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

We consider a class of stochastic evolution equations that include 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 Hs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^s$$\end{document} with s>3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>3/2$$\end{document}. Then we show that strong enough nonlinear noise can prevent blow-up almost surely. To analyze the effects of weaker noise, we consider a linearly multiplicative noise with non-autonomous pre-factor. Then, we formulate precise conditions on the initial data that lead to global existence of strong solutions or to blow-up. The blow-up occurs as wave breaking. For blow-up with positive probability, we derive lower bounds for these probabilities. Finally, the blow-up rate of these solutions is precisely analyzed.


Introduction
The Dullin-Gottwald-Holm (DGH) equation is a third-order dispersive evolution equation given by Both (1.2) and (1.3) have been studied widely in the literature. We notice that the CH equation exhibits two interesting phenomenon, namely (peaked) soliton interaction and wave breaking (the solution remains bounded but its slope becomes unbounded in finite time, cf. [12]), while the KdV equation does not model breaking waves [35] (when c 0 = 0, (1.2) admits a smooth soliton). For the CH equation, wave breaking and the necessary and sufficient criterion for the occurrence of breaking waves in the Cauchy problem with smooth initial data have been analyzed [10,12,13,43]. As pointed out in [11,14,15], the essential feature of the CH equation is the occurrence of traveling waves with a peak at their crest, exactly like that the governing equations for water waves admit the so-called Stokes waves of the greatest height. Bressan&Constantin [5,6]  Holden&Raynaud [31,32] also obtained global conservative and dissipative solutions using a Lagrangian point of view.
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 [20]) and admits also soliton solutions.
Here, we are interested in stochastic variants of the DGH equation to model energy consuming/exchanging mechanisms in (1.1) that are driven by external stochastic influences. Adding multiplicative noise has also been connected to the prevailing hypotheses that the onset of turbulence in fluid models involves randomness, cf. [7,21,39]. Precisely, our stochastic evolution equation is rewritten as (1.4) where W is a standard 1-D Brownian motion and h = (t, u) is a typically nonlinear function. We notice that the deterministic counterpart of (1.4) is the weakly dissipative CH equation Equation (1.5) has been introduced and studied for h(t, u) = u in [40,53], In (1.5), the operator λ(1 − ∂ 2 xx ) is linear and only models the (weak) energy dissipation. In order to model more general random energy exchanges, we consider the possibly nonlinear noise term −Ẇ (1 − α 2 ∂ 2 xx )h(t, u) in (1.4). To compare our model with deterministic weakly dissipative CH type equations (see [40,52,53] and the references therein), we focus our attention on the case that α = 0. 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 (1. 6) In (1.6), the operator (1 − ∂ 2 xx ) −1 in torus T = R/2πZ is understood as where [x] stands for the integer part of x. Here we remark that for additive noise, (1.6) has been studied in [42]. In this paper we will consider a more general context with noise driven by a cylindrical Wiener process W, rather than a standard Brownian motion W . It is assumed that W is defined on an auxiliary Hilbert space U which is adapted to a right-continuous filtration {F t } t≥0 , see Sect. 2 for more details.
With the above notations, the first goal of the present paper is to analyze the existence and uniqueness of pathwise solutions and to determine possible (1.9) Under generic assumptions on h(t, u), we will show that (1.8) has a local unique pathwise solution (see Theorem 2.1 below). Here we remark that Chen et al. in [9] have considered the stochastic CH equation with additive noise. For the linear multiplicative noise case, we refer to [48] for the stochastic CH equation, and to [8] for a stochastic modified CH equation.
For stochastic nonlinear evolution equations, the noise effect is a crucial question to study. Can the noise prevent blow-up or does it even drive the formation of singularities? For example, it is known that the well-posedness of linear stochastic transport equations with noise can be established under weaker hypotheses than its deterministic counterpart (cf. [22,24]). For stochastic scalar conservation laws, noise on the flux may bring some regularization effects [27], 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 [37]. Moreover, we refer to [29,36,46,48] for the dissipation of energy caused by the linear multiplicative noise.
However, most existing results on the regularization effects by noise for transport type equations are for linear equations or restricted to linear growing noise. Much less is known concerning the cases of nonlinear equations with nonlinear noises. Indeed, the interplay between regularization provided by noise and the nonlinearities of the governing equation is more complicated. For example, singularities can be prevented in some cases (cf. [25]: coalescence of vortices disappears in stochastic 2-D Euler equations). On the other hand, it is known that noise does not prevent shock formation in the Burgers equation, see [23].
Therefore the second goal of this work is to study the case of strong nonlinear noise and consider its effect. As we will see in (2.5) below, for the solution to (1.8), its H s -norm blows up if and only if its W 1,∞ -norm blows up. This suggests choosing a noise coefficient involving the W 1,∞ -norm of u. Therefore in this work we consider the case that h(t, u) dW = a (1 + u W 1,∞ ) θ u dW , where θ > 0, a ∈ R and W is a standard 1-D Brownian motion. We will try to determine the range of θ and a such that the solution to the following problem (1.10) As is shown in Theorem 2.2 below, if the noise is strong enough (either θ > 1/2, a = 0 or θ = 1/2, a 2 1), then the global existence holds true for (1.10) almost surely. This result justifies the idea that large nonlinear noise can actually prevent blow-up.
On the other hand, as put forward by e.g. Whitham in [51], 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 [12,13,43] for example. Particularly, for equations with dissipation term λ(u − u xx ), we refer to [53] for the phenomenon of wave breaking. When random noise is involved, as far as we know, we can only refer to [17,49] for wave breaking. In [17] the authors proved that temporal stochasticity (in the sense of Stratonovich) in the diffeomorphic flow map for the stochastic CH equation does not prevent the wave breaking process. In [49], wave breaking in the stochastic CH equation with multiplicative Itô noise is considered.
Thus, the third goal of this paper is to consider noise effects associated with the phenomenon of wave breaking. Due to Theorem 2.2, we see that if wave breaking occurs, the noise term does not grow fast. Hence we consider θ = 0 in (1.10) but introduce a non-autonomous pre-factor depending on time t. Precisely, we consider the DGH equation with linear multiplicative noise given by (1.11) This case can be formally reformulated as the following stochastic evolution equation when s > 3 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. See Theorems 2.3-2.7 for the statements.
The precise statements of all the results above can be found in Sect. 2 jointly with the necessary assumptions on the noise coefficient.

Definitions, assumptions and main results
We begin by introducing some notations. L 2 (T) is the usual space of squareintegrable functions on T. For s ∈ R, 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 We briefly recall some aspects of the theory of infinite dimensional stochastic analysis which we will use below. We refer the readers to [18,26,33] 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 Pnegligible subsets and W(t) = W(ω, t)(ω ∈ Ω) is a cylindrical Wiener process adapted to {F t } t≥0 . More precisely, we consider a separable Hilbert space U as well as a larger Hilbert space U 0 such that the embedding U → U 0 is Hilbert-Schmidt. Therefore we define where {W k } k≥1 is a sequence of mutually independent 1-D Brownian motions and {e k } k∈N is a complete orthonormal basis of U .
For a predictable stochastic process G taking values in the space of Hilbert-Schmidt operators from U to H s , denoted by L 2 (U ; H s ), the Itô stochastic integral [18,44] for example). Remember that 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 E sup , p ≥ 1. (2.1)

Definitions of the solutions
We now precise the notion of pathwise solutions to (1.8).

Assumptions
Next, we prescribe some conditions on the noise coefficient h in (1.8) and on b in (1.12).
. Moreover, we assume the following: • There is a non-decreasing locally bounded function f (·) Particularly, in the additive noise case, we assume h : such that for any t > 0, After the regularization effect of strong noise is established in Theorem 2.2, to analyze the effect of noise on the regularity of pathwise solutions, we restrict ourselves to the linear-noise case (1.12) imposing the following bounds on the coefficient b.

Assumption 2.2. When considering (1.12) with non-autonomous linear noise
Remark 2.1. Let us give some brief explanations for the assumptions.
• The function h : . This will be essential to pass to the limit when establishing the existence of a martingale solution as an intermediate step, cf. [3,48,49]. • The uniform-in-time assumption (2.2) bounds the growth of the L 2 (U ; H s )-norm of the noise coefficient in terms of a product of a nonlinear function of the W 1,∞ -norm and the H s -norm. This allows us to control the W 1,∞ -norm by some cut-off later. • Formula (2.3) ensures local Lipschitz continuity in H s , which will be used to obtain (local) existence and uniqueness. • Let us outline that we will use a Girsanov-type transformation to study (1.11) (see Remark 2.6 and Sect. 6). The assumption b 2 (t) ≤ b * is used to guarantee that such transformation is well-defined and the condition b(t) = 0, t ≥ 0 is needed to establish certain estimates for the Girsanov-

Main results and remarks
Now we present our results. For the general case (1.8), we have the following local existence result which moreover relates the possible blow-up in the H snorm to simultaneous blow up in the W 1,∞ -norm. (

2.4)
Besides, (u, τ ) can be extended to a unique maximal solution (u, τ * ) in the sense of Definition 2.1 and the following blow-up criterion holds true: For the proof of Theorem 2.1 one can follow the ideas in e.g. [2][3][4]16,19,29,48] by constructing a sequence of approximations for a problem with cut-off for the W 1,∞ -norm. Such a cut-off implies at-most linear growth of u and guarantees the global existence of an approximate solution. Otherwise we have to find a positive lower bound for the lifespan of the approximate solutions, which is a priori not clear. Besides, with the cut-off, one can close the a priori L 2 (Ω; H s ) estimate by splitting E( u 2 H s u W 1,∞ ). Turning to noise-driven regularization effects, the blow-up criterion (2.5) suggests relating the noise coefficient to the W 1,∞ -norm of u. Therefore we NoDEA On the stochastic DGH equation Page 9 of 34 5 consider (1.10) with scalable noise impact, i.e., we assume h(t, u) = a(1 + u ) θ W 1,∞ u for some θ > 0 and a ∈ R. When a and θ satisfy certain strengthconditions, the noise term counteracts the formation of singularities and we have where Q = Q(s, c 0 , γ) is a constant that will be specified in Lemma 3.5. Then the corresponding maximal solution (u, τ * ) to (1.10) satisfies Theorem 2.2 implies that blow-up of pathwise solutions might only be observed if the noise is weak. To detect such noise, we analyze the simpler ansatz h(t, u) = b(t)u as in (1.11). Even in this linear noise case the situation is quite subtle allowing for global existence as well as blow-up of solutions. For global existence, we can identify two cases.
then (1.11) has a maximal solution (u, τ * ) satisfying for any 1], then the corresponding maximal solution (u, τ * ) to (1.11) satisfies On the other hand, since the proof of Theorem 2.4 relies on the analysis of a PDE with random coefficient (see (6.2) below), the deterministic case can be included by formally letting the random coefficient be 1. Therefore, in this sense, Theorem 2.4 covers the corresponding deterministic result, cf. [13,41]. Indeed, by letting β ≡ 1 in (6.2) and taking (p, q) = (1, 0) or (p, q) = (0, 1) in Theorem 2.4, we obtain the global existence for the deterministic DGH equation.
According to (2.5) in Theorem 2.1, a blow-up comes along with an explosion of the W 1,∞ -norm. For the special noise in (1.11) we can improve the result by showing that a blow-up is related to the first spatial derivative only and corresponds to the wave-breaking phenomenon with exploding negative slope.
Then the singularities can arise only in the form of wave breaking, i.e., s., and Still we have not identified initial data for (1.11) that lead to a blow-up. A precise condition in terms of probability is given in the next theorem. To formulate it, we introduce the number λ > 0 such that for any f ∈ H 3 , the estimate holds.

Theorem 2.6. (Wave breaking and its probability)
where b * is given in Assumption 2.2 and λ is given in (2.11), then the maximal solution (u, τ * ) to (1.11) (or (1.12), equivalently) satisfies . We conclude this section with a result refining Theorem 2.5. It is possible to quantify the blow-up rate.
dt . Remark 2.5. As a corollary of Theorems 2.5 and 2.7, we have that as long as singularities occurs, they can arise only in the form of wave breaking and the breaking rate is given by (2.12). This result is optimal in the sense that it is consistent with the result for the corresponding deterministic case. Indeed, for the deterministic DGH equation (cf. [41,Theorem 4.2]), the blow-up rate is Formally, since the deterministic DGH equation can be viewed as (6.2) with β ≡ 1, we see that the blow-up estimate (2.12) coincides with the above deterministic result when β ≡ 1.
Remark 2.6. Let us make a comment on the idea for the subsequent analysis of (1.11), which is motivated by [29,46,48]. By introducing the Girsanov-type transformation we obtain an equation for v (see Sect. 6 for the detailed calculation), namely Although the above equation for v does not depend on a stochastic integral on v itself, 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 (see e.g., (6.6), (6.8) and (6.10)). With the help of certain estimates and asymptotic limits of Girsanov-type processes (see Lemma 3.7), we are able to apply the energy estimate pathwisely (for a.e. ω ∈ Ω) to study the global existence and possible blow-up of solutions.
We outline the rest of the paper. In the next section, we briefly recall some relevant preliminaries. In Sect. 4, we prove Theorem 2.1. For the large noise case, we prove Theorem 2.2 in Sect. 5. 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 2.3, 2.4, 2.6 and 2.7 in Sect. 6.

Preliminary results
We summarize some auxiliary results, which will be used to prove our main results from Sect. 2. Define the regularizing operator T ε on T as Since T ε can be characterized by its Fourier multipliers, it is easy to see Furthermore, we have Then for some C > 0, The following estimates are classical for Sobolev spaces.
Specifically, for our problem (1.8), we have introduced the nonlocal term F (·) in (1.9). Using the Moser estimate from Lemma 3.3, we can obtain the next statement on F (·) (see [50]).  The following estimate will be used in the proof of the blow-up criterion (2.5) and of Theorem 2.2. Lemma 3.5. Let s > 3/2, c 0 , γ ∈ R. Let F (·) and T ε be given in (1.9) and (3.1), respectively. There is a constant Q = Q(s, c 0 , γ) > 0 such that for all ε > 0, Due to (3.2) and (3.3), we commute the operator to derive

Using Lemma 3.4 and (3.4) directly, we have
H s . Combining the above two inequalities gives rise to the desired estimate of the lemma.
The following lemma has been established for the real-line case in [12] and [10], respectively. They hold likewise for x ∈ T, using the periodicity on T.

Moreover, the function
We conclude this preparatory section with some results from [47], which are needed to establish the theorems on global existence.

Lemma 3.7. Let Assumption 2.2 hold true and assume that a(t) ∈ C([0, ∞)) is a bounded function. For
dt the following properties hold true.

Proof of Theorem 2.1
We consider the initial value problem (1.8). The proof of existence and uniqueness of pathwise solutions can be carried out by standard procedures used in many works, see [2,3,28,29,[47][48][49] for more details. Therefore we only give a sketch.
(1) Firstly, one constructs a suitable approximation scheme using a cut-off function to control the W 1,∞ -norm (arising from (u − γ)u x , (2.2) and Lemma 3.4). With such cut-off, both the drift and diffusion coefficients in the problem become locally Lipschitz continuous and grow linearly in u (cf. [47][48][49]). Thus the approximation solutions exist globally. Besides, such cut-off enables us to close the a priori L 2 (Ω; H s ) estimate by splitting E( u 2 H s u W 1,∞ ). Therefore by using Lemma 3.2 and (2.2), uniform estimates for the approximation solutions can be established. We refer the readers to [48,49] for some closely related models; . We refer to [45,48,49] for example. Applying the probabilistic compactness arguments, i.e., the Prokhorov theorem and the Skorokhod theorem, and using some technical convergence results as in [1][2][3]19] [30]) can be applied to show the existence and uniqueness of a pathwise solution in H s with s > 3, cf. [3,29,49]; (4) Finally, mollifying initial data, analyzing the convergence and employing the argument as in [28,29,48,49] lead to a local pathwise solution (u, τ ) To finish the proof of Theorem 2.1, we only need to verify the blow-up criterion (2.5). Motivated by [16,47], we first consider, in next lemma, the relationship between the explosion time of u(t) H s and the explosion time of u(t) W 1,∞ for (1.8). The results of the lemma will not only immediately imply the blow-up criterion (2.5) but also be used in the next sections.  (1.8). Then the real-valued stochastic process u W 1,∞ is also F t -adapted. Besides, for any m, n ∈ N, define For τ 1 = lim m→∞ τ 1,m and τ 2 = lim n→∞ τ 2,n , we have then Proof. To begin with, since u(· ∧ τ ) ∈ C([0, ∞); H s ) almost surely, we see that for any t ∈ [0, τ], Therefore u(t), as a W 1,∞ -valued process, is also F t -adapted. Moreover, the embedding H s → W 1,∞ for s > 3/2 means that there is a K = K(s) > 0 such that · W 1,∞ < K · H s . Then for every m ∈ N,   [44,Theorem 4.2.5]) requires the dual product H s−1 (u − γ) u x , u H s+1 to be well-defined. In our case we only have u ∈ H s and (u − γ) u x ∈ H s−1 such that neither requirement is fulfilled. Therefore we utilize the mollifier operator T ε defined in (3.1). We first apply T ε to (1.8), and then use the Itô formula for T ε u 2 H s = D s T ε u 2 L 2 to deduce that for any n, k > 1 and t ∈ [0, τ 2,n ∧ k], On account of the Burkholder-Davis-Gundy inequality (2.1), for the expectation of the H s -norm of T ε u, we arrive at We can infer from (3.4) and Assumption 2.1 that For L 2 and L 3 , we use Lemma 3.5 to find Similarly, it follows from the assumption (2.2) that If we combine the above estimates and use (3.4), we are led for some constant Since the right hand side of the last estimate does not depend on ε, and T ε u tends to u in C ([0, T ]; H s ) for any T > 0 almost surely as ε → 0, one can send Then Grönwall's inequality shows that for each n, k ∈ N, there is a constant C = C(n, k, u 0 ) > 0 such that which gives (4.2).
We finish the section with the proof of the blow-up criterion in Theorem (2.1).

Proof of Theorem 2.2: strong nonlinear noise
To begin with, we note the following algebraic inequality.
There is a C > 0 such that for all 0 ≤ x ≤ My < ∞, Proof. Since My ≥ x, we have When η > 1 and a, b > 0 or η = 1 and b > a > 0, the latter expression tends to −∞ for x → +∞, which implies the statement of the lemma.
We are now ready to prove Theorem 2.2 following [45] to large extent.

Proof of Theorem 2.2. Assume s > 5/2 and let
θ u with θ ≥ 1/2 and a = 0. For r > 3/2, the embedding H r → W 1,∞ implies that we have for any u, v ∈ H r the estimate This means that one can establish the pathwise uniqueness for (1.10) in H r with r > 3/2. Hence, in the same way as proving Theorem 2.1, one can show that (1.10) admits a unique pathwise solution u in H s with s > 5/2 and maximal existence time τ * . We recall the definition of the mollifier T ε from Sect. 3 and define Applying the Itô formula to Lemma 3.5 and (3.4) imply that there is a Q = Q(s, c 0 , γ) > 0 such that for any t > 0 we have Notice that for any T > 0, T ε u tends to u in C ([0, T ]; H s ) almost surely as ε → 0. Then, by (3.4) and the dominated convergence theorem, the last estimate leads to Since we have assumed (2.6), Lemma 5.1 immediately shows that there are constants K 1 , K 2 > 0 such that and Next, we notice that there is a function δ : holds. Therefore, for any T > 0, by using Lemma 5.

Proofs of Theorems 2.3-2.7: non-autonomous linear noise case
In this section, we study (1.12) with linear noise. Depending on the strength of the noise in (1.12), we provide either the global existence of pathewise solutions or the precise blow-up scenarios for the maximal pathwise solution.
As discussed in Remark 2.6, we rely on the Girsanov-type transform We first collect some properties of v.  (1.11), then for any c 0 , γ ∈ R and for t ∈ [0, τ * ), the process v defined by (6.1) solves the following problem on T almost surely, Therefore we arrive at Multiplying both sides of (6.5) 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 (6.3).

Theorem 2.3: global existence for weak noise I
Now we prove the first global existence result, which is motivated by [29,[46][47][48].
Proof of Theorem 2.3. To begin with, we apply the operator D s to (6.4), multiply both sides of the resulting equation by D s v and integrate over T to obtain for a.e. ω ∈ Ω 1 2 where β is given in (6.1) (If necessary, T ε can be used as in Lemma 4.1). Then Let R > 1 and λ 1 > 2. Assume u 0 H s < b * CKλ1R < b * CKλ1 almost surely and define Then it follows from the embedding The above inequality and w = e − t 0 b(t )dW t u imply that for a.e. ω ∈ Ω, for any λ 2 > 2λ1 λ1−2 and for t ∈ [0, τ 1 ), Define the stopping time dt > R . (6.8) Notice that P{τ 2 > 0} = 1. From (6.7), we have that almost surely By Assumption 2.2, (6.9) and (6.6), we find that on [0, τ 1 ∧ τ 2 ), which means Therefore it follows from (6.9) that We apply (ii) in Lemma 3.7 to find that , which completes the proof.
By the embedding H 2 → W 1,∞ and the blow-up criterion in Theorem 2.1, almost surely we have that u(t) H s can be extended beyond τ * . Therefore we obtain a contradiction and hence ω ∈ Ω 2 . Therefore we obtain (2.10).

Theorem 2.6: wave breaking and its probability
The proof of Theorem 2.6 relies on certain properties of the solution v to the problem (6.2).