On the stochastic Dullin-Gottwald-Holm equation: Global existence and wave-breaking phenomena

We consider a class of stochastic evolution equations that includes in particular the stochastic Camassa--Holm equation. For the initial value problem on a torus, we first establish the local existence and uniqueness of pathwise solutions in the Sobolev spaces $H^s$ with $s>3/2$. For non-autonomous linear multiplicative noise, we formulate then conditions that lead to global existence of strong solutions and wave-breaking phenomena. Wave-breaking solutions appear in fact with positive probability, and lower bounds for these probabilities are derived. Finally, the blow-up rate of these solutions is precisely analyzed.

(1. 1) It has been derived by Dullin et al. in [18]  Both, (1.2) and (1.3) have been studied widely in the literature. For this paper it is important to outline that wave breaking effect as an interesting solution phenomena. Wave breaking means that the solution remains bounded but develops an unbounded slope in finite time, cf. [10]. We note that the KdV equation does admit breaking waves but allows for globally smooth soliton solutions [34]. For the CH equation wave breaking has been analyzed in [8,11,41] including necessary and sufficient criteria for the occurrence of breaking waves in the Cauchy problem with smooth initial data [10,41]. Bressan&Constantin and later Holden&Raynaud [30,31] advanced the analysis of the CH equation developping new techniques that permit to prove the existence of global conservative and dissipative solutions. Besides wave breaking, another feature of the CH equations is (peaked) soliton interaction. Moreover, as pointed out in [9,12,13], the CH equation admits the occurrence of the traveling waves with a peak at their crest, exactly like the waves of the greatest height solutions of the governing equations for water waves. Combining the linear dispersion of the KdV equation with the nonlocal dispersion of the CH equation, the DGH equation (1.1) preserves its bi-Hamiltonian structure, is completely integrable (via the inverse scattering transform method [18]) and admits also soliton solutions.
Here, we are interested in stochastic variants of the DGH equation since the energy consuming/exchanging mechanisms of (1.1) may be connected with randomness through external stochastic influences. We are also motivated by the prevailing hypotheses that the onset of turbulence in fluid models involves randomness, cf. [5,37,19]. In this work, our stochastic evolution equation writes as Compared to the literature, the following deterministic weakly dissipative CH equation has been introduced and studied, cf. [38,49], u t − u xxt + 3uu x + λ(1 − ∂ 2 xx )u = 2u x u xx + uu xxx , λ > 0. (1.5) In (1.5), the operator λ(1 − ∂ 2 xx ) is linear and only models the (weak) energy dissipation, whereas the noise term −Ẇ (1 − α 2 ∂ 2 xx )h(t, u) in (1.4). is nonlinear and models random energy exchange. For convenience, we assume α = 1 in this paper. When α = 1, applying the operator (1 − ∂ 2 xx ) −1 to (1.4) gives rise to the following nonlocal equation du + [(u − γ) ∂ x u + F (u)] dt = h(t, u)dW, (1.6) where F (u) = F 1 (u) + F 2 (u) + F 3 (u) and (1.7) In (1.7), the operator (1 − ∂ 2 xx ) −1 in F (·) is understood as where [x] stands for the integer part of x. Here we remark that for additive noise, i.e., h(t, u) ≡ 1, the equation (1.6) has been studied in [40]. In this paper we will consider a more general context with noise driven by a cylindrical Wiener process W. It is assumed that this process is defined on an auxiliary Hilbert space U which is adapted to a right-continuous filtration {F t } t≥0 , rather than a white noise. The first goal of the present paper is to analyze existence, uniqueness and blow-up criteria of pathwise solutions to the periodic boundary value problem (1.9) Under generic assumptions on h(t, u), we will show that (1.9) has a unique pathwise solution. Here we remark that Chen et al. in [7] have considered the stochastic CH equation with additive noise. For the linear multiplicative noise case, we refer to [46] what concerns the stochastic CH equation, and to [6] for a stochastic modified CH equation. For stochastic evolution equations, the noise can have a regularization effect but also can lead to the contrary behaviour. For example, it is known that the well-posedness of linear stochastic transport equations with noise can be established under weaker hypotheses than for its deterministic counterpart (cf. [20,22]). For stochastic scalar conservation laws, noise on the flux may bring some regularization effects [25], and it may also trigger the discrete entropy dissipation in the numerical schemes for conservation laws such that the schemes enjoy some stability properties not present in the deterministic case [36]. For stochastic Euler equations, certain noise may prevent the coalescence of vortex singularities in two-dimensional space [23]. Moreover, we refer to [35,27,44,46] for the dissipation of energy caused by the linear multiplicative noise. In this paper, we will consider noise effects associated with the phenomenon of wave breaking. As put forward by e.g. Whitham in [48], the wave breaking phenomenon is one of the most intriguing long-standing problems of water wave theory. For the deterministic CH type equations, the wave breaking phenomenon has been extensively studied, see [10,11,41] for example. Particularly, for equations with dissipation of the term λ(u − u xx ), we refer to [49] for the phenomenon of wave breaking. For the stochastic Camassa-Holm equation, Crisan&Holm [15] proved that temporal stochasticity (in the sense of Stratonovich) in the diffeomorphic flow map for the stochastic Camassa-Holm equation does not prevent the wave breaking process. In fact they show a weaker statement, namely that wave breaking occurs with positive probability. In [45], wave breaking was studied for the non-autonomous linear multiplicative noise case (in the Itô sense).
The second goal of this work is to study the global existence and wave breaking phenomenon of the solutions to the following equation with non-autonomous linear multiplicative noise, namely (1.10) In (1.10) W is a standard 1-D Brownian motion. This case can be formally reformulated as the following SPDE, When c 0 + γ = 0, we give two conditions on the initial data that guarantee the global existence of the solutions. Besides, we also estimate the probability that the solution breaks and describe its breaking rate.
The precise statements of all the results above can be found in Section 2 jointly with the necessary assumptions on the data.

Definitions, assumptions and main results
We begin by introducing some notations. Throughout the paper, (Ω, F , P) denotes a complete probability space, with P being a probability measure on Ω and F being a σ-algebra. Let t > 0 and τ ∈ [0, t]. Then σ{X(τ ), Y (τ )} τ ∈[0,t] stands for the completion of the union σ-algebra generated by (X(τ ), Y (τ )). All stochastic integrals are defined in the Itô sense and E· is the mathematical expectation of · with respect to P. For some separable Banach space X, B(X) denotes the Borel sets of X, and Pr(X) stands for the collection of Borel probability measures on X. For E ⊆ X, When the function space refers to T, we will drop T if there is no ambiguity. We will use to denote estimates that hold up to some universal deterministic constant which may change from line to line but whose meaning is clear from the context. For linear operators A and B, we denote the Lie bracket [A, B] = AB − BA.
We briefly recall some aspects of the theory of infinite dimensional stochastic analysis which we use below. We refer the readers to [16,24,32] for an extended treatment of this subject.
We call S = (Ω, F , P, {F t } t≥0 , W) a stochastic basis, where {F t } t≥0 is a right-continuous filtration on (Ω, F ) such that {F 0 } contains all the P-negligible subsets and W(t) = W(ω, t), ω ∈ Ω is a cylindrical Brownian motion, defined on an auxiliary Hilbert space U , which is adapted to {F t } t≥0 . More precisely, we consider a separable Hilbert space U as well as a larger Hilbert U 0 such that the embedding U ֒→ U 0 is Hilbert-Schmidt. Therefore we have where {W k } k≥1 is a sequence of mutually independent one-dimensional Brownian motions.
To define the Itô stochastic integral Ge k dW k on H s , it is required (see [16,42] for example) that the predictable stochastic process G takes values in the space of Hilbert-Schmidt operators from U to H s , denoted by L 2 (U, H s ). Remember For predictable G ∈ L 2 (U, H s ), one can define the Itô stochastic integral Actually, the stochastic integral t 0 G dW is an H s -valued square-integrable martingale. In our case we have the Burkholder-Davis-Gundy inequality 1) or in terms of the coefficients, and for all t > 0, (2) Local pathwise solutions are said to be pathwise unique, if given any two pairs of local pathwise solutions (u 1 , τ 1 ) and (u 2 , τ 2 ) with P {u 1 (0) = u 2 (0)} = 1, we have (3) Additionally, (u, τ * ) is called a maximal solution to (1.9) if τ * > 0 almost surely and if there is an increasing sequence τ n → τ * such that for any n ∈ N, (u, τ n ) is a pathwise solution to (1.9) and on the set {τ * < ∞}, (4) If τ * = ∞ almost surely, then we say that the pathwise solution exists globally.

2.2.
Assumptions. Then we prescribe some conditions on the noise coefficient h.
• There is an increasing locally bounded function g(·) : [0, +∞) → [0, +∞), such that for any t > 0 and u ∈ H s , 2.3. Main results. Now we present our results. For the general case (1.9), we have (2.5) Moreover, (u, τ ) can be extended to a unique maximal solution (u, τ * ) in the sense of Definition 2.1 and Remark 2.1. The proof of Theorem 2.1 can follow the ideas in [27,17,2,4,3,46,14]. However, the Faedo-Galerkin method used in [17,27] is hard to be used here directly since we do not have the additional incompressibility, which guarantees the global existence of an approximate solution (see, e.g. [21,27]). Without this, we need to find a positive lower bound for the lifespan of the approximate solutions, which is generally not clear. Particularly, for our case in, this difficulty can be overcome by constructing a suitable approximation scheme and establishing a blow-up criteria, which is not only available for u, but also for the approximate solution u ε . We borrow the idea from the recent work [14] to achieve such a blow-up criteria. If h does not depend on u, then this result covers the existence and uniqueness results in [40].
Finally, we focus on (1.10). For the issue of global existence, we have H s < ∞. Let K ≥ 0 be a constant such that the embedding · W 1,∞ < K · H s holds. Then there is a C = C(s) > 1 such that for any R > 1 and λ 1 > 2, if u 0 H s < b * CKλ1R almost surely, then (1.11) has a maximal solution (u, τ * ) satisfying for any λ 2 > 2λ1 λ1−2 the estimate In other words, That is to say, P {u exists globally} ≥ p + q.
Then we obtain the following blow-up scenario, which is more precise than the common one given by (2.6): is the corresponding unique maximal solution to (1.10) (or to (1.11), equivalently), then the singularities can arise only in the breaking form. More precisely, s., and where Now we are in the position to answer the question on regularity from the introduction.
where b * is given in Assumption A 2 and λ is given in Lemma 3.5, then the maximal solution (u, τ * ) to (1.11) satisfies We notice that the conditions in Theorem 2.2 and Theorem 2.5 are not contradicting.
almost surely, where 0 < c < 1, then u blows up in finite time with positive probability. At this time, . What concerns a quantitative estimate on the wave breaking we have Theorem 2.6 (Wave breaking rate). Let the conditions in Theorem 2.4 hold true. Then As a corollary, if all the conditions in Theorem 2.5 hold true, then wave breaking occurs with the breaking rates given by (2.8).
Remark 2.3. Motivated by [27,44,46], where the linear noise βudW with β ∈ R \ {0} is considered, we focus on the non-autonomous linear multiplicative noise case, namely (1.10). We transform (1.10) to a non-autonomous random system (5.2). Although the stochastic integral is absent in (5.2), to extend the deterministic results to the stochastic setting, we need to overcome a few technical difficulties since the system is not only random but also non-autonomous. In this work, we manage to gain some estimates and asymptotic limits of the Girsanov type processes (see e.g., (5.6), (5.8), (5.10) and Lemma 3.6), which enable us to apply the energy estimate pathwisely (namely for a.e. ω ∈ Ω) and obtain Theorem 2.2.
Remark 2.4. Formally, if b(t) ≡ 0 in (1.11), then everything is deterministic and β ≡ 1. We see that the blow-up estimate (2.8) turns out to be which covers the deterministic case, cf. Theorem 4.2 in [39].
We outline the rest of the paper. In the next section, we briefly recall some relevant preliminaries. In Section 4, we prove Theorem 2.1. For the non-autonomous linear multiplicative noise case, we consider the global existence, decay, wave breaking and the blow-up rate of the pathwise solutions and prove Theorems 2.2 2.3, 2.5 and 2.6 in Section 5.

Preliminary Results
Now we state some relevant preliminaries, which will be used later. We first recall some classical estimates. .
Now we establish some estimates on the nonlocal term F (u) in (1.7). Lemma 3.3. For the F (·) defined in (1.7) and for any v 1 , v 2 ∈ H s with s > 3/2, we have Proof. Since s > 3/2, the desired result follows from Lemma 3.2 immediately. We omit the details and we refer to [47] for the CH case.
The following lemma has been established for the whole line case in [10]. When x ∈ T, using the periodic property of v in the proof as in Theorem 2.1 in [10], one also has (cf. [8]):  ). There is a λ > 0 such that for any f ∈ H 3 , Lemma 3.6. Assume Assumption A 2 holds true and a(t) ∈ C([0, ∞)) is a bounded function. Let Then we have the following properties: (ii) Let a(t) = λb 2 (t) with λ < 1 2 and τ R = inf{t ≥ 0 : X(t) > R} with R > 1, then is oneto-one (since φ is strictly increasing) and onto (by Assumption A 2 ), φ −1 (t) is well defined and lim t→∞ Q(t) = lim t→∞ Q(φ −1 (t)) P − a.s. Direct computation (see [1, Exercise 7.7 and Theorem 8.2] for example) show that is itself a Brownian motion. Then we use the law of the iterated logarithm to get lim sup which leads to lim t→∞ X(t) = e lim t→∞ Q(φ −1 (t)) = 0 P − a.s.
We apply the Itô formula to X p with p > 0 to obtain that Particularly, if p = 1 − 2λ, then Hence by the definition of τ R , continuity of measures and the Chebyshev inequality, we have which is (3.5).

Proof of Theorem 2.1
Let us postpone the proof for the existence and uniqueness of pathwise solutions to a later section. Here we will prove the blow-up criteria first, since some of the used estimates will be used later.

4.1.
Blow-up criteria. Motivated by [14], we first consider the relationship between the explosion time of u(t) H s and the explosion time of u(t) W 1,∞ for (1.9). Lemma 4.1. Let (u, τ * ) be the unique maximal solution to (1.9). Then the real valued stochastic process u W 1,∞ is also F t -adapted. Besides, for any m, n ∈ Z + , define Moreover, let τ 1 = lim m→∞ τ 1,m and τ 2 = lim n→∞ τ 2,n , then we have Proof. To begin with, we see that u(· ∧ τ ) ∈ C([0, ∞); H s ) meaning that for any t ∈ [0, τ ] we have Therefore u(t), as a W 1,∞ -valued process, is also F t -adapted. We then infer from the embedding H s ֒→ W holds true, then for all n, k ∈ Z + , P {τ 2,n ∧ k ≤ τ 1 } = 1 and Since (4.2) is a consequence of the assumption in (4.1), it suffices to prove (4.1). Apply the Itô formula for u 2 H s = D s u 2 L 2 to deduce that for any n, k > 1 and t ∈ [0, τ 2,n ∧ k], Then (2.3) and the stochastic Fubini theorem [16] imply For L 2 , we first notice that Then we commute the operator D s with u to derive, By utilizing Lemma 3.1 and integration by parts, we find that Similarly, it follows from Lemma 3.3 and the assumption (2.3) that Therefore we combine the above estimates to have where C depends on n through n + f 2 (n). Then the Gronwall's inequality shows that for each n, k ∈ Z + , there is a constant C = C(n, k, u 0 ) > 0 such that u(t) 2 H s < C(n, k, u 0 ), which gives (4.1). If u is the unique pathwise solution with maximal existence time τ * , for fixed m, n > 0, even if P{τ 1,m = 0} or P{τ 2,n = 0} is larger than 0, for a.e. ω ∈ Ω, there is m > 0 or n > 0 such that τ 1,m , τ 2,n > 0. By continuity of u(t) H s and the uniqueness of u, it is easy to check that τ 1 = τ 2 = τ * . Consequently, we obtain the desired blow-up criteria.

Existence and uniqueness.
(1) (Approximate solutions) The first step is to construct a suitable approximation scheme. For any R > 1, we let and χ R (x) = 0 for x > 2R. Then we consider the following cut-off problem on T, From Lemma 3.3, we see that the nonlinear term F (u) reserves the H s -regularity for any s > 3/2. However, to apply the theory of SDEs in Hilbert space to (4.3), we will have to mollify the transport term (u − γ)∂ x u since the product (u − γ)∂ x u loses one regularity.
To this end, we consider the following approximation scheme: where J ε is the Friedrichs mollifier defined as J ε f (x) = j ε * f (x), where * stands for the convolution, j ε (x) = k∈Z j(εk)e ixk and j(x) is a Schwartz function satisfying 0 ≤ j(ξ) ≤ 1 for all the ξ ∈ R and j(ξ) = 1 for any ξ ∈ [−1, 1]. From the construction, we see that J ε enjoys some useful estimates, see [46] for example. With a uniform L ∞ (Ω; W 1,∞ ) condition provided by the cut-off function χ R ( u W 1,∞ ), we can split the expectation E( u ε 2 H s u ε W 1,∞ ). Otherwise, the a priori L 2 (Ω; H s ) estimate for u ε is not closed. After introducing the mollifying of the transport term (u − γ)u x , for a fixed stochastic basis S = (Ω, F , P, {F t } t≥0 , W ) and for u 0 ∈ L 2 (Ω; H s ) with s > 3, according to the existence theory of SDE in Hilbert space [24,32,16], (4.4) admits a unique solution u ε ∈ C([0, T ε ), H s ) P − a.s. Using the same way as we prove Lemma 4.1, we see that for each fixed ε, if T ε < ∞, then lim sup t→Tε u ε (t) W 1,∞ = ∞. Due to the cut-off in (4.4), for a.e. ω ∈ Ω, u ε (t) W 1,∞ is always bounded and hence u ε is actually a global in time solution, that is, u ε ∈ C([0, ∞), H s ) P − a.s. We remark here that the global existence of u e is necessary in our framework due to the lack of life-span estimate in the stochastic setting, otherwise we will have to prove P{inf ε>0 T ε > 0} = 1, which is generally not clear.
(2) (Pathwise solution to the cut-off problem in H s with s > 3) We pass the limit ε → 0 by applying the stochastic compact argument, namely Prokhorov theorem and Skorokhod theorem to obtain the almost sure convergence for a new approximate solution ( u ε , W ε ) defined on a new probability space. By virtue of a refined martingale representation theorem [29, Theorem A.1], we may send ε → 0 in ( u ε , W ε ) to build a global martingale solution in H s with s > d/2 + 3 to the cut-off problem. Finally, we apply the powerful Gÿongy-Krylov characterization [28] of the convergence in probability to prove the convergence of the original approximate solutions. For more details, we refer to [46] and here we omit the details. Though the Yamada-Watanabe type result in infinite dimensional space has been established in [43] for SPDEs with variational structure, however, the conditions therein are unverifiable for our problem. (3) (Remove the cut-off and extend the range of s to s > 3/2) When u 0 ∈ L ∞ (Ω, H s ) with s > 3/2, by mollifying the initial data, we obtain a sequence of regular solutions {u k } k∈N to (1.9). Motivated by [26,46], one can prove that there is a subsequence such that for some almost surely positive stopping time τ , u H s ≤ u 0 H s + 2 P − a.s. (4.5) Then we can pass limit to prove that (u, τ ) is a solution to (1.9). Besides, using a cutting argument, as in [27,26,2], enables us to remove the boundedness assumption on u 0 . More precisely, when E u 0 2 H s < ∞, we consider the decomposition Ω k = {k − 1 ≤ u 0 H s < k}, k ∈ N, k ≥ 1.

Global existence and wave breaking
In this section, we study the global existence and the blow-up of solution to (1.11), and estimate the associated probabilities. Motivated by [27,44,46], we introduce the following Girsanov type transform is the corresponding unique maximal solution to (1.10), then for any c 0 , γ ∈ R and for t ∈ [0, τ * ), the function v defined by (5.1) solves the following problem on T almost surely, Consequently, Theorem 2.1 implies that (1.10) has a unique maximal solution (u, τ * ). Direct computation with the Itô formula yields Therefore we arrive at Multiplying both sides of the above equation by v and then integrating the resulting equation on x ∈ T, we see that for a.e. ω ∈ Ω and for all t > 0 which implies (5.3).

5.1.
Global existence: Case I. Now we prove the first global existence result, which is motivated by [27,44,46].
Proof for Theorem 2.2. To begin with, we apply the operator D s to (5.4), multiply both sides of the resulting equation by D s v and integrate over T to obtain that for a.e. ω ∈ Ω, Using Lemma 3.1, integration by parts and Lemma 3.3, we conclude that there is a C = C(s) > 1 such that for a.e. ω ∈ Ω, where β is given in (5.1).
Via (5.1), we obtain the desired estimate.
5.4. Wave breaking. Since no stochastic integral appears in (5.2), motivated by [8], we first establish the following result.
Since b 2 (t) < b * for all t > 0, we have e t 0 b(t ′ )dW t ′ > c for all t ⊆ β(t) ≥ ce − b * 2 t for all t .
Therefore we arrive at P {τ * < ∞} ≥ P e t 0 b(t ′ )dW t ′ > c for all t > 0, which gives the desired estimate in Theorem 2.5.

5.5.
Wave breaking rate. Now we prove Theorem 2.6, which provides a precise bound on the wave breaking rate.
Proof of Theorem 2.6. Recalling (5.22), we have that almost surely