The Burgers’ equation with stochastic transport: shock formation, local and global existence of smooth solutions
 19 Downloads
Abstract
In this work, we examine the solution properties of the Burgers’ equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine–Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blowup criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.
1 Introduction
 (1)
For \(\nu = 0\), Eq. (1.1) has a unique solution of class \(H^s\) for \(s > 3/2\) until some stopping time \(\tau > 0\).
 (2)
However, shock formation cannot be avoided a.s. in the case \(\xi (x) = \alpha x + \beta \) and for a broader class of \(\{\xi _k(\cdot )\}_{k \in {\mathbb {N}}}\), we can prove that it occurs in expectation.
 (3)
For \(\nu > 0\), we have global existence and uniqueness in \(H^2\).
if \(\partial _x u(0,X_0) > 0\), then \(\partial _x u(t,X_t) < \infty \) almost surely for all \(t>0\) and
if \(\partial _x u(0,X_0)\) is sufficiently negative, then there exists \(0< t_* < \infty \) such that \(\lim _{t \rightarrow t_*} {\mathbb {E}}[\partial _x u (t,X_t)] = \infty \).
We finally address the question of wellposedness. We will prove that by choosing a sufficiently regular initial condition, equation (1.1) admits a unique local solution that is smooth enough, such that the arguments employed in the previous section on shock formation are valid (in fact, we show this for a noise of the type \(Qu \circ {{\,\mathrm{d\!}\,}}W_t,\) where \(Qu = a(x) \partial _x u + b(x) u,\) which generalises the one considered in (1.1)). For Burgers’ equation with additive spacetime white noise, however, there have been many previous works showing wellposedness [3, 11, 15, 16]. The techniques used in these works are primarily based on reformulating the equations by a change of variable or by studying its linear part. The main difference in our work is that the multiplicative noise we consider depends on the solution and its gradient. Therefore, the effect of the noise hinges on its spatial gradient and the solution, giving rise to several complications. For instance, when deriving a priori estimates, certain high order terms appear, which need to be treated carefully. Recently, the same type of multiplicative noise has been treated for the Euler equation [8, 22] and the Boussinesq system [2], whose techniques we follow closely in our proof. We note that the wellposedness analysis of a more general stochastic conservation law, which includes the inviscid stochastic Burgers’ equation as a special case, has also been considered, for instance in [18, 19, 27]. However, these works deal with the wellposedness analysis of weak kinetic and entropy solutions, in contrast to classical solutions, which we consider here. There is also the recent work [31] showing the local wellposedness of weak solutions in the viscous Burgers’ equation (\(\nu > 0\)) driven by rough paths in the transport velocity. An important contribution of this paper is showing the global wellposedness of strong solutions in the viscous case by proving that the maximum principle is retained under perturbation by stochastic transport of type \(Qu \circ {{\,\mathrm{d\!}\,}}W_t\).
1.1 Main results
Let us state here the main results of the article:
Theorem 1.1
 (1)
Let \(\xi _1(x) = \alpha x + \beta \), \(x \in {\mathbb {R}}\) and \(\xi _k \equiv 0\) for \(k=2,3,\ldots \) and assume that u(0, x) has a negative slope. Then, there exists two characteristics satisfying (1.5) with different initial conditions that cross in finite time almost surely.
 (2)
Let \(X_t\) be a characteristic solving (1.5) with \(\{\xi _k(\cdot )\}_{k \in {\mathbb {N}}}\) satisfying the conditions in Assumption A1 below and let \(\partial _x u(0,X_0) \ge 0\). Then, if \(\psi (x) < \infty \) for all \(x \in {\mathbb {T}}\) or \({\mathbb {R}}\), we have that \(\partial _t u(t,X_t) < \infty \) almost surely for all \(t>0\).
 (3)
Again, let \(X_t\) be a characteristic solving (1.5) with \(\{\xi _k(\cdot )\}_{k \in {\mathbb {N}}}\) satisfying the conditions in Assumption A1 and let \(\partial _x u(0,X_0) < 0\). Also assume that \(\partial _x u(0,X_0) < \psi (x)\) for all \(x \in {\mathbb {T}}\) or \({\mathbb {R}}\). Then there exists \(0<t_*<\infty \) such that \(\lim _{t \rightarrow t_*} {\mathbb {E}} [\partial _x u(t,X_t)] =  \infty \).
Theorem 1.2
Theorem 1.3
(Wellposedness in the inviscid case) Let \(u_{0}\in H^{s} (\mathbb {T},\mathbb {R}),\) for \(s>3/2\) fixed. Then there exists a unique maximal solution \((\tau _{max},u)\) of the 1D stochastic Burgers’ equation (1.1) with \(\nu = 0\). Therefore, if \((\tau ',u')\) is another maximal solution, then necessarily \(\tau _{max}=\tau '\), \(u=u'\) on \([0,\tau _{max})\). Moreover, either \(\tau _{max}=\infty \) or \(\displaystyle \lim \sup _{s\nearrow \tau _{max}} u(s)_{H^{s}}= \infty \).
Theorem 1.4
(Global wellposedness in the viscous case) Let \(u_{0} \in H^{2} (\mathbb {T},\mathbb {R}).\) Then there exists a unique maximal strong global solution \(u:[0,\infty ) \times \mathbb {T} \times \Xi \rightarrow \mathbb {R}\) of the viscous stochastic Burgers’ equation (1.1) with \(\nu > 0\) in \(H^{2} (\mathbb {T},\mathbb {R})\).
Remark 1.5
Theorems 1.3 and 1.4 can be extended in a straightforward manner to the full line \({\mathbb {R}}\) and to higher dimensions.
Remark 1.6
1.2 Structure of the paper
This manuscript is organised as follows. In Sect. 2 we review some classical mathematical deterministic and stochastic background. We also fix the notations we will employ and state some definitions. Section 3 contains the main results regarding shock formation in the stochastic Burgers’ equation. Using a characteristic argument, we show that noise cannot prevent shocks from occurring for certain classes of \(\{\xi _k(\cdot )\}_{k \in {\mathbb {N}}}\). Moreover, we prove that these shocks satisfy a Rankine–Hugoniot type condition in the weak formulation of the problem. In Sect. 4, we show local wellposedness of the stochastic Burgers’ equation in Sobolev spaces and a blowup criterion. We also establish global existence of smooth solutions of a viscous version of the stochastic Burgers’ equation, which is achieved by proving a stochastic analogue of the maximum principle. In Sect. 5, we provide conclusions, propose possible future research lines, and comment on several open problems that are left to study.
2 Preliminaries and notation
Lemma 2.1
We will also use the following result as main tool for proving the existence results and blowup criterion:
Theorem 2.2
Remark 2.3
Theorem 2.2 is fundamental for closing the energy estimates when showing wellposedness of the stochastic Burgers’ equation. It permits reducing the order of a sum of terms which in principle seems hopelessly singular.
Next, we briefly recall some aspects of the theory of stochastic analysis. Fix a stochastic basis \(\mathcal {S}=(\Xi ,\mathcal {F},\lbrace \mathcal {F}_{t}\rbrace _{t\ge 0}, \mathbb {P},\lbrace W^{k}\rbrace _{k\in \mathbb {N}}),\) that is, a filtered probability space together with a sequence \(\lbrace W^{k} \rbrace _{k\in \mathbb {N}}\) of scalar independent Brownian motions relative to the filtration \(\lbrace \mathcal {F}_{t}\rbrace _{t\ge 0}\) satisfying the usual conditions.
We also state the celebrated Itô–Wentzell formula, which we use throughout this work.
Theorem 2.4
Let us also introduce three different notions of solutions:
Definition 2.5
Definition 2.6
\(\mathbb {P} (\tau _{max} >0) = 1, \ \tau _{max} = lim_{n \rightarrow \infty } \tau _n,\) where \(\tau _n\) is an increasing sequence of stopping times, i.e. \(\tau _{n+1}\ge \tau _{n}\), \(\mathbb {P}\) almost surely.
\((\tau _{n},u)\) is a local solution for every \(n\in \mathbb {N}\).
If \((\tau ',u')\) is another pair satisfying the above conditions and \(u'=u\) on \([0,\tau '\wedge \tau _{max} )\), then \(\tau '\le \tau _{max}\), \(\mathbb {P}\) almost surely.
A maximal solution is said to be global if \(\tau _{max}=\infty \), \(\mathbb {P}\) almost surely.
Definition 2.7
Notations: Let us stress some notations that we will use throughout this work. We will denote the Sobolev \(L^{2}\) based spaces by \(H^{s}(\text {domain}, \text {target space})\). However, we will sometimes omit the domain and target space and just write \(H^s\), when these are clear from the context. \(a\lesssim b\) means there exists C such that \(a \le Cb\), where C is a positive universal constant that may depend on fixed parameters and constant quantities. Note also that this constant might differ from line to line. It is also important to remind that the condition “almost surely” is not always indicated, since in some cases it is obvious from the context.
3 Shocks in Burgers’ equation with stochastic transport
3.1 Inviscid Burgers’ equation with stochastic transport
Assumption A1
 Lipschitz continuity$$\begin{aligned} \varphi (x)  \varphi (y) \le C_0 xy, \quad \xi _k(x)  \xi _k(y) \le C_k xy, \quad k \in {\mathbb {N}}, \end{aligned}$$(3.5)
 Linear growth condition$$\begin{aligned} \varphi (x) \le D_0 (1 + x), \quad \xi _k(x) \le D_k (1+x), \quad k \in {\mathbb {N}} \end{aligned}$$(3.6)
Provided \(u(t,\cdot )\) is sufficiently smooth and bounded (hence satisfying Lipschitz continuity and linear growth) until some stopping time \(\tau \), and \(\{\xi _k(\cdot )\}_{k \in {\mathbb {N}}}\) satisfies the conditions in Assumption A1, the characteristic equation (3.4) is locally wellposed. One feature of the multiplicative noise in (3.1) is that u is transported along the characteristics, that is, we can show that \(u(t,x) = (\Phi _t)_*u_0(x)\) for \(0\le t < \tau _{max},\) where \(\Phi _t\) is the stochastic flow of the SDE (3.4), \((\Phi _t)_*\) represents the pushforward by \(\Phi _t,\) and \((\tau _{max}, X_t)\) is the maximal solution of (3.4). This is an easy corollary of the ItôWentzell formula (2.9).
Corollary 3.0.1
Let \(u(t,\cdot )\) be \(C^3 \cap L^\infty \) in space for \(0<t<\tau \). Assume also that \(u(\cdot ,x)\) is a continuous semimartingale satisfying (3.2), \(\partial _x u (\cdot ,x)\) is a continuous semimartingale satisfying the spatial derivative of (3.2), and \(\{\xi _k(\cdot )\}_{k \in {\mathbb {N}}}\) satisfies the conditions in Assumption A1. If \((\tau _{max},X_t)\) is a maximal solution to (3.4), then \(u(t,X_t) = u(0,X_0)\) almost surely for \(0<t<\tau _{max}\).
Remark 3.1
Notice that due to our local wellposedness result (Theorem 1.3) and the maximum principle (Proposition 4.10), one has \(u_t \in C^3 \cap L^\infty \) for \(t < \tau _{max}\) provided \(u_0\) is smooth enough and bounded. For instance, \(u_0 \in H^4 \cap L^\infty \) is sufficient.
Proof of Corollary 3.0.1
3.2 Results on shock formation
Proposition 3.2
The first crossing time of the inviscid stochastic Burgers’ equation (3.1) with \(\xi _1(x) = \alpha x + \beta \) for constants \(\alpha , \beta \in {\mathbb {R}}\) and \(\xi _k(\cdot ) \equiv 0\) for \(k = 2,3,\ldots \) is equivalent to the first hitting time for the integrated geometric Brownian motion \(I_t := \int ^t_0 e^{\alpha W_s}ds\).
Proof
Remark 3.3
Note that the constant \(\beta \) does not affect the first crossing time, hence we can set \(\beta =0\) without loss of generality. Also in the following, we simply write \(\xi (\cdot )\) without the index when we only consider one noise term.
As an immediate consequence of Proposition 3.2, we prove that the transport noise with \(\xi (x) = \alpha x\) cannot prevent shocks from forming almost surely in the stochastic Burgers’ equation (3.1).
Corollary 3.3.1
Let \(\xi (x) = \alpha x\) for some \(\alpha \in {\mathbb {R}}\). If the initial profile \(u_0\) has a negative slope, then \(\tau < \infty \) almost surely.
Proof
In the following, we show that for a broader class of \(\{\xi _k(\cdot )\}_{k \in {\mathbb {N}}}\), shock formation occurs in expectation provided the initial profile has a sufficiently negative slope. Moreover, no new shocks can develop from positive slopes. We show this by looking at how the slope \(\partial _x u\) evolves along the characteristics \(X_t\), which resembles the argument given in [10] for the stochastic Camassa–Holm equation.
Theorem 3.4
Proof
Remark 3.5
Blowup in expectation does not imply pathwise blowup. It is merely a necessary condition, which suggests that the law of \(\partial _x u\) becomes increasingly fattailed with time, making it more likely for it to take extreme values. Nonetheless, it is a good indication of blowup occurring with some probability.
Example 3.6
3.3 Weak solutions
We saw that if the initial profile \(u_0\) has a negative slope, then shocks may form in finite time (almost surely in the linear case \(\xi (x) = \alpha x\)), so solutions to (3.1) cannot exist in the classical sense. This motivates us to consider weak solutions to (3.1) in the sense of Definition 2.7.
Suppose that the profile u is differentiable everywhere except for a discontinuity along the curve \(\gamma = \left\{ (t,s(t)) \in [0,\infty ) \times M \right\} \), where \(M = {\mathbb {T}}\) or \({\mathbb {R}}\). Then the curve of discontinuity must satisfy the following for u to be a solution of the integral equation (2.10).
Proposition 3.7
The main obstacle here is that the curve s(t) is not piecewise smooth and therefore we cannot apply the standard divergence theorem, which is how the Rankine–Hugoniot condition is usually derived. Extending classical calculus identities such as Green’s theorem on domains with nonsmooth boundaries is a tricky issue, but fortunately, there have been several works that extend this result to nonsmooth but rectifiable boundaries in [40], and to nonrectifiable boundaries in [28, 29, 30, 38].
Lemma 3.8
Remark 3.9
\(\partial \Omega \) has boxcounting dimension \(d < 2\) and u, v is \(\alpha \)Hölder continuous for any \(\alpha > d1\) (Harrison and Norton [30]).
Proof of Theorem 3.7
4 Wellposedness results
4.1 Local wellposedness of a stochastic Burgers’ equation
Theorem 4.1
Let \(u_{0}\in H^{s} (\mathbb {T},\mathbb {R}),\) for \(s>3/2\) fixed, then there exists a unique maximal solution \((\tau _{max},u)\) of the 1D stochastic Burgers’ equation (4.1). Therefore, if \((\tau ',u')\) is another maximal solution, then necessarily \(\tau _{max}=\tau '\), \(u=u'\) on \([0,\tau _{max})\). Moreover, either \(\tau _{max}=\infty \) or \(\displaystyle \lim \sup _{s\nearrow \tau _{max}} u(s)_{H^{s}}= \infty \).
Step 1: Uniqueness of local solutions. To show uniqueness of local solutions, one argues by contradiction. More concretely, one can prove that any two different solutions to (4.1) defined up to a stopping time must coincide, as explained in the following Proposition.
Proposition 4.2
Let \(\tau \) be a stopping time, and \(u^1,u^2: [0,\tau ] \times \mathbb {T}\times \Xi \rightarrow \mathbb {R}\) be two solutions with same initial data \(u_{0}\) and continuous paths of class \(C\left( [0,\tau ]; H^{s}(\mathbb {T},\mathbb {R})\right) \). Then \(u^{1}=u^{2}\) on \([0,\tau ],\) almost surely.
Proof
For this, we refer the reader to [2, 8]. It suffices to define \(\bar{u} = u^1  u^2,\) and perform standard estimates for the evolution of the \(L^2\) norm of \(\bar{u}\). \(\square \)
 Step 2: Existence and uniqueness of truncated maximal solutions. Consider the truncated stochastic Burgers’ equationwhere \(\theta _{r}:[0,\infty )\rightarrow [0,1]\) is a smooth function such that$$\begin{aligned} {{\,\mathrm{d\!}\,}}u_{r} + \theta _{r}(\partial _{x}u_\infty )u_{r}\partial _{x}u_{r} {{\,\mathrm{d\!}\,}}t+ \mathcal {Q}u_{r} {{\,\mathrm{d\!}\,}}W_{t}= \frac{1}{2} \mathcal {Q}^{2}u_{r} {{\,\mathrm{d\!}\,}}t, \end{aligned}$$(4.2)for some \(r>0\). Let us state the result which is the cornerstone for proving existence and uniqueness of maximal local solutions of the stochastic Burgers’ equation (4.1).$$\begin{aligned} \theta _{r}(x)= {\left\{ \begin{array}{ll} 1, \ \text {for } x \le r, \\ 0, \ \text {for } x \ge 2r, \end{array}\right. } \end{aligned}$$
Proposition 4.3
Given \(r>0\) and \(u_{0}\in H^{s}(\mathbb {T},\mathbb {R})\) for \(s>3/2\), there exists a unique global solution u in \(H^{s}\) of the truncated stochastic Burgers’ equation (4.2).
 Step 3: Global existence of solutions of the hyperregularised truncated stochastic Burgers’ equation. Let us consider the following hyperregularisation of our truncated equationwhere \(\nu >0\) is a positive parameter and \(s'= 2s+1\). Notice that we have added dissipation in order to be able to carry out the calculations rigorously. Equation (4.3) is understood in the mild sense, i.e., as a solution to an integrodifferential equation (see (4.4)).$$\begin{aligned} {{\,\mathrm{d\!}\,}}u^{\nu }_{r} + \theta _{r}(\partial _{x}u_\infty )u^{\nu }_{r}\partial _{x}u^{\nu }_{r} {{\,\mathrm{d\!}\,}}t+\mathcal {Q}u^{\nu }_{r} {{\,\mathrm{d\!}\,}}W_{t}= \nu \partial ^{s'}_{xx} u^{\nu }_{r} {{\,\mathrm{d\!}\,}}t +\frac{1}{2} \mathcal {Q}^{2}u^{\nu }_{r} {{\,\mathrm{d\!}\,}}t, \end{aligned}$$(4.3)
Proposition 4.4
For every \(\nu ,r >0\) and initial data \(u_{0}\in H^{s}(\mathbb {T},\mathbb {R})\) for \(s>3/2\), there exists a unique global strong solution \(u^{\nu }_{r}\) of Eq. (4.3) in the class \(L^{2}(\Xi ; C([0,T]; H^{s}(\mathbb {T},\mathbb {R})))\), for all \(T>0\). Moreover, its paths will gain extra regularity, namely \(C([\delta ,T]; H^{s+2}(\mathbb {T},\mathbb {R}) )\), for every \(T>\delta >0\).
Proof of Proposition 4.4

Step 4: Limiting and compactness argument. The main objective of this step is to show that the family of solutions \(\{u^{\nu }_{r} \}_{\nu >0}\) of the hyperregularised stochastic Burgers’ equation (4.3) is compact in a particular sense and therefore we are able to extract a subsequence converging strongly to a solution of the truncated stochastic Burgers’ equation (4.2). The main idea behind this argument relies on proving that the probability laws of this family are tight in some metric space. Once this is proven, one only needs to invoke standard stochastic partial differential equations arguments based on the Skorokhod’s representation and Prokhorov’s theorem. A more thorough approach can be found in [8, 24]. In the next Proposition, we present the main argument to show that the sequence of laws are indeed tight.
Proposition 4.5
 Step 5: Hypothesis estimates. We are left to show that hypothesis (4.6)–(4.7) hold. First, we will show that condition (4.6) implies condition (4.7). Applying Minkowski’s and Jensen’s inequalities, and carrying out some standard computations, we obtain$$\begin{aligned} \mathbb {E} [ u^{\nu }_{r}(t)u^{\nu }_{r}(s)^2_{H^{N}} ]&\lesssim (ts) \int _s^t \mathbb {E} [\theta _{r}(\partial _{x}u_\infty )u^{\nu }_{r}\partial _{x}u^{\nu }_{r}(\gamma )^2_{H^{N}}] {{\,\mathrm{d\!}\,}}\gamma \\&\quad + (ts)\int _s^t \mathbb {E} [\nu \partial ^{s'}_{xx} u^{\nu }_{r} (\gamma )^2_{H^{N}}] {{\,\mathrm{d\!}\,}}\gamma \\&\quad + (ts) \int _s^t \mathbb {E} [ {}^{\mathrm{a}}\mathcal {Q}^{2}u^{\nu }_{r} (\gamma )^2_{H^{1}}] {{\,\mathrm{d\!}\,}}\gamma \\&\quad + \mathbb {E} \left[ \left \left \int _s^t \mathcal {Q} u^{\nu }_{r} (\gamma ) {{\,\mathrm{d\!}\,}}W_{\gamma } \right \right ^2_{L^2} \right] . \end{aligned}$$
4.2 Blowup criterion
Theorem 4.6
Remark 4.7
The norm in the definition of \(\tau _n^2\) in Theorem 4.6 could be replaced with \(u(t, \cdot )_{H^s},\) for any \(s> 3/2,\) but we choose \(s=2\) for the sake of simplicity.
Proof of Theorem 4.6
We show both \(\tau ^{2}\le \tau ^{\infty }\) and \(\tau ^{\infty }\le \tau ^{2}\) in two steps.
4.3 Global wellposedness of a viscous stochastic Burgers’ equation
Theorem 4.8
Let \(u_{0} \in H^{2} (\mathbb {T},\mathbb {R}).\) Then there exists a unique maximal strong global solution \(u:[0,\infty ) \times \mathbb {T} \times \Xi \rightarrow \mathbb {R}\) of the viscous stochastic Burgers’ equation (4.29) with \(\nu > 0\) in \(H^{2} (\mathbb {T},\mathbb {R})\).
For our purpose, we prove the following result:
Proposition 4.9
Once we have proven the a priori estimate (4.30) we can repeat the arguments in Sect. 4.1 to obtain Theorem 4.8. However, since this is repetitive and tedious, we do not explicitly carry out these arguments here. Hence, we just provide a proof of Proposition 4.9.
Proof
Lemma 4.10
The maximum principle (4.36) is satisfied.
Proof
5 Conclusion and outlook
In this paper, we studied the solution properties of a stochastic Burgers’ equation on the torus and the real line, with the noise appearing in the transport velocity. We have shown that this stochastic Burgers’ equation is locally wellposed in \(H^s(\mathbb {T},\mathbb {R}),\) for \(s>3/2,\) and furthermore, found a blowup criterion which extends to the stochastic case. We also proved that if the noise is of the form \(\xi (x)\partial _x u \circ dW_t\) where \(\xi (x) = \alpha x + \beta \), then shocks form almost surely from a negative slope. Moreover, for a more general type of noise, we showed that blowup occurs in expectation, which follows from the previously mentioned stochastic blowup criterion. Also, in the weak formulation of the problem, we provided a Rankine–Hugoniot type condition that is satisfied by the shocks, analogous to the deterministic shocks. Finally, we also studied the stochastic Burgers’ equation with a viscous term, which we proved to be globally wellposed in \(H^2\).
Regarding shock formation, it is natural to ask whether our results can be extended to show that shock formation occurs almost surely for more general types of noise.
Another possible question is whether our global wellposedness result can be extended for the viscous Burgers’ equation with the Laplacian replaced by a fractional Laplacian \((\Delta )^{\alpha },\) \(\alpha \in (0,1)\). The main difficulty here is that in the stochastic case, the proof of the maximum principle (Proposition (4.10)) does not follow immediately since the pointwise chain rule for the fractional Laplacian is not available. In the deterministic case, this question has been settled and it is known that the solution exhibits a very different behaviour depending on the value of \(\alpha \): for \(\alpha \in [1/2,1]\), the solution is global in time, and for \(\alpha \in [0,1/2)\), the solution develops singularities in finite time [33, 35]. Interestingly, when an Itô noise of type \(\beta u {{\,\mathrm{d\!}\,}}W_t\) is added, it is shown in [39] that the probability of solutions blowing up for small initial conditions tends to zero when \(\beta > 0\) is sufficiently large. It would be interesting to investigate whether the transport noise considered in this paper can also have a similar regularising effect on the equation.
Similar results could be derived for other onedimensional equations with nonlocal transport velocity [5, 13, 14]. For instance, the so called CCF model [5] is also known to develop singularities in finite time, although by a different mechanism to that of Burgers’. To our knowledge, investigating these types of equations with transport noise is new.
Notes
Acknowledgements
The authors would like to thank José Antonio Carillo de la Plata, Dan Crisan, Theodore Drivas, Franco Flandoli, Darryl Holm, JamesMichael Leahy, Erwin Luesink, and Wei Pan for encouraging comments and discussions that helped put together this work. DAO has been partially supported by the Grant MTM201783496P from the Spanish Ministry of Economy and Competitiveness, and through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV20150554). ABdL has been supported by the Mathematics of Planet Earth Centre of Doctoral Training (MPE CDT). ST acknowledges the Schrödinger scholarship scheme for funding during this work.
References
 1.Attanasio, S., Flandoli, F.: Zeronoise solutions of linear transport equations without uniqueness: an example. C.R. Math. 347(13–14), 753–756 (2009)MathSciNetCrossRefGoogle Scholar
 2.AlonsoOrán, D., Bethencourt de León, A.: On the well posedness of a stochastic Boussinesq equation (2018). arXiv:1807.09493
 3.Bertini, L., Cancrini, N., JonaLasinio, G.: The stochastic Burgers equation. Commun. Math. Phys. 165, 211–231 (1994)MathSciNetCrossRefGoogle Scholar
 4.Beck, L., Flandoli, F., Gubinelli, M., Maurelli, M.: Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness (2014). arXiv:1401.1530
 5.Córdoba, A., Córdoba, D., Fontelos, M.: Formation of singularities for a transport equation with nonlocal velocity. Ann. Math. 162, 1377–1389 (2005)MathSciNetCrossRefGoogle Scholar
 6.Cotter, C., Crisan, D., Holm, D.D., Pan, W., Shevchenko, I.: Modelling uncertainty using circulationpreserving stochastic transport noise in a 2layer quasigeostrophic model (2018). arXiv:1802.05711
 7.Cotter, C.J., Crisan, D., Holm, D.D., Pan, W., Shevchenko, I.: Numerically modelling stochastic lie transport in fluid dynamics (2018). arXiv:1801.09729
 8.Crisan, D., Flandoli, F., Holm, D.D.: Solution properties of a 3D stochastic Euler fluid equation (2017). arXiv:1704.06989
 9.Cotter, C.J., Gottwald, G.A., Holm, D.D.: Stochastic partial differential fluid equations as a diffusive limit of deterministic lagrangian multitime dynamics. Proc. R. Soc. A 473(2205), 20170388 (2017)MathSciNetCrossRefGoogle Scholar
 10.Crisan, D., Holm, D.D.: Wave breaking for the stochastic Camassa–Holm equation. Physica D 376, 138–143 (2018)MathSciNetCrossRefGoogle Scholar
 11.Catuogno, P., Olivera, C.: Strong solution of the stochastic Burgers equation. Appl. Anal. Int. J. 93(3), 646–652 (2013). https://doi.org/10.1080/00036811.2013.797074 MathSciNetCrossRefzbMATHGoogle Scholar
 12.Delarue, F., Flandoli, F., Vincenzi, D.: Noise prevents collapse of Vlasov–Poisson point charges. Commun. Pure Appl. Math. 67(10), 1700–1736 (2014)MathSciNetCrossRefGoogle Scholar
 13.De Gregorio, S.: On a onedimensional model for the threedimensional vorticity equation. J. Stat. Phys. 59, 1251–1263 (1990)MathSciNetCrossRefGoogle Scholar
 14.De Gregorio, S.: A partial differential equation arising in a 1D model for the 3D vorticity equation. Math. Methods Appl. Sci. 19, 1233–1255 (1996)MathSciNetCrossRefGoogle Scholar
 15.Da Prato, G., Debussche, A., Temam, R.: Stochastic Burgers’ equation. Nonlinear Differ. Equ. Appl. 1, 389–402 (1994)MathSciNetCrossRefGoogle Scholar
 16.Da Prato, G., Gatarek, D.: Stochastic Burgers equation with correlated noise. Stoch. Stoch. Rep. 52(1–2), 29–41 (2007)MathSciNetzbMATHGoogle Scholar
 17.Fedrizzi, E., Flandoli, F.: Noise prevents singularities in linear transport equations. J. Funct. Anal. 264, 1329–1354 (2013) MathSciNetCrossRefGoogle Scholar
 18.Friz, P.K., Gess, B.: Stochastic scalar conservation laws driven by rough paths. In: Annales de l’Institut Henri Poincare (C) Non Linear Analysis, vol. 33, pp. 933–963. Elsevier (2016) Google Scholar
 19.Funaki, T., Gao, Y., Hilhorst, D.: Uniqueness of the entropy solution of a stochastic conservation law with a Qbrownian motion. hal02159743 (2019) Google Scholar
 20.Flandoli, F., Gubinelli, M., Priola, E.: Wellposedness of the transport equation by stochastic perturbation. Inventiones mathematicae 180, 1–53 (2010)MathSciNetCrossRefGoogle Scholar
 21.Flandoli, F., Gubinelli, M., Priola, E.: Full wellposedness of point vortex dynamics corresponding to stochastic 2D Euler equations. Stoch. Process. Appl. 121, 1445–1463 (2011)MathSciNetCrossRefGoogle Scholar
 22.Flandoli, F., Luo, D.: Euler–Lagrangian approach to 3D stochastic Euler equations (2018). arXiv:1803.05319
 23.Flandoli, F.: Random Perturbation of PDEs and Fluid Dynamic Models: École d’été de Probabilités de SaintFlour XL2010, vol. 2015. Springer, Berlin (2011)CrossRefGoogle Scholar
 24.GlattHoltz, N., Vicol, V.: Local and global existence of smooth solutions for the stochastic Euler equation with multiplicative noise. Ann. Probab. 42, 80–145 (2014). https://doi.org/10.1214/12AOP773 MathSciNetCrossRefzbMATHGoogle Scholar
 25.Gess, B., Maurelli, M.: Wellposedness by noise for scalar conservation laws (2017). arXiv:1701.05393
 26.Goldstein, J.A.: Semigroups of Linear Operators and Applications. Courier Dover Publications, New York (1985)zbMATHGoogle Scholar
 27.Gess, B., Souganidis, P.E.: Stochastic nonisotropic degenerate parabolic–hyperbolic equations. Stoch. Process. Appl. 127(9), 2961–3004 (2017)MathSciNetCrossRefGoogle Scholar
 28.Harrison, J.: Stokes’ theorem for nonsmooth chains. Bull Am. Math. Soc. 29(2), 235–242 (1993)MathSciNetCrossRefGoogle Scholar
 29.Harrison, J.: Flux across nonsmooth boundaries and fractal Gauss/Green/Stokes’ theorems. J. Phys. A 32, 5317 (1999)MathSciNetCrossRefGoogle Scholar
 30.Harrison, J., Norton, A.: The Gauss–Green theorem for fractal boundaries. Duke Math. J 67(3), 575–588 (1992)MathSciNetCrossRefGoogle Scholar
 31.Hocquet, A., Nilssen, T., Stannat, W.: Generalized Burgers equation with rough transport noise (2018). arXiv:1804.01335
 32.Holm, D.D.: Variational principles for stochastic fluid dynamics. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 471(2176) (2015) MathSciNetCrossRefGoogle Scholar
 33.Kiselev, A.: Regularity and blow up for active scalars. Math. Model. Nat. Phenom. 5(4), 225–255 (2010)MathSciNetCrossRefGoogle Scholar
 34.Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41(7), 891–907 (1988)MathSciNetCrossRefGoogle Scholar
 35.Kiselev, N.F.A., Shterenberg, R.: Blow up and regularity for fractal Burgers equation. Dyn. PDE 5(3), 211–240 (2008)MathSciNetzbMATHGoogle Scholar
 36.Kunita, H.: Some extensions of Itô’s formula. Séminaire de probabilités (1981)Google Scholar
 37.Kunita, H.: Stochastic Flows and Stochastic Differential Equations, vol. 24. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
 38.Lyons, T.J., Yam, P.S.C.: On Gauss–Green theorem and boundaries of a class of Hölder domains. Journal de mathématiques pures et appliquées 85(1), 38–53 (2006)MathSciNetCrossRefGoogle Scholar
 39.Röckner, M., Zhu, R., Zhu, X.: Local existence and nonexplosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise. Stoch. Process. Appl. 124(5), 1974–2002 (2014)MathSciNetCrossRefGoogle Scholar
 40.Shapiro, V.L.: The divergence theorem without differentiability conditions. Proc. Natl. Acad. Sci. USA (1957) Google Scholar
 41.Veretennikov, A.J.: On strong solutions and explicit formulas for solutions of stochastic integral equations. Sb. Math. 39(3), 387–403 (1981)CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.