) a.s., then one also has a.s.ũ ∈ C([0, T ], C l+1,α ). Moreover, if the process u is adapted, then so isũ. Now we consider the random ODEẎ Applying the generalized Itô formula, we see that (cf. [1,19]) Once we have the stochastic flow {X t } t≥0 , we can use the second formula in (2.3) to obtain a new random vector fieldû. Our purpose is to show that this series of transforms have a fixed point.
From the above discussions, we see that we can fix a random element ω ∈ Ω 0 , where Ω 0 is some full measure set, and consider ϕ t (ω, ·), u t (ω, ·) and so on as deterministic objects. Hence in Section 3.1 we solve a deterministic fixed-point problem, and apply this result in Section 3.2 to prove the local existence of the system (2.3).

Deterministic case
In this section, we assume that we are given a deterministic family of diffeomorphisms we consider the following system: Following the arguments in [3, Section 4] and [16, Section 4], we shall prove that the map defined byû = Φ(u) has a fixed point in for some small τ and big U . Here is the main result of this section.
We need some preparations. The following result is taken from [16, Lemma 4.1].
Lemma 3.2. If l ≥ 1 and α ∈ (0, 1), then there exists C = C(l, α) such that Denote by U t = sup s≤t u s l+1,α and λ = Y − I, ℓ = Y −1 − I. Then there exists a continuous function f l,α : [0, T ] × R + → R + , which is increasing in both variables and f l,α (0, θ) = 0 for all θ ≥ 0, such that Proof. We first prove the case for l = 0. Using the integral form of the ODE, it is clear that (3.5) where I is the 3 × 3 identity matrix. Therefore, The Gronwall inequality implies that Now for x, y ∈ T 3 , x = y, we deduce from (3.5) that (3.7) We have where the third inequality follows from (3.6). Moreover, Substituting the estimates J 1 and J 2 into (3.7), we deduce that Gronwall's inequality leads to Combining this estimate with (3.6) and using the simple inequality e t − 1 ≤ te t (t ≥ 0), we conclude that ∇λ t 0,α ≤ 2tU t e (2+α)tUt , t ≤ T.
Again by Gronwall's inequality, The proof of the first estimate is complete.
To prove the second assertion, note that the inverse flow Y −1 t can be obtained by reversing the time. More precisely, fix any t ∈ (0, T ] and consideṙ Then Y t t = Y −1 t . Similar to the above arguments, we can prove estimates for λ t s = Y t s − I, and hence for ℓ t = Y −1 which only depends on the C l+1,α -norm of u s for s ∈ [0, t], the latter being dominated by U t . In this way, we obtain the second result. We need the following key technical result, see [ Lemma 3.4. For l ≥ 1, the operator W : (ℓ, v) → P[(I+ ∇ℓ) * v] is well defined on C l,α × C l,α with values in C l,α ; moreover, there is C > 0 depending only on l and α such that

Now we can prove
Proof. Let U > 0 be a constant which will be determined later. We divide the proof into four steps.
Step 1. Take u ∈ U T,U . By the definition ofũ in (3.4), we have Note that where C l,ϕ,t = sup Step 2. Let Y be the flow generated byũ, and denote by ℓ = Y −1 − I. Then applying Lemma 3.3 with u replaced byũ gives us ∇ℓ t l,α ≤ f l,α (t,Ũ t ) ≤ f l,α (t, C l,ϕ,t U t ), t ≤ T.
Step 3. Let X t = ϕ t • Y t and denote by m t = ϕ −1 t − I, 0 ≤ t ≤ T . Then we have
The next estimate is need for establishing contraction property of Φ.
Let Y,Ȳ be the flows generated by u andū, respectively. Then there exists a continuous functionf l,α : [0, T ] × R + → R + which is increasing in both variables, such that Therefore, by Lemma 3.2, Similarly, ∇Ȳ s l−1,α ≤ 1 + f l,α (s, U s ). Substituting these estimates into the above inequality yields From this and Gronwall's inequality we obtain the first assertion. The proof of the second one follows analogously by reversing the time.
Before proving that the map Φ is a contraction in a certain space, we introduce the following property of the operator W defined in Lemma 3.4 (see [16,Proposition 3.1]).
Finally we are ready to prove Theorem 3.1.
Proof of Theorem 3.1. With the above preparations, the proof is the same as that in [16]. The existence of a fixed point of Φ follows by successive iteration. We define u n+1 = Φ(u n ). The sequence converges strongly with respect to the C l,α -norm. Since U τ,U is closed and convex, and the sequence {u n } n≥1 is uniformly bounded in the C l+1,α -norm, it must have a weak limit u ∈ U τ,U . Since Φ is continuous with respect to the weaker C l,α -norm, this limit must be a fixed point of Φ, and thus a solution of the deterministic system (3.4).

Discussions
The inverse flow A t (x) = X −1 t (x) is a random vector field, solution of the stochastic transport equation This equation does not contain stretching terms of the form A t (x) · ∇u t (x) dt and A t (x) · ∇σ k (x) • dW k t , hence the quantity A t (x) is only transported. Therefore we do not expect a blow-up of A t (x) itself. We may however expect, in analogy with shocks appearing in nonlinear transport equations (like Burgers equation), that space derivatives of A t (x) may blow-up. This is the potential mechanism which could lead to blow-up in the formula as discussed in [3,Section 5].
The question posed by the presence of noise is: could the noise prevent or mitigate blow-up of ∇A t ? If u t (x), in the SPDE above, would be given (passive field A t ) and deterministic, several results of regularization due to noise have been proved for similar equations, for instance the absence of shocks for the scalar transport equation with u of class L q 0, T ; L p R d , R d with d p + 2 q < 1 (see [9]), or the absence of singularity for a passive magnetic field proved in [12,13] under various assumptions. The intuitive reason is that noise prevents A t (x) to stretch for too much time around the more singular points of u t (x), because A t (x) is continuosly randomly displaced. However, no result of this form has been proved until now in the case when u t (x) is random (see [6] for a related work), as it is in the nonlinear case; the obstruction is not technical but conceptual: the singularities of u t (x) move accordingly to noise and to A t (x) itself, hence there is no straightforward reason why noise should displace A t (x) to avoid those singularities.
Thus the question of singularities remains open, as it is in the deterministic case but here, thanks to the noise, new intuitions may develop.