The Burgers' equation with stochastic transport: shock formation, local and global existence of smooth solutions

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 blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.


Introduction
We prove the well-posedness of a stochastic Burgers' equation of the form where x ∈ T or R, ν ≥ 0 is constant, {W k t } k∈N is a countable set of independent Brownian motions, {ξ k (·)} k∈N is a countable set of prescribed functions depending only on the spatial variable, and • means that the stochastic integral is interpreted in the Stratonovich sense. If the set {ξ k (·)} k∈N forms a basis of some separable Hilbert space H (for example L 2 (T)), then the process dW := ∞ k=1 ξ k (x) • dW k t is a cylindrical Wiener process on H, generalising the notion of a standard Wiener process to infinite dimensions.
The multiplicative noise in (1.1) makes the transport velocity stochastic, which allows the Burgers' equation to retain the form of a transport equation ∂ t u +ũ ∂ x u = 0, whereũ(t, x) := u(t, x) +Ẇ is a stochastic vector field with noiseẆ that is smooth in space and rough in time. Compared with the well-studied Burgers' equation with additive noise, where the noise appears as an external random forcing, this type of noise arises by taking the diffusive limit of the Lagrangian flow map regarded as a composition of a slow mean flow and a rapidly fluctuating one [CGH17]. In several recent works, this type of noise, which we call stochastic transport, has been used to stochastically parametrise unresolved scales in fluid models while retaining the essential physics of the system [Hol15, CCH + 18, CCH + 19]. On the other hand, it has also been shown to have a regularising effect on certain PDEs that are ill-posed [FGP10,FGP11,FF13,GM18]. Therefore, it is of interest to investigate how the stochastic transport in (1.1) affects the Burgers' equation, which in the inviscid case ν = 0 is a prototypical model for shock formation. In particular, we ask whether this noise can prevent the system from developing shocks or, on the contrary, produce new shocks. We also ask whether this system is well-posed or not. In this paper, we will show that: (1) For ν = 0, equation (1.1) has a unique solution of class H s for s > 2 until some stopping time τ > 0.
(2) However, shock formation cannot be avoided a.s. in the case ξ(x) = αx + β and for a broader class of {ξ k (·)} k∈N , we can prove that it occurs in expectation.
(3) For ν > 0, we have global existence and uniqueness in H s for s > 2.
On top of this, we prove a continuation criterion for the inviscid equation (ν = 0), which generalises the result for the deterministic case. The above results are not immediately evident for reasons we will discuss below. Although we cannot prove this here, we believe that shocks in Burgers' equation are too robust and ubiquitous to be prevented by noise, regardless of what {ξ k (·)} k∈N is chosen. Our results provide rigorous evidence to support this claim.
The question of whether noise can regularise PDEs is not new. In finite dimensions, it is well-known that additive noise can restore the well-posedness of ODEs whose vector fields are merely bounded and measurable (see [Ver81]). For PDEs, a general result is not known; however, there has been a significant effort in recent years to generalise this celebrated result to PDEs. In a remarkable paper, Flandoli, Gubinelli, and Priola [FGP10] demonstrated that the linear transport equation ∂ t u + b(t, x) · ∇u = 0, which is ill-posed if b is sufficiently irregular, can recover existence and uniqueness of L ∞ solutions that is strong in the probabilistic sense, by the addition of a "simple" transport noise, where the drift b is bounded, measurable, Hölder continuous, and satisfies an integrability condition on the divergence ∇ · b ∈ L p ([0, T ] × R d ). In a subsequent paper [FF13], the same noise was shown to retain some regularity of the initial condition, thus restoring wellposedness of strong solutions, and a selection principle based on taking the zero-noise limit as opposed to the inviscid limit was considered in [AF09].
However, for nonlinear transport equations such as Burgers', the same type of noise du + u ∂ x u dt + ∂ x u • dW t = 0 does not help, since a simple change of variables v(t, x) := u(t, x − W t ) will lead us back to the original equation ∂ t v + v ∂ x v = 0. Hence, if noise were to prevent shock formation, a more general class could be required, such as the cylindrical transport noise ∞ k=1 ξ k (x)∂ x u • dW k t that we consider in this paper. In [FGP11] and [DFV14], it was shown that collapse in Lagrangian point particle solutions of certain nonlinear PDEs (point vortices in 2D Euler and point charges in the Vlasov-Poisson system), can be prevented by this cylindrical transport noise with ξ k (x) satisfying a certain hypoellipticity condition, thus providing hope for regularisation of nonlinear transport equation by noise. More recently, Gess and Maurelli [GM18] showed that adding a simple stochastic transport term into a nonlinear transport equation du + b(x, u(t, x))∇u dt + ∇u • dW t = 0, (1.3) which in the deterministic case admits non-unique entropy solutions for sufficiently irregular b, can restore uniqueness of entropy solutions, providing a first example of a nonlinear transport equation that becomes well-posed when adding a suitable noise.
We should now stress the difference between the present work and previous works. First, we acknowledge that in Flandoli [Fla11], Chapter 5.1.4, it is argued that shock formation does not occur even with the most general cylindrical transport noise, by writing the characteristic equation as an Itô SDE which is a martingale perturbation of straight lines that will cross without noise. Thus, using the property that a martingale M t grows slower than t almost surely as t → ∞, it is shown that the characteristics cross almost surely. However, the characteristic equation for the system (1.1) is in fact a Stratonovich SDE, In summary, shock formation occurs in expectation if the initial profile has a sufficiently negative slope and no new shocks can form from a positive slope.
We finally address the question of well-posedness. 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 • dW t , where Qu = a(x)∂ x u+b(x)u, which generalises the one considered in (1.1)). For Burgers' equation with additive space-time white noise, however, there have been many previous works showing well-posedness [BCJL94,DPDT94,DPG07,CO13]. 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 [CFH19,FL18] and the Boussinesq system [AOB20], whose techniques we follow closely in our proof. We note that the well-posedness 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 [FG16,GS17,FGH20]. However, these works deal with the well-posedness analysis of weak kinetic and entropy solutions, in contrast to classical solutions, which we consider here. There is also the recent work [HNS19] showing the local well-posedness of weak solutions in the viscous Burgers' equation (ν > 0) driven by rough paths in the transport velocity. An important contribution of this paper is showing the global well-posedness of strong solutions in the viscous case by proving that the maximum principle is retained under perturbation by stochastic transport of type ξ(x)∂ x u(t, x) • dW t .
1.1. Main results. Let us state here the main results of the article: Theorem 1.1 (Shock formation in the stochastic Burgers' equation). In the following, we use the notation ψ( The main results regarding shock formation in (1.1) are as follows: (1) Let ξ 1 (x) = αx + β, x ∈ R and ξ k ≡ 0 for k = 2, 3, . . . 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 {ξ k (·)} k∈N satisfying the conditions in Assumption A1 below and let ∂ x u(0, X 0 ) ≥ 0. Then, if ψ(x) < ∞ for all x ∈ T or R, we have that ∂ t u(t, X t ) < ∞ almost surely for all t > 0.
(3) Again, let X t be a characteristic solving (1.5) with {ξ k (·)} k∈N satisfying the conditions in Assumption A1 and let ∂ x u(0, Theorem 1.2 (Stochastic Rankine-Hugoniot condition). The curve of discontinuity (t, s(t)) ∈ [0, ∞)×T (or R) of the stochastic Burgers' equation (1.1) satisfies the following: where u ± (t, s(t)) := lim x→s(t) ± u(t, x) are the left and right limits of u. Remark 1.6. We prove Theorem 1.3 for a more general noise Qu • dW t , where Q is a first order linear differential operator, which includes the transport noise as a special case. For the sake of clarity, our proof deals only with one noise term Qu•dW t , however, we can readily extend this to cylindrical noise with countable set of first order linear differential operators by imposing certain smoothness and boundedness conditions for the sum of the coefficients. We also prove Theorem 1.4 for one noise term.
1.2. Structure of the paper. This manuscript is organised as follows. In Section 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 {ξ k (·)} k∈N . Moreover, we prove that these shocks satisfy a Rankine-Hugoniot type condition in the weak formulation of the problem. In Section 4, we show local well-posedness of the stochastic Burgers' equation in Sobolev spaces and a blow-up 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 Section 5, we provide conclusions, propose possible future research lines, and comment on several open problems that are left to study.

Preliminaries and notation
Let us begin by reviewing some standard functional spaces and mathematical background that will be used throughout this article. Sobolev spaces are given by for any s ≥ 0 and p ∈ [1, ∞), equipped with the norm ||f || W s,p = ||(I − ∂ xx ) s/2 f || L p . We will also use the notation Λ s = (−∂ xx ) s/2 . Recall that L 2 based spaces are Hilbert spaces and may alternatively be denoted by H s = W s,2 . For s > 0, we also define H −s := (H s ) , i.e. the dual space of H s . Let us gather here some well-known Sobolev embedding inequalities: Let us also recall the well-known commutator estimate of Kato and Ponce: ). If s ≥ 0 and 1 < p < ∞, then We will also use the following result as main tool for proving the existence results and blow-up criterion: Theorem 2.2 ([AOB20]). Let Q be a linear differential operator of first order where the coefficients are smooth and bounded. Then for f ∈ H 2 (T, R) we have Moreover, if f ∈ H 2+s (T, R), and P is a pseudodifferential operator of order s, then , it is fundamental to show the cancellation property provided by Theorem 2.2 for such noises. This can be done under some mild Sobolev regularity assumption on the coefficients ξ k (respectively a k , b k ). In particular, one has to precisely compute the constants C k hidden on the right hand-side of (2.5) and (2.6), whose sum a priori does not have to converge. We refer the reader to Lemma A.5 in [AOHR21] for a detailed calculation to deal with this extension.
Next, we briefly recall some aspects of the theory of stochastic analysis. Fix a stochastic basis S = (Ξ, F, {F t } t≥0 , P, {W k } k∈N ), that is, a filtered probability space together with a sequence {W k } k∈N of scalar independent Brownian motions relative to the filtration {F t } t≥0 satisfying the usual conditions. Given a stochastic process X ∈ L 2 (Ξ; L 2 ([0, ∞); L 2 (T, R))), the Burkholder-Davis-Gundy inequality is given by for any p ≥ 1 and C p an absolute constant depending on p.
We also state the celebrated Itô-Wentzell formula, which we use throughout this work.
Theorem 2.5 ([Kun81], Theorem 1.2). For 0 ≤ t < τ , let u(t, ·) be C 3 almost surely, and u(·, x) be a continuous semimartingale satisfying the SPDE is also a family of continuous semimartingales that are C 2 in space for 0 ≤ t < τ . Also, let X t be a continuous semimartingale. Then, we have the following Let us also introduce three different notions of solutions: is a stopping time such that u t∧τ is adapted to (F t ) t and we have: • a.s. u has paths of class C([0, τ ]; H s (T)).
When the stopping time is clear from the context, we simply write that u is a solution.
, satisfying the following conditions: • P(τ max > 0) = 1, where τ max = lim n→∞ τ n for an increasing sequence of stopping times is a local solution in the sense of Definition 2.1, for all n ∈ N.
A maximal solution is said to be global if τ max = ∞, a.s.
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 (domain, 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 b means there exists C such that a ≤ 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.

Shocks in Burgers' equation with stochastic transport
Recall that we are dealing with a stochastic Burgers' equation of the form is an orthonormal basis of some separable Hilbert space H, and • means that the integration is carried out in the Stratonovich sense. In this section, we study the problem of whether shocks can form in the inviscid Burgers' equation with stochastic transport. By using a characteristic argument, we prove that for some classes {ξ k (x)} k∈N , the transport noise cannot prevent shock formation. We also consider a weak formulation of the problem and prove that the shocks satisfy a stochastic version of the Rankine-Hugoniot condition.
3.1. Inviscid Burgers' equation with stochastic transport. The inviscid Burgers' equation with stochastic transport is given by which in integral form is interpreted as for all x ∈ T or R. Also, we will assume throughout this paper that the initial condition is positive, that is, Consider a process X t that satisfies the Stratonovich SDE We call this process the characteristic of (3.1), analogous to the characteristic lines in the deterministic Burgers' equation. We assume the following conditions on {ξ k (·)} k∈N .
Assumption A1. ξ k is smooth for all k ∈ N and together with the Stratonovich-to-Itô correction term ϕ( , satisfy the following: (3.5) • Linear growth condition for real constants C 0 , C 1 , C 2 , . . . and D 0 , Provided u(t, ·) is sufficiently smooth and bounded (hence satisfying Lipschitz continuity and linear growth) until some stopping time τ , and {ξ k (·)} k∈N satisfies the conditions in Assumption A1, the characteristic equation (3.4) is locally well-posed. One feature of the multiplicative noise in (3.1) is that u is transported along the characteristics, that is, where Φ t is the stochastic flow of the SDE (3.4), (Φ t ) * represents the pushforward by Φ t , and (τ max , X t ) is the maximal solution of (3.4). This is an easy corollary of the Itô-Wentzell formula (2.9).
Remark 3.1. Notice that due to our local well-posedness result (Theorem 1.3) and the maximum principle (Proposition 4.14), one has u t ∈ C 3 ∩ L ∞ for t < τ max provided u 0 is smooth enough and bounded. For instance, u 0 ∈ H 4 ∩ L ∞ is sufficient.
Proof of Corollary 3.0.1. Note that under the given assumptions, σ 0 (t, Using the Itô-Wentzell formula (2.9) for the stochastic field u(t, x) satisfying (3.2), and the semimartingale X t , we obtain Now, we have I 1 = I 2 almost surely so indeed, u(t, X t ) = u(0, X 0 ) almost surely for 0 < t < τ max .
3.2. Results on shock formation. In order to investigate the crossing of characteristics in the stochastic Burgers' equation (3.1) with transport noise, we define the first crossing time τ as where X a t , X b t are two characteristics that solve the SDE (3.3) with initial conditions X a 0 = a and X b 0 = b. This gives us the first time when two characteristics intersect. In the following, we will show that in the special case ξ 1 (x) = αx+β (where we only consider one noise term and the other terms ξ k are identically zero for k = 2, 3, . . .), the first crossing time is equivalent to the first hitting time of the integrated geometric Brownian motion. We note that in this case, equation (3.4) is explicitly solvable, where the general solution is given by Proposition 3.2. The first crossing time of the inviscid stochastic Burgers' equation (3.1) with ξ 1 (x) = αx + β for constants α, β ∈ R and ξ k (·) ≡ 0 for k = 2, 3, . . . is equivalent to the first hitting time for the integrated geometric Brownian motion I t := t 0 e −αWs ds.
Proof. Consider two arbitrary characteristics X a t and X b t with X a 0 = a and X b 0 = b. From (3.9), one can check that X a t = X b t if and only if Now, since the left-hand side is continuous, strictly increasing with I 0 = 0, and independent of a and b, we have is the steepest negative slope of u 0 . Hence, the first crossing time is equivalent to the first hitting time of the process I t .
Remark 3.3. Note that the constant β does not affect the first crossing time, hence we can set β = 0 without loss of generality. Also in the following, we simply write ξ(·) 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 ξ(x) = αx cannot prevent shocks from forming almost surely in the stochastic Burgers' equation (3.1).
Proof. To prove this, it is enough to show that lim t→∞ t 0 e αWs ds = ∞ a.s.
where we have assumed α > 0, without loss of generality, and W • : R ≥0 × Ξ → R is the standard Wiener process on the Wiener space (Ξ, F, P), adapted to the natural filtration F t . This implies that τ < ∞ a.s. by Proposition 3.2.
Since the integrand is strictly positive, this implies lim t→∞ e αWt(ω) = 0, and hence W t (ω) → −∞. On the other hand, for ω ∈ Ξ such that W t (ω) → −∞, it is easy to see that ω ∈ A. This implies that under the identification Ξ ∼ = C([0, ∞); R), the set A is equivalent to the set of Wiener processes W t with W t → −∞, which is open in C([0, ∞); R) endowed with the norm · ∞ and therefore measurable. In particular, for ω ∈ A, we have lim sup t→∞ W t (ω) = −∞, but since lim sup t→∞ W t = +∞, a.s., this implies P(A) = 0.
In the following, we show that for a broader class of {ξ k (·)} k∈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 ∂ x u evolves along the characteristics X t , which resembles the argument given in [CH18] for the stochastic Camassa-Holm equation.
Proof. Taking the spatial derivative of (3.2), and evaluating the stochastic field ∂ x u(t, x) along the semimartingale X t by the Ito-Wentzell formula (2.9) (again, this is valid due to the local well-posedness result, Theorem 1.3), the process Y t := ∂ x u(t, X t ) together with X t satisfy the following coupled Stratonovich SDEs (3.12) In Itô form, this reads (3.14) Taking the expectation of (3.14) on both sides, we obtain Now, assume that there exists a constant C ∈ R such that for all x ∈ R. If Y 0 = −σ < 0, we have Y t < 0 for all t > 0, since Y = 0 is a fixed line in the phase space (X, Y ) and therefore cannot be crossed. Hence from (3.16), we have Solving this differential inequality, we get The right-hand side tends to −∞ in finite time provided −σ < C/2.
for all x ∈ R, then there exists t * < ∞ such that lim t→t * E[u x (t, X t )] = −∞.
One can check that E[Y t ] < ∞ for all t > 0, which implies Y t < ∞ almost surely.
Remark 3.5. Blow-up in expectation does not imply pathwise blow-up. It is merely a necessary condition, which suggests that the law of ∂ x u becomes increasingly fat-tailed with time, making it more likely for it to take extreme values. Nonetheless, it is a good indication of blow-up occurring with some probability.
Example 3.6. Consider the set {ξ k (x)} k∈N = 1 k 2 sin(kx), 1 k 2 cos(kx) k∈N , which forms an orthogonal basis for L 2 (T). Then, one can easily check that for all x ∈ T, so blow-up occurs in expectation for any initial profile with negative slope, but no new shocks can form from positive slopes.

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 ξ(x) = αx), 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 2.3.
Suppose that the profile u is differentiable everywhere except for a discontinuity along the curve γ = {(t, s(t)) ∈ [0, ∞) × M }, where M = T or 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 (Stochastic Rankine-Hugoniot condition). The curve of discontinuity s(t) of the stochastic Burgers' equation in weak form (2.10) satisfies the following SDE where u ± (t, s(t)) := lim x→s(t) ± u(t, x) are the left and right limits of u.
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 non-smooth boundaries is a tricky issue, but fortunately, there have been several works that extend this result to non-smooth but rectifiable boundaries in [Sha57], and to non-rectifiable boundaries in [HN92,Har93,Har99,LY06].

Lemma 3.8 (Green's theorem for non-smooth boundaries).
Let Ω be a bounded domain in the (x, y)-plane such that its boundary ∂Ω is a Jordan curve and let u, v be sufficiently regular functions in Ω (see remark 3.9 below). Then where the contour integral on the right-hand side can be understood as a limit of a standard contour integral along a smooth approximation of the boundary. Here, the integral is taken in the anti-clockwise direction of the contour.
Remark 3.9. For the above to hold, there must be a pay-off between the regularity of ∂Ω and the functions u, v (i.e. the less regular the boundary, the more regular the integrand).
Proof of Theorem 3.7. We provide a proof in the case M = T with only one noise term.
Here Q represents a first order differential operator where the coefficients a(x), b(x) are smooth and bounded. We state the main result of this section: Theorem 4.1. Let u 0 ∈ H s (T) for some s > 2 fixed. Then there exists a pathwise unique H s -maximal solution (τ max , u) of the 1D Burgers' equation (4.1) in the sense of Definition 2.2 with initial datum u 0 . Moreover, either τ max = ∞ or lim sup t→τmax |u(t)| H s = ∞, a.s.
We will provide a sketch of the proof, which follows closely the approach developed in [AOB20,CFH19]. For clarity of exposition, let us divide the argument into several steps.  Proof. For this, we refer the reader to [AOB20,CFH19]. It suffices to defineū = u 1 −u 2 , and perform standard estimates for the evolution of the L 2 norm ofū. • Step 2: Existence and uniqueness of truncated maximal solutions. Following the techniques in [CFH19] for the Euler equation with Lie transport noise, for r > 0 we consider the truncated stochastic Burgers' equation where θ r : [0, ∞) → [0, 1] is a smooth function such that θ r (x) = 1, |x| ≤ r, 0, |x| ≥ 2r.
As we explain in the following lemma, local solutions of (4.1) can be constructed by restricting global solutions of (4.2) to a certain stopping time.
Lemma 4.3. Fix r > 0 and u 0 ∈ H s (T), s > 2. Let u r : Ξ × [0, ∞) × T → R be an H s -global solution of (4.2) with initial datum u 0 . Consider the stopping time where C is chosen in such a way that the following inequality holds: which can be guaranteed thanks to the Sobolev embedding (2.3). Then (τ r , u r ) is a local solution to the stochastic Burgers' equation (4.1).
Proof of Lemma 4.3. The proof is straightforward by construction. For any t ∈ [0, τ r ], we have that It is very easy to check that once Proposition 4.4 is proven, Theorem 4.1 follows immediately (cf. [CFH19]). Therefore, we focus our efforts on showing Proposition 4.4.

•
Step 3: Global existence of solutions of the hyper-regularised truncated stochastic Burgers' equation. In order to show the global well-posedness of the truncated equation, we consider the following hyper-regularisation of (4.2) where ν > 0 is a positive parameter and s = 2[s] + 1. Notice that we have added dissipation in order to be able to perform our computations rigorously. Equation (4.3) is understood in the mild sense, i.e., as a solution to an integro-differential equation, which we discuss in the next step (see (4.4)).
Proof of Proposition 4.5. The proof is based on a classical fixed point iteration argument which employs Duhamel's principle (see [CFH19]). We omit the subscripts ν and r throughout the proof for simplicity, but it should be kept in mind that our functions depend on those parameters. Given u 0 ∈ H s (T), consider the mild formulation of the hyper-regularised truncated equation (4.3) t > 0 and we have employed the notation A = ν∂ s xx , W θ u = θ r (|∂ x u| L ∞ )u∂ x u, Lu = 1 2 Q 2 u, and Ru = Qu. Define the space W T = L 2 (Ξ; C([0, T ]; H s (T))). One can show that Υ is a contraction on W T by following classical arguments as in [CFH19]. Therefore, by applying Picard's iteration, a local solution can be constructed. To extend it to a global one, it is sufficient to show that for any given T > 0 and initial datum u 0 ∈ H s (T), s > 2, the following bound is available for a finite constant C(T ) < ∞, so that one can patch together each local solution to cover any finite time interval [0, T ]. We will prove estimate (4.5) further below. Furthermore, by standard properties of the semigroup e tA (cf. [Gol85]), one can prove that for positive times T > δ > 0, each term in the mild equation (4.4) enjoys higher regularity, namely, u ∈ L 2 (Ξ; C([δ, T ]; H s+2 (T))). All the computations are omitted and can be carried out easily by mimicking the same ideas as in [CFH19, AOB20]. • Step 4: Limiting and compactness argument. The main objective of this step is to show that the family of solutions {u ν r } ν>0 of the hyper-regularised 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) in a convenient space. The central idea for proving this is to show compactness of the probability laws of this family. Consequently, we demonstrate that these laws are tight in a suitable metric space. Let T > 0 and define the Polish space E by (4.6) Assume that the laws of {u ν r } ν>0 are tight in E. 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 to conclude that there exists a subsequence of {u ν r } ν>0 such that solutions of equation (4.3) converge to solutions of (4.2) in the weak limit in the Polish space E (4.6). A more thorough approach can be found in [CFH19,GHV14]. In the next proposition, we present the main argument to show that the sequence of laws are indeed tight.
Proposition 4.6. Assume that for some α > 0, M ∈ N, there exist constants C 1 (T ) and C 2 (T ) such that uniformly in ν. Then the sequence {u ν r } ν>0 is tight in E. Proof of Proposition 4.6. We employ the following lemma, which can be found in [CFH19], was originally proved in [Sim86], and constitutes a variation of the classical Aubin-Lions Lemma.
Lemma 4.7. Suppose that X, Y, Z are separable Hilbert spaces with continuous dense embedding X → Y → Z such that there exists θ ∈ (0, 1) and M > 0 verifying is a compact embedding.
In Lemma 4.7 we select where β ≥ 2 as specified before and we impose the extra condition β < s so that the embedding of X into Y is compact. We also choose α ∈ (0, 1). Therefore we obtain that L ∞ ([0, T ]; H s (T)) ∩ W α,4 ([0, T ]; H −M (T)) is compactly embedded in E. By hypotheses (4.7)-(4.8) [CFH19], the family of laws of {u ν r } ν>0 is P-a.s. supported on the space and it suffices to prove that this family is tight in E 0 . For A 1 , A 2 , A 3 > 0, define the set which is relatively compact in E 0 . It is enough to show that for every , there exist A 1 , A 2 , A 3 > 0 such that P(u ν r ∈ B c ) ≤ . Fix > 0. Invoking Markov's inequality and taking into account hypothesis (4.7), we have that and this is smaller than /3 if we choose A 1 sufficiently large. Moreover, since |f | H −M |f | H s , which can also be made arbitrarily small. A similar argument applies to the set Hence for 0 < α < 1/2, We are left to prove that hypothesis (4.7) holds, i.e.

E sup
We drop the parameter dependence on u ν r to make the notation simpler. By putting together the previous estimates, we have where the constant in the last inequality depends on r. Integrating the above expression against exp(t) and squaring as in [CFH19], we obtain . By applying supremum and expectation on both sides of the equation above, we have where we remind that t ∈ [0, T ]. We apply Burkholder-Davis-Gundy inequality (2.7) in order to obtain the bound Integrating by parts, we can derive as in the proof of (2.6) in [AOB20] or in the appendix of [AOB20]. The constant in the last inequality depends on the W s+1,∞ -norm of the coefficients. Therefore, (4.14) Hence, combining (4.11)-(4.14), together with application of Grönwall's inequality yields

4.2.
Blow-up criterion. We are now interested in deriving a blow-up criterion for the stochastic Burgers' equation (1.1) with ν = 0. First of all, we note that for the deterministic Burgers' equation there is a well-known blow-up criterion available. In the deterministic case, the following theorem of local existence and uniqueness of strong solutions holds. The above result is classic and numerous proofs are available in the literature (see [KNS08] for a related cutting edge result and the references therein). The blow-up criterion for equation (4.15) is the following.
Theorem 4.9 (Blow-up criterion for deterministic Burgers'). Assume that u 0 ∈ H s (T), s > 3/2, T * > 0, and u : [0, T * ) × T → R is a local solution of (4.15). Then the following statements are equivalent In the rest of this subsection, we focus on proving the following stochastic version of Theorem 4.9.
Theorem 4.10 (Blow-up criterion for stochastic Burgers'). Assume that u 0 ∈ H s (T), s > 2, and u : Ξ × [0, τ max ) × T → R is a maximal solution of (4.1). If τ max < ∞, then Proof of Theorem 4.10. By following the argument in [CFH19], we start by noting that it is clear that if 0 ≤ τ < τ max is a smaller stopping time, necessarily τ 0 |∂ x u(t, ·)| L ∞ dt < ∞ a.s. This is guaranteed by the embedding (4.16) Proving τmax 0 |∂ x u(t, ·)| L ∞ dt = ∞ is more involved. For this, we consider the hyperregularised truncated Burgers' equation (4.3) and define the following stopping times: We claim that τ ∞ = τ s . By again making use of (4.16), it is easy to check τ s ≤ τ ∞ . For simplicity, we omit the subscripts ν and r throughout the proof. Imitating the techniques from the previous subsection, we arrive at By integration by parts and standard estimates (use (2.6) for estimating the last two terms), one gets d|Λ s u| 2 L 2 + 2 Λ s Qu, Λ s u L 2 dW t (1 + |∂ x u| L ∞ ) |Λ s u| 2 L 2 dt.
By also deriving a similar estimate for s = 0 (which follows in a simpler manner by application of the techniques in the previous subsection), we obtain Finally, we treat the stochastic term. Without loss of generality, we assume |u| H s ≥ > 0 (otherwise add a positive constant to this function) and by applying Itô's formula in L 2 [KR81] to the logarithm, we obtain where N t is defined by Notice that we have the bound where M t is the local martingale for any t > 0. Expressing (4.20) in integral form, we have (4.21) By applying Burkholder-Davis-Gundy inequality (see (2.7)), we can control the local martingale term by estimating Here, we have employed again the arguments in the appendix of [AOB20]  for every n, m ∈ N, which from [CFH19] implies τ ∞ ≤ τ s . Therefore τ ∞ = τ s . Recall that we have omitted subscripts throughout this proof but u = u ν r . By application of Fatou's Lemma we also obtain (4.23) in the limit ν → 0, r → ∞, and we can show that τ ∞ = τ s also holds in the limit, concluding our argument. The few steps missing can be checked in [CFH19] We assume Q = ξ(x)∂ x , s > 2.
Remark 4.11. Observe that from our assumption on Q, (4.24) is simply (1.1) with one noise term, but we wish to keep the Q-notation for convenience.
We establish the global well-posedness of strong solutions of (4.24). More concretely, we prove the following theorem. The proof follows the same strategy as the local existence proof for the Burgers' equation without viscosity (4.1) that we provided in Subsection 4.1. The most important part is proving the following a priori estimate. In this estimate, we assume that u is smooth enough, but the rigorous way to do this is to regularise the equation first as we did in Subsection 4.1. Once the a priori estimate (4.25) is established, one can repeat the arguments in Subsection 4.1 to obtain Theorem 4.12. However, since this is repetitive and tedious, we do not explicitly carry out these arguments here. From now on, we focus on proving Proposition 4.13.
Proof. We start by computing the evolution of the L 2 -norm of the solution u. First note that the viscosity term has the good sign since ν ∂ xx u, u L 2 = −ν |∂ x u| 2 L 2 . By applying the same techniques as in Subsection 4.1 (take into account estimate (2.5)), we obtain d|u| 2 L 2 + 2 Qu, u L 2 dW t + 2ν |∂ x u| 2 L 2 dt |u| 2 L 2 dt, and we have the bounds We compute the evolution of theḢ s -norm of u (4.28) Integrating by parts the first term of the right-hand side above and observing that u∂ x u = (1/2)∂ x (u 2 ), we obtain where we have employed Kato-Ponce (see Lemma 2.4) in the second inequality. The second term on the right-hand side after the equality of equation (4.28) can be integrated as As usual, applying inequality (2.6) we also have the estimate (4.31) Putting together (4.29)-(4.31), one derives d|Λ s u| 2 L 2 + 2 Λ s (Qu), Λ s u L 2 dW t + ν Λ s+1 u 2 L 2 dt 1 ν (1 + |u| 2 L ∞ ) |Λ s u| 2 L 2 dt.
By applying expectation in the above equation and taking into account that the expectation of the Itô integral vanishes due to the martingale property, one obtains and Now we claim that the following maximum principle holds which we show in Lemma 4.14. Once (4.34) is proven, we can infer from (4.32) and (4.33) together with Grönwall's inequality that there exist constants This concludes the proof, so we are only left to show the maximum principle (4.34).
Hence (4.37) is satisfied. Since ψ t is a diffeomorphism, we have |u t | L ∞ = |w t | L ∞ so the maximum principle also follows for u.
With this, we conclude the proof of Theorem 4.12.

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 well-posed in H s (T, R), for s > 2, and furthermore, established a blow-up criterion which extends the deterministic one to the stochastic case. We also proved that if the noise is of the form ξ(x)∂ x u • dW t where ξ(x) = αx + β, then shocks form almost surely from a negative slope. Moreover, for a more general type of noise, we showed that blow-up occurs in expectation, which follows from the previously mentioned stochastic blow-up 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 well-posed in H s for s > 2.
Let us conclude by proposing some future research directions and open problems that have emerged during the course of this work: • 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 well-posedness result can be extended for the viscous Burgers' equation with the Laplacian replaced by a fractional Laplacian (−∆) α , α ∈ (0, 1). The main difficulty here is that in the stochastic case, the proof of the maximum principle (Proposition (4.14)) 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 α: for α ∈ [1/2, 1], the solution is global in time, and for α ∈ [0, 1/2), the solution develops singularities in finite time [KNS08,Kis10]. Interestingly, when an Itô noise of type βu dW t is added, it is shown in [RZZ14] that the probability of solutions blowing up for small initial conditions tends to zero when β > 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 one-dimensional equations with non-local transport velocity [CCF05,DG90,DG96]. For instance, the so called CCF model [CCF05] 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.