A local-in-time theory for singular SDEs with applications to fluid models with transport noise

In this paper, we establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in some Hilbert space. The key requirement is an approximation property that allows us to embed the singular drift and diffusion mappings into a hierarchy of regular mappings that are invariant with respect to the Hilbert space and enjoy a cancellation property. Various nonlinear models in fluid dynamics with transport noise belong to this type of singular SDEs. With a cancellation estimate for generalized Lie derivative operators, we can construct such regular approximations for cases involving the Lie derivative operators, or more generally, differential operators of order one with suitable coefficients. In particular, we apply the abstract theory to achieve novel local-in-time results for the stochastic two-component Camassa--Holm (CH) system and for the stochastic C\'ordoba-C\'ordoba-Fontelos (CCF) model.


Introduction
Let X , Y and Z be three separable Hilbert spaces such that X ⊂ Y ⊂ Z. (1.1) We consider the initial value problem for a stochastic differential equation (SDE) with unknown process X = X(t), t ≥ 0, given by dX = b(t, X) + g(t, X) dt + h(t, X) dW, X(0) = X 0 ∈ X .
(1. 2) In (1.2), W denotes a cylindrical Wiener process defined on some separable Hilbert space U. The drift is given by the sum of the mappings b : [0, ∞) × X → X and g : [0, ∞) × X → Z. The operator h : [0, ∞) × X → L 2 (U; Y) stands for the diffusion coefficient with L 2 (U; Y) being the space of Hilbert-Schmidt operators from U to Y. We call (1.2) a singular initial value problem because g and h map X to the larger spaces Z and Y, i.e. they are not invariant in X . We refer to Sections 2.1, 2.2 for the precise setting.
For the entirely regular case X = Y = Z, it is well-known that (local) Lipschitz conditions on b(t, ·) + g(t, ·) and h(t, ·) ensure that (1.2) admits unique (local) pathwise solution in X . If additional monotonicity properties on the coefficients are imposed, then the Itô formula for Gelfand-triple Hilbert spaces can be exploited to assure global existence and continuity of solutions, cf. [42,48,46,50] and the references therein. Notably this covers also the case when the Hilbert spaces form a Gelfand triple.
In this work, we focus on the singular case which appears in particular for ideal fluid models. Indeed, when we consider particular examples in Sobolev spaces X = H s , if g(t, X) and h(t, X) involve ∇X in a nonlinear way, or more generally, general Lie-type derivatives of X (see our examples (3.7) and (3.11) in the second part of the paper), then g(t, X) and h(t, X) can not be expected to be in X = H s , either. Likewise, the concept of monotonicity fails to apply as it relies on self-embedding drift and diffusion mappings. Working with the abstract framework in (1.2) entails another difficulty as compared to the regular or the Gelfand-triple case: the Itô formula is no longer available. To highlight the latter difficulty, let us recall the classical Itô formula for a Gelfand triplet V ֒→ H ֒→ V * , where H is a separable Hilbert space with inner product (·, ·) and H * is its dual; V is a Banach space such that the embedding V ֒→ H is dense. Then the following result is classical, see [46,Theorem I.3.1] or [50,Theorem 4.2.5].
Assume that U is a continuous V * -valued stochastic process given by where G ∈ L 2 (Ω × [0, T ]; L 2 (U; H)) and g ∈ L 2 (Ω × [0, T ]; V * ) are both progressively measurable and U(0) ∈ L 2 (Ω; H) is F 0 -measurable. If U ∈ L 2 (Ω × [0, T ]; V ), then U is an H-valued continuous stochastic process and the Itô formula  H)). However, if G is singular (not invariant in H), then G ∈ L 2 (U; H) is ambiguous. Besides, even though g is allowed to be less regular, (1.3) requires U(t) to be more regular than U(0), i.e., U ∈ V ֒→ H ∋ U(0). In many cases (for example, stochastic ideal fluid models), we do not know that this holds true. Hence, (1.3) is not applicable in singular cases, and then the time continuity of the solution cannot be obtained directly.
The first major goal of this paper is to establish a local-in-time theory for (1.2) generalizing classical results for e.g. the completely regular case X = Y = Z. The second goal of this work is to show that the abstract theory for (1.2) can be used to establish new results for ideal fluid systems with noise.
(1) To achieve the first goal we fix in Section 2.2 the precise assumptions on the regular drift b and in particular on the singular drift g and diffusion h (see Assumption (A)). Then we provide our main results for (1.2), including the existence, uniqueness, time regularity, and a result characterizing the possible blow-up of pathwise solutions (see Theorem 2.1). The key requirements for the proof are the assumption on the existence of appropriate Lipschitz-continuous and monotone regularizations for the singular mappings. This allows us to exploit Itô-like formulas as above. (2) With the abstract framework at hand, for a large number of nonlinear SPDE models, we are able to construct such regular approximation schemes by using convolution operators and establishing a cancellation property for generalized Lie derivatives (cf. Lemma A.5). To set the stage, in Section 3, we consider two models governing ideal flows with particularly interesting stochastic perturbation, namely • the two-component Camassa-Holm (CH) system with transport noise [39], see (3.4) below, • a nonlinear transport equation with non-local velocity, referred to the Córdoba-Córdoba-Fontelos (CCF) model [19], with transport noise, see (3.10) below. In both cases, we obtain a local-tin-time theory in the sense of the abstract-framework Theorem 2.1. These results for both models are new up to our knowledge and can be found in Section 3.2. Finally, we will explain in Section 3.5 how our abstract framework and the regular approximation schemes can be applied to a broader class of fluid dynamics equations including the surface quasigeostrophic (SQG) equation with transport noise.
2. An abstract framework for a class of singular SDEs 2.1. Notations and definitions. To begin with, we introduce some notations. We consider a probability space (Ω, F , P), where P is a probability measure on Ω and F is a σ-algebra. We endow the probability space (Ω, F , P) with an increasing filtration {F t } t≥0 , which is a right-continuous filtration on (Ω, F ) such that {F 0 } contains all the P-negligible subsets. For some separable Hilbert space U with a complete orthonormal basis {e i } i∈N the noise W in (1.2) is a cylindrical Wiener process, i.e., it is defined by W k e k P − a.s, (2.1) where {W k } k≥1 is a sequence of mutually independent standard 1-D Brownian motions. To guarantee the convergence of the above formal summation, we consider a larger separable Hilbert space U 0 such that the canonical injection U ֒→ U 0 is Hilbert-Schmidt. Therefore, for any T > 0, we have, cf. [25,34,43], W ∈ C([0, T ], U 0 ) P − a.s.
Note that the choice of the auxiliary Hilbert spaces U and U 0 is not crucial for our analysis. Thus we let U and U 0 be arbitrary but fixed in the sequel. For some time t > 0, the family σ{x 1 (τ ), · · · , x n (τ )} τ ∈[0,t] stands for the completion of the union σ-algebra generated by (x 1 (τ ), · · · , x n (τ )) for τ ∈ [0, t]. EY stands for the mathematical expectation of a random variable Y with respect to P. From now on S = (Ω, F , P, {F t } t≥0 , W) is called a stochastic basis. For any Hilbert space H the inner product is denoted by (·, ·) H . Furthermore, the space L 2 (U; H) contains all Hilbert-Schmidt operators Z : H . As in [11,Theorem 2.3.1], we see that for an H-valued progressively measurable stochastic process Z with Z ∈ L 2 Ω; L 2 loc ([0, ∞); L 2 (U; H)) , one can define the Itô stochastic integral Most notably for the analysis here, if Z ∈ L 2 (U; H) and W is given as above, we have the Burkholder- or in terms of the coefficients, Let X be a separable Banach space. B(X) denotes the Borel sets of X and P(X) stands for the collection of Borel probability measures on (X, B(X)). We denote P r (X) the family of probability measures in P(X) with finite moment of order r ∈ [1, ∞), i.e., P r (X) = µ : X x r X µ(dx) < ∞ . For two Banach spaces X and Y, X ֒→ Y means that X is embedded continuously into Y, and X ֒→֒→ Y means that the embedding is compact.
For some set E, 1 E denotes the indicator function on E.
Next, let us make precise two different notions of solutions in the Hilbert space X from (1.1) for the Cauchy problem (1.2). Definition 2.1 (Martingale solutions). Let µ 0 ∈ P(X ). A triple (S, X, τ ) is said to be a martingale solution to (1.2) if (1) S = (Ω, F , P, {F t } t≥0 , W) is a stochastic basis and τ is a stopping time with respect to {F t } t≥0 ; (2) X(· ∧ τ ) : Ω × [0, ∞) → X is an F t -progressively measurable process such that it is continuous in Z, µ 0 (·) = P{X 0 ∈ ·} for all · ∈ B(X ) and for every t > 0, s., then we say that the martingale solution is global.
It follows from Definition 2.1 that, if a martingale solution exists, then (2.2) implies that Particularly, if τ * = ∞ almost surely, then such a solution is called global.

Assumptions and main results.
To study the existence of martingale and pathwise solutions, we need the following assumptions on the three separable Hilbert spaces X , Y, Z from (1.1) and on the coefficients b, g and h in (1.2). Recall that {e i } i∈N is a complete orthonormal basis of U.

4)
and for all N ∈ N, For ε ∈ (0, 1) and N ≥ 1 there exist progressively measurable maps and constants C ε,N > 0 such that for all t ≥ 0 the bounds

8)
and hold. Moreover, for any bounded sequence {X ε } ⊂ X such that X ε → X in Z and for any t > 0, we have Here Z ·, · Z * denotes the dual pairing in Z. and The embedding X ֒→ Z is dense, and there is a family of continuous linear operators {T ε : Z → X } ε∈(0,1) such that and for all t ≥ 0, N ≥ 1 hold true for t ≥ 0.
Note that for the singular mappings the constants C ε,N in Assumption (A) are non-decreasing in N for ε fixed and explode for ε → 0 with N fixed. Then we can state our main results for the initial value problem (1.2).
Remark 2.1. We first remark that the singular terms g and h are in general not monotone in the sense of [49,50]. So, the well-known approximation scheme under a Gelfand triple developed for quasi-linear SPDEs does not work for the present model. Motivated by [51], we will employ a regularization argument to overcome this difficulty. Let us give some explanations on Assumption (A) that makes precise the required regularization procedure.
• The condition (A 1 ) provides the local Lipschitz continuity for the regular drift coefficient b(t, X) and bounds its growth. Assumption (A 2 ) requires the local Lipschitz continuity on the approximations g ε and h ε of the singular terms g and h, which together with (A 1 ) will ensure local-in-time existence for some approximate problem. In Section 3.1 we will show how to construct such approximations using mollifiers.
• (A 3 ) can be viewed as a renormalization type condition in the following sense. Formally speaking, even though g and h are not invariant with respect to X (hence (g(t, X), X) X and h(t, X) L2(U;X ) may be infinite), we require that (g(t, X), X) X + h(t, X) L2(U;X ) can be controlled. In fact, (A 3 ) specifies this relationship for g ε and h ε such that (g ε (t, X), X) X and h ε (t, X) L2(U;X ) make sense. • Since g and h are singular, we need (A 4 ) on the joint space Z to guarantee pathwise uniqueness.
• As explained in the introduction, we can not use the Itô formula (1.3) to obtain the time continuity of the solution directly. This is why we need to assume (A 5 ) and (A 6 ) to establish time continuity and blow-up criterion, respectively. (A 6 ) is stronger than (A 5 ) because we need both, the validity of the Itô formula and the growth condition. However, the dense embedding X ֒→ Z is not necessary for deriving the blow-up criterion. Moreover, in applications, usually one can take T ε = Q ε . • In view of Assumption (A), it is worthwhile noticing that the regular drift b will not be used to control the singular terms, i.e. our result covers the case b ≡ 0, where both the drift and diffusion in (1.2) are singular. However, we assume that the problem (1.2) has a regular part to cover more ideal fluid models.
Remark 2.2. We remark that in [26], an abstract fluid model involving a Stokes operator (viscous term) and a regular noise coefficient is studied. Here, we are able to deal with ideal fluid models without viscosity and noise of transport type. Moreover, in [26], the martingale solution exists under the condition that the initial measure has finite moment of order r > 8 (See [26,Theorem 6.1]). In the present work, we only require r = 2, i.e., µ 0 ∈ P 2 (X ) in (i) in Theorem 2.1.

Remark 2.3.
We also remark that when the noise coefficient h(t, X) is as regular as the solution X and the singularity of (1.2) only arises in g, namely b : [0, ∞) × X → X , h : [0, ∞) × X → L 2 (U; X ) and g : [0, ∞) × X → Z, one can also obtain a local theory as in Theorem 2.1 even under weaker conditions as in Assumption (A).

2.3.
Proof of (i) in Theorem 2.1. For the sake of clarity, we split the proof into the following subsections.
2.3.1. Approximation scheme and uniform estimates. For µ 0 ∈ P 2 (X ), we first fix a stochastic basis S and a random variable X 0 such that the distribution law of X 0 is µ 0 . For any R > 1, we let χ R (x) : [0, ∞) → [0, 1] be a C ∞ -function such that χ R (x) = 1 for x ∈ [0, R] and χ R (x) = 0 for x > 2R. Then we consider a cut-off version of (1.2) given by (2.23) We have not posed any structural properties like monotonicity on the singular mappings g, h that ensure the existence of solutions for (2.23). Therefore we employ the regular approximations g ε and h ε from Assumption (A) which leads us to the regular approximative version dX = H 1,ε (t, X) dt + H 2,ε (t, X) dW, For (2.24) we can obtain the following global existence result. Lemma 2.1. For µ 0 ∈ P 2 (X ), we fix a stochastic basis S and a F 0 -measurable random variable X 0 such that the distribution of X 0 is µ 0 . Let R > 1 be fixed. For each ε ∈ (0, 1), the problem (2.24) has a global solution X ε . Moreover, for any sequence {ε n } n∈N and for any T > 0, we have that Proof. From (A 1 ), (A 2 ), it is easy to see that for each n ≥ 1, H 1,ε (t, X) and H 2,ε (t, X) are locally Lipschitz in X ∈ X . Moreover, the growth of H 1,ε (·, X) X and H 2,ε (·, X) L2(U;X ) is controlled by the continuous function k(t). Therefore, for each ε ∈ (0, 1), there is a stopping time τ * ε > 0 almost surely such that the problem (2.24) has a unique solution X ε ∈ L 2 (Ω; C([0, τ * ε ); X )), see [48] or [42, Theorem 5.1.1]. Next, we prove that the solution is actually a global solution. Using the Itô formula in X for the regular mappings g ε , h ε , we find For any T > 0, we integrate (2.26), take a supremum for t ∈ [0, T ] and then use the BDG inequality, (A 1 ) and (A 3 ) to find a constant C = C R > 0 depending on R such that Via Grönwall's inequality, we arrive at the ε-independent bound Since T > 0 can be chosen arbitrarily, we see in particular that X ε is a global solution for each ε ∈ (0, 1). Moreover, the bound (2.27) implies that the stopping times Now we turn to prove the tightness result on the Borel measure in (2.25). For any given δ ∈ (0, 1), we get that holds. Note that we used the ε-independent bound (2.29) for the last inequality. To estimate the expectation term in (2.30) we utilize the approximative problem (2.24) directly. We start with the drift term H 1,ε . On account of (2.28), (A 1 ) and (A 2 ) and the BDG inequality, there are a non-decreasing, locally bounded function a(·) ∈ C ([0, +∞); [0, +∞)) and a constant C > 0 independent of ε such that we have For the diffusion operator H 2,ε and the stochastic integral, the bound (2.28), (A 2 ) and the BDG inequality imply Combining the estimates (2.31), (2.32), for any δ ∈ (0, 1), one has Therefore, returning to (2.30), the last estimate implies that for all δ ∈ (0, 1), Because a(·) is non-decreasing, we have Thus, we obtain that, for any δ > 0, the limit there is a sequence of random variables X ε , W ε and a pair X, W such that we have and Moreover, for t ∈ [0, T ], the following results hold. Proof.

2.3.3.
Concluding the proof of (i) in Theorem 2.1. To begin with, we notice that the embedding X ֒→ Z is continuous, which means there exist continuous maps π m : Z → X , m ≥ 1 such that As before, it follows from (2.36), (2.38), Therefore we derive that for all φ ∈ Z * and dt ⊗ P − a.s., ) dt ′ is a continuous process on Z as well.
Hence, we obtain that X is a global martingale solution to (2.23). Moreover, (2.36) and (2.38) imply that . We have finished the proof.
2.4. Proof of (ii) in Theorem 2.1. To obtain a pathwise solution to (1.2), we will use (i) in Theorem 2.1 and the Gyöngy-Krylov Lemma, cf. Lemma A.9. The proof can naturally be broken down into several subsections.
2.4.1. Pathwise uniqueness of the cut-off problem. We first state the following result which indicates that for L ∞ (Ω)-initial values, the solution map is time locally Lipschitz in the less regular space Z.
Then Z satisfies the stochastic differential equation , Itô's formula (which holds true on the entire space Z), and the BDG inequality, we find for some C > 0 depending on b, g, h the estimate If we apply Grönwall's inequality to the estimate above, we get (2.39).
Proof. We first assume that X 0 X < M P − a.s. for some deterministic M > 0. For any K > 2M and T > 0, we define Then one can repeat all steps in the proof of (2.39) by using It is easy to see that Sending K → ∞, using the monotone convergence theorem and (2.41) with noticing T > 0 is arbitrary, we obtain the desired result for X 0 being almost surely bounded. It remains to remove this restriction. Motivated by [35,36], for general Notice that Then we can proceed with which completes the proof.
For the cut-off problem (2.23), we also have pathwise uniqueness. Indeed, since Z ֒→ V, the additional terms coming from the cut-off function χ R (·) can be handled by the mean value theorem as Then one can modify the proof of Lemma 2.4 in a straightforward way to get

2.4.2.
Pathwise solution to the cut-off problem. Now we prove the existence and uniqueness of a pathwise solution to (2.23). To be more precise, we are going to show the following result.

(2.42)
Proof. Uniqueness is a direct consequence of Lemma 2.5. The proof of the other assertions is divided into two steps.
Step 1: Existence. Let S = (Ω, F , P, {F t } t≥0 , W) be given and let X ε be the global pathwise solution to (2.24). We define sequences of measures ν ε (1) ,ε (2) and µ ε (1) ,ε (2) as can be obtained. Similar to Lemma 2.2, one can find a probability space Ω, F , P on which there is a sequence of random variables , W k and a random variable X, X, W such that Going along the lines as in Section 2.3.3, we see that both S, X, T and S, X, T are martingale solutions to (2.23) such that X, X ∈ L 2 Ω; L ∞ (0, T ; X ) ∩ C([0, T ]; Z) . Moreover, since X ε (0) ≡ X 0 for all n, we have that X(0) = X(0) almost surely in Ω. Then we use Lemma 2.5 to see Lemma A.9 implies that the original sequence {X ε } defined on the initial probability space (Ω, F , P) has a subsequence (still labeled in the same way) satisfying ). Since for each n, X ε is {F t } t≥0 progressive measurable, so is X. Using (2.43) and the embedding Z ֒→ V, we obtain a global pathwise solution to (2.23).
Step 2: Time continuity. As X ∈ L 2 (Ω; L ∞ (0, T ; X ) ∩ C([0, T ]; Z)), now we only need to prove that X(t) is continuous in X . Since X ֒→ Z is dense, we see that X is weakly continuous in X (cf. [56, page 263, Lemma 1.4]). It suffices to prove the continuity of [0, T ] ∋ t → X(t) X . The difficulty here is that the problem (1.2) is singular, i.e., g(t, X) is only a Z-valued process and h(t, X) is only an L 2 (U; Y)-valued process, hence the products (g(t, X), X) X and (h(t, X)e i , X) X might not exist and the classical Itô formula in the Hilbert space X (see [25,Theorem 4.32] or [34, Theorem 2.10]) can not be used directly here. At this point the regularization operator T ε from (A 5 ) is invoked to consider the Itô formula for T ε X 2 X instead. Then we have By (2.44), Thus, we only need to prove the continuity up to time τ N ∧ T for each N ≥ 1. Using (A 5 ), (A 1 ) and the bound Using Fatou's lemma, we arrive at which together with Kolmogorov's continuity theorem ensures the continuity of t → X(t ∧ τ N ) X .
With Lemma 2.6 at hand, we are in the position to finish the proof of (ii) in Theorem 2.1.

2.4.3.
Concluding the proof of (ii) in Theorem 2.1. Similar to Lemma 2.4, for X 0 (ω, x) ∈ L 2 (Ω; X ), we let Since E X 0 2 X < ∞, we have 1 = k≥1 1 Ω k P − a.s., which means that On account of Lemma 2.6, we let X k,R be the global pathwise solution to the cut-off problem (2.23) with initial value X 0,k and cut-off function χ R (·). Define Since X k,R is continuous in time (cf. Lemma 2.6), for any R > 0, we have P{τ k,R > 0, ∀k ≥ 1} = 1. Now we let R = R k be discrete and then denote ( , ∀k ≥ 1} = 1. Therefore (X k , τ k ) is the pathwise solution to (1.2) with initial value X 0,k . As has been shown in Lemma 2.4, 1 Ω k X k also solves (1.2) with initial value X 0,k on [0, 1 Ω k τ k ]. Then uniqueness means X k = 1 Ω k X k on [0, 1 Ω k τ k ] P − a.s. Therefore we infer from P{ k≥1 Ω k } = 1 that the pair is a pathwise solution to (1.2) corresponding to the initial condition X 0 . Since for each k, X k is continuous in time (cf. Lemma 2.6), so is X. Then we have Taking expectation gives rise to (2.21) and we have finished the proof of (ii) in Theorem 2.1. Hence it suffices to prove (2.48). Because X ֒→ V, it is obvious that τ 1 ≤ τ 2 P − a.s. Therefore, the proof reduces further to checking only τ 1 ≥ τ 2 P − a.s. We first notice that for all M, l ∈ N, we have P (τ 2,l ∧ M ≤ τ 1 ) = 1 for all M, l ∈ N, and As a result, it remains to prove (2.49). However, as mentioned before, we can not directly apply the Itô formula to X 2 X to get control of E X(t) 2 X . As in (2.45), but now with Q ε , we use Itô formula for Q ε X 2 X , apply the BDG inequality, (A 1 ) and (A 6 ) to find constants C 1 > 0 and This, together with (A 6 ), yields Since the right hand side of the inequality above does not depend on ε, and since Q ε satisfies (2.16), we can send ε → 0 to find Then Grönwall's inequality shows that for each l, M ∈ N, which gives (2.49). We conclude the proof of (iii) in Theorem 2.1.
3. Applications to nonlinear ideal fluid models with transport noise 3.1. Stochastic advection by Lie transport in fluid dynamics. Starting with the pioneering works [29,31] for linear scalar transport equations, many achievements have been made in recent years for stochastic fluid equations with noise of transport type. Transport-type noise refers to noise depending linearly on the gradient of the solution. In [38], stochastic equations governing the dynamics of some ideal fluid regimes have been derived by employing a novel variational principle for stochastic Lagrangian particle dynamics. Later, the same stochastic evolution equations were rediscovered in [21] using a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean, and a rapidly fluctuating small-scale map. In [38], the extension of geometric mechanics to include stochasticity in nonlinear fluid theories was accomplished by using Hamilton's variational principle. This extension motivates us to study stochastic Lagrangian fluid trajectories, denoted as X t (x, t), arising from the stochastic Eulerian vector field with a noise in the Stratonovich sense, i.e., In (3.1) u(x, t) means the drift velocity, {W k = W k (t)} k=1,2,··· ,M is a family of standard 1-D independent Brownian motions, and M can be determined via the amount of variance required from a principal component analysis, or via empirical orthogonal function analysis. Deriving continuum-scale equations taking into account noise as in (3.1) is known as the Stochastic Advection by Lie Transport (SALT) approach, see [20] and the references therein. The SALT approach combines stochasticity in the velocity of the fluid material loop in Kelvin's circulation theorem with ensemble forecasting and meets the important challenge of incorporating stochastic parameterisation at the fundamental level, see for example [10,47,57].
Many subsequent investigations of the properties of the equations of fluid dynamics with the SALT modification have appeared in the literature recently. For example, local existence in Sobolev spaces and a Beale-Kato-Majda type blow-up criterion were derived in [22,32] for the incompressible 3-D SALT Euler equations. For the 2-D version, global existence of solutions has been shown in [23]. In [2], the authors provide a local existence result for the incompressible 2-D SALT Boussinesq equations. For a simpler but still nonlinear equation as the SALT Burgers equation, we refer to [3,30].
3.1.1. The two-component CH system with transport noise. The Camassa-Holm (CH) equation was proposed independently by Fokas and Fuchssteiner in [33] and by Camassa and Holm in [14]. In [33], it was proposed to consider some completely integrable generalizations of the Korteweg-de-Vries equation with bi-Hamiltonian structures, and in [14], it was derived to describe the unidirectional propagation of shallow water waves over a flat bottom. Solutions of equation (3.2) exhibit the wave-breaking phenomenon, i.e., smooth global existence may fail [16,17]. Global conservative solutions to the CH equation (3.2) were obtained in [12,40]. Different stochastic versions of the CH equation have been studied including additive noise [15] and multiplicative noise [1,52,53,54]. Following the approach in [38], the corresponding stochastic version of the CH equation with transport noise was introduced in [8,24]. Transforming the equation into a partial differential equation with random coefficients, the well-posedness of the stochastic CH equation with some special transport noise has been studied in [1]. We can extend this result to a far more complex system: the stochastic two-component CH system which has been derived in [39], i.e., (3.4) The differential operator L ξ is given by (3.5) We use the notation L ξ k since it coincides with the Lie derivative operator acting on one-forms. However, our analysis is valid for general linear differential operators with suitable coefficients. Calculating the cross-variation term in the general transformation formula we obtain the corresponding Itô formulation of (3.4), given by Note that the operator L 2 ξ k in (3.6) is the second-order operator In this paper, we will consider (3.6) on the periodic torus T = R/2πZ in terms of the unknowns (u, η). Therefore, for any real number s, we define D s = (I − ∆) s/2 as D s f (k) = (1 + |k| 2 ) s/2 f (k). Then we apply (1 − ∂ 2 xx ) −1 = D −2 to (3.6) and consider for (u, η) the nonlocal Cauchy problem Here we remark that in (3.7), f = D −2 g = (I − ∂ 2 xx ) −1 g means f = G ⋆ g, where G is the Green function of the Helmholtz operator (I − ∂ 2 xx ) and ⋆ stands for the convolution. The local theory for (3.7) is stated in Theorem 3.1 below.
3.1.2. The CCF model with transport noise. As the second application of the abstract framework, we will consider a stochastic transport equation with non-local velocity on the periodic torus T. In the deterministic case, it reads θ t + (Hθ)θ x = 0, (3.8) where H is the periodic Hilbert transform defined by Equation (3.8) was proposed by Córdoba, Córdoba and Fontelos in [19] to consider advective transport with non-local velocity. It is deeply connected to the 2-D SQG equation and hence with the 3-D Euler equations (cf. [6] and the references therein). Notice that, if we replace the non-local Hilbert transform by the identity operator we recover the classical Burgers equation. In [19], the breakdown of classical solutions to (3.8) for a generic class of smooth initial data was discovered.
To the best of our knowledge, the stochastic counterpart of the CCF model (3.8) has not been studied yet.
In this paper, we will consider the stochastic CCF model with transport noise, i.e., where {W k = W k (t)} k∈N is a sequence of standard 1-D independent Brownian motions and L ξ k is given as in (3.5). Using the corresponding Itô formulation, we are led to the Cauchy problem . For the sake of simplicity, we omit the parentheses in the above notations from now on if there is no ambiguity. Similarly, for two spaces H s1 and H s2 (s 1 , s 2 > 0) and (f, The commutator for two operators P, Q is denoted by [P, Q] := P Q − QP. The space of linear operators from U to some separable Hilbert space X is denoted by L(U; X).
To obtain a local theory for (3.7) and (3.11), we have to impose natural regularity assumptions on {ξ k (x)} k∈N to give a reasonable meaning to the stochastic integral and to show certain estimates. For this reason, we make the following assumption: Besides, we do not require that {ξ k } k∈N is an orthogonal system.
The main results for (3.7) and (3.11) are the following:  , η), τ ) such that (3.12) Moreover, the maximal solution ((u, η), τ * ) to (3.7) satisfies Theorem 3.2. Let s > 7 2 and S = (Ω, F , P, {F t } t≥0 , W) be a stochastic basis fixed in advance. Let Assumption (B) hold. If θ 0 ∈ L 2 (Ω; H s ) is an F 0 -measurable random variable, then (3.11) has a local unique pathwise solution (θ, τ ) such that (3.13) Moreover, the maximal solution (θ, τ * ) to (3.11) satisfies Remark 3.2. We require s > 11/2 in Theorem 3.1. This is because, if (u, η) ∈ H s × H s−1 , then To apply Theorem 2.1 to (3.7) with X = H s × H s−1 , we have to verify (2.15) with using Lemma A.5. Therefore s − 4 > 3 2 , which means s > 11/2. Similarly, s > 7/2 is needed in Theorem 3.2. As mentioned before, the scalar stochastic CH equation with transport noise has been analyzed in [1] with a completely different approach. The authors obtain the local existence of pathwise solutions in a less regular space but without a blow-up criterion. We note that our approach can be also applied to this equation to give local existence, uniqueness and the blow-up criterion.

Remark 3.3. Notice that in the deterministic case, one can use the estimate
to improve the blow-up criterion (3.14) into (cf. [27]) To achieve this in the stochastic setting, we have an essential difficulty in closing the H s -estimate. That is, one has to split the expectation E Hθ x L ∞ θ 2 H s . If we use (3.15), so far we have not known how to close the estimate for E θ 2 3.3. The stochastic two-component CH system: Proof of Theorem 3.1. Now we consider (3.7) on the periodic torus T, and we will apply the abstract framework developed in Section 2 to obtain Theorem 3.1. To put (3.7) into the abstract framework, we define (3.16) Now we recall that U is a fixed separable Hilbert space and {e i } i∈N is a complete orthonormal basis of U such that the cylindrical Wiener process W is defined as in (2.1). Then we define h(X) ∈ L(U; (3.17) Altogether we can rewrite the problem (3.7) as dX = (b(X) + g(X)) dt + h(X) dW, In order to prove Theorem 3.1 by applying Theorem 2.1, we need to check that Assumption (A) is satisfied. To ease notation, we define X s = H s × H s−1 (3.19) and make the following choice for the spaces X ⊂ Y ⊂ Z and Z ⊂ V, Using the fact that H s−1 is an algebra, we can infer that Lemma 3.2. Let Assumption (B) hold true and s > 7/2. If X = (u, η) ∈ X s , then g : X s → X s−2 and h : X s → L 2 (U; X s−1 ) obey which implies the first estimate. Similarly, from the definition of h in (3.17), and the definition of L ξ in (3.5), one has which gives the second estimate.
Lemma 3.3. Let s > 11 2 , X = (u, η) ∈ X s and Y = (v, ρ) ∈ X s . Then we have Proof. Recalling (3.16) and (3.17), we have Because H s−2 ֒→ W 1,∞ , we can use Lemma A.4 and integration by parts to arrive at Similarly, we have Since s − 4 > 3/2, we can invoke Lemma A.5 to obtain In the same way, we have Collecting the above estimates, we obtain the desired result.

3.3.2.
Proof of Theorem 3.1. Now we will prove that all the requirements in Assumption (A) hold true. We first fix regular mappings g ε and h ε using the mollification operators from (A.1) and (A.2) in the Appendix A by Similar to (3.17), here we define h ε (X) ∈ L(U; X s ) such that We choose functions k(·) ≡ 1, f (·) = C(1 + ·), q(·) = C(1 + · 5 ) for some C > 1 large enough depending only on b, g, h. Finally we let T ε = Q ε =J ε , whereJ ε is given in (A.2). Let s > 11/2. Obviously, X ֒→ Y ֒→֒→ Z ֒→ V. Then Lemma 3.1 shows b : X s → X s , and Lemma 3.2 implies g : X s → X s−2 and h : X s → L 2 (U; X s−1 ). Hence the stochastic integral in (3.18) is a well defined X s−1 -valued local martingale. It is straightforward to verify that all of them are continuous in X ∈ X s . Checking Let v = D 2 J ε u. From the definition of the operator L ξ in (3.5), we have It follows from (A.5), (A.7), Lemma A.4 and integration by parts that By (A.5), (A.6) and the fact that D s−2 = D s D −2 , we obtain Since P = D s−2 J ε ∈ OPS s−2 1,0 (cf. Lemma A.1), we apply Lemma A.5 to arrive at Combining the above estimates, we arrive at Checking (A 4 ): It is clear that X = X s is dense in Z = X s−2 . Since s − 2 > 5 2 , inequality (2.14) follows directly from Lemma 3.1. Applying Lemma 3.3 yields (2.15).
Checking (A 6 ): It is easy to prove (2.16) and we omit the details here. Then we notice that For the first term we have that Using Lemma A.5 yields Gathering together the above estimates and noticing (A.7), we get (2.20). We are just left to show (2.19) to conclude the proof of Theorem 3.1. To this end, we recall (3.16) and consider where P 1 := T ε D s−2 ∈ OPS s−2 1,0 , P 2 := T ε D s−1 ∈ OPS s−1 1,0 (cf. Lemma A.1), and T 1 = [P 1 , L ξ k ], T 2 = [P 2 , L ξ k ]. Using integration by parts, (3.5) and (A.5), we have that Using (A.6) and (A.5), we have On account of H s−1 ֒→ W 1,∞ and integration by parts, it holds that and Then we apply Lemma A.4 to K 1 to find For K 2 , we use Lemma A.3 and integration by parts to derive Therefore, The form J 4 = T 1 D 2 u, P 1 D 2 u L 2 can be handled in the same way using H s−2 ֒→ W 1,∞ . Hence we have Now we summarize the above estimates, and use (3.17) and Assumption (B) to arrive at Hence we obtain inequality (2.19) and complete the proof.
3.4. Stochastic CCF model: Proof of Theorem 3.2. In this section we will apply Theorem 2.1 to (3.11) with x ∈ T to obtain Theorem 3.2. To that purpose, we set X = θ and b(t, X) = b(X) = 0, With the above notations, we reformulate (3.11) in the abstract form, i.e., dX = (b(X) + g(X)) dt + h(X) dW, To prove Theorem 3.2, we would like to invoke Theorem 2.1 to this setting. To do that, we just need to check the Assumption (A). Now we let r ∈ (3/2, s − 2), and then let X s = H s and V = H r .
(3.28) 3.4.1. Estimates on nonlinear terms. Analogously to Section 3.3.1 we will need the following auxiliary lemmas.
Lemma 3.4. Let Assumption (B) hold true and s > 5/2. If X = θ ∈ X s , then g : X s → X s−2 and h : X s → L 2 (U; X s−1 ) such that Proof. Using H s−2 ֒→ W 1,∞ , the continuity of the Hilbert transform for s ≥ 0 and Remark 3.1, one can prove the above estimates directly. We omit the details for exposition clearness.
Lemma 3.5. Let X = θ ∈ X s and Y = ρ ∈ X s . Then we have that for s > 7/2, Proof. Recalling (3.25) and (3.26), we have Because H s−2 ֒→ W 1,∞ , we use Remark 3.1, Lemma A.4, the continuity of the Hilbert transform and integration by parts to bound the first term as The last two terms can be bounded by invoking Lemma A.5 to obtain Collecting the above estimates, we obtain the desired result.
Let s > 7/2. Obviously, X ֒→ Y ֒→֒→ Z ֒→ V. Moreover, Lemma 3.4 implies g : X s → X s−2 and h : X s → L 2 (U; X s−1 ). Hence the stochastic integral in (3.27) is a well defined X s−1 -valued local martingale. It is easy to check that g and h are continuous in X ∈ X s .
Checking (A 3 ): Since (3.30) enjoys similar estimates as we established for (3.22), the first part (2.12) can be proved as before. Therefore, we just need to show (2.13). For all X = θ ∈ X s , we have Invoking Lemma A.5 with P = D s J ε ∈ OPS s 1,0 (cf. Lemma A.1), we have that To bound the first term, we notice that H r ֒→ W 1,∞ , then we use Lemma A.4, integration by parts, (A.7) and (A.8) to find Combining the above estimates, we arrive at which implies (2.13).
Checking (A 5 ): As before, this is a direct consequence of (A 6 ), which will be shown next.
Checking (A 6 ): Following the same way as we proved (3.24), we have that for some C > 1, Using Lemma A.4, (A.8), (A.9), integration by parts, Lemma A.3, and (A.7), we have Using Lemma (A.5) with P = D s T ε ∈ OPS s 1,0 (cf. Lemma A.1), we have that Combining the above estimates, we find some C > 1 such that, Due to V = H r ֒→ W 1,∞ and (A.9), (2.20) holds true. Therefore, we can apply Theorem 2.1 to obtain the existence, uniqueness of pathwise solutions, together with the blow-up criterion where r ∈ (3/2, s − 2) is arbitrary. Now we only need to improve the above blow-up criterion to (3.14).
To this end, we proceed as in the proof of (2.22) (cf. (2.48)). For m, l ∈ N, we define where inf ∅ = ∞. Denote σ 1 = lim m→∞ σ 1,m and σ 2 = lim l→∞ σ 2,l . Now we fix a r ∈ (3/2, s − 2). Then To apply Theorem 2.1 to (3.35) to get a local theory, we introduce some notations. For any real number s, Λ s = (−∆) s/2 are defined by Λ s f (k) = |k| s f (k). Then we let X s = H s ∩ f : We notice that with the mean-zero condition, X s is Hilbert space for s > 0 with inner product (f, g) X s = (Λ s f, Λ s g) L 2 and homogeneous Sobolev norm f X s = Λ s f L 2 . However, it can be shown that if f ∈ X s for s > 0, then, cf. [7], Then we have the following local results for (3.35): , W) be a stochastic basis fixed in advance and X s be given in (3.36). Let Assumption (C) hold true. If θ 0 ∈ L 2 (Ω; X s ) is an F 0 -measurable random variable, then (3.35) has a local unique pathwise solution θ starting from θ 0 such that Moreover, the maximal solution (θ, τ * ) to (3.35) satisfies Proof. We only give a very quick sketch. The approximation of (3.35) can be constructed as in the proof of Theorem 3.2. We only notice that if Assumption (C) is verified and θ 0 has mean-zero, then the approximate solution θ ε has also mean-zero. Recalling that U is fixed in advance to define (2.1), we take X = X s , Y = X s−1 , Z = X s−2 , V = X r with 2 < r < s − 2 and T ε = Q ε = T ε . One can basically go along the lines as in the proof of Theorem 3.2 with using the Λ s -version of Lemma A.4 (see also in [44,45]) to estimate the nonlinear term. For the noise term, after writing it into the Itô form, one can use Lemma A.5 and (3.37) to estimate the corresponding two terms. For the sake of brevity, we omit the details. For the deterministic incompressible porous medium equation, we refer to [13]. Both of them with SALT noise ∞ k=1 (ξ k · ∇θ) • dW k have not been studied. Similar to Theorem 3.1, our general framework (ii) is also applicable to them. Remark 3.5. It is worthwhile remarking that, a new framework called Lagrangian-Averaged Stochastic Advection by Lie Transport (LA SALT) has been developed for a class of stochastic partial differential equations in [4,28]. For LA SALT the velocity field is randomly transported by white-noise vector fields as well as by its own average over realizations of this noise. For the even more general distribution-path dependent case of transport type equations, we refer to [51]. Generally speaking, the distribution of the solution is a global object on the path space, and it does not exist for explosive stochastic processes whose paths are killed at the life time. For a local theory of distribution dependent SDEs/SPDEs, we have to either consider the non-explosive setting or modify the "distribution" by a local notion (for example, conditional distribution given by solution does not blow up at present time). Here, we focus our attention to the abstract framework for SPDEs with SALT noise. The general case with LA SALT is left as future work.
Lemma A.4 ( [44,45]). If f, g ∈ H s W 1,∞ with s > 0, then for p, p i ∈ (1, ∞) with i = 2, 3 and and D s (f g) L p ≤ C( f L p 1 D s g L p 2 + D s f L p 3 g L p 4 ).
If Assumption (B) holds, then we have Proof. The essential part of the desired estimate lies in the following result in [2]: Let Q be a first-order linear operator with smooth coefficients and P ∈ OPS s 1,0 . Then f ∈ H s with s > d 2 + 1 we have that PQ 2 f, Pf L 2 + (PQf, PQf ) L 2 f 2 H s . In particular, if we choose Q = L ξ k we have that: Since we want to calculate this estimate for ∞ k=1 L 2 ξ k , we need to precise the constant of the right hand side of (A.12). To this end, mimicking the proof of [2] we can rewrite the left hand side of (A.12) as Pf, E 2 Pf L 2 + (R 1 f, EPf ) L 2 =: where E = divξ k ∈ OPS 0 1,0 , R 0 = [L ξ k , E] ∈ OPS 1 1,0 , R 1 = [P, L ξ k ] and R 2 = [R 1 , L ξ k ]. By Lemma A.2, we have R 1 , R 2 , [R 1 , ∂ x ] ∈ OPS s 1,0 . To derive (A.11) we will invoke the following commutator estimates (see [55, (3.6.1) and (3.6.2)]): • If P ∈ OPS s 1,0 , s > 0, then there is a C > 0 such that P (gu) − gP u L 2 ≤ C ( g W 1,∞ u H s−1 + g H s u L ∞ ) . (A.13) • If P ∈ OPS 1 1,0 , then there is a C > 0 such that P (gu) − gP u L 2 ≤ C g W 1,∞ u H s−1 . (A.14) For I 1 , we have that Applying (A.13) with P = R 1 , g = ξ k , u = ∇f , and using H s ֒→ W 1,∞ , we arrive at For the second term, we have Applying (A.13) with P = R 1 , g = divξ k and u = f yields Hence, we have show that |I 1 | ≤ C ξ k H s+1 f 2 H s . Repeat the above procedure as we estimate R 2 f L 2 = [R 1 , L ξ k ]f L 2 with replacing R 1 by P, we have For the third term, using the Cauchy-Schwarz inequality and the fact that E = divξ k ∈ OPS 1 1,0 gives rise to For I 4 , we notice that L ξ k ∈ OPS 1 1,0 . Hence it follows from (A.14) with P = L ξ k , g = divξ k and u = Pf that Gathering all the above estimates implies that for some C > 0, Using Assumption (B) to the above estimates, we obtain (A.11).
We conclude this appendix with some useful tools in stochastic analysis.
Lemma A.6 (Prokhorov Theorem, [25]). Let X be a complete and separable metric space. A sequence of measures {µ n } ⊂ P(X) is tight if and only if it is relatively compact, i.e., there is a subsequence {µ n k } converging to a probability measure µ weakly.
Lemma A.7 (Skorokhod Theorem, [25]). Let X be a complete and separable metric space. For an arbitrary sequence {µ n } ⊂ P(X) such that {µ n } is tight on (X, B(X)), there exists a subsequence {µ n k } converging weakly to a probability measure µ, and a probability space (Ω, F , P) with X-valued Borel measurable random variables x n and x, such that µ n is the distribution of x n , µ is the distribution of x, and x n n→∞ − −−− → x P−a.s. Lemma A.8 ( [11,26]). Let (Ω, F , P) be a complete probability space and X be a separable Hilbert space. Let S n = Ω, F , {F n t } t≥0 , P, W n be a sequence of stochastic bases such that for each n ≥ 1, W n is cylindrical Brownian motion (over U with the canonical embedding U ֒→ U 0 being Hilbert-Schmidt) with respect to {F n t } t≥0 . Let G n be an F n t predictable process ranging in L 2 (U; X). Finally consider S = (Ω, F , P, {F t } t≥0 , W) and G ∈ L 2 (0, T ; L 2 (U; X)), which is F t predictable. Suppose that in probability we have W n → W in C ([0, T ]; U 0 ) and G n → G in L 2 (0, T ; L 2 (U; X)) . Then · 0 G n dW n → · 0 GdW in L 2 (0, T ; X) in probability.
Then {Y j } j≥0 converges in probability if and only if for every subsequence of {µ j k ,l k } k≥0 , there exists a further subsequence which weakly converges to some µ ∈ P(X × X) satisfying µ ({(u, v) ∈ X × X, u = v}) = 1.