Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure
We are interested in viscous scalar conservation laws with a white-in-time but spatially correlated stochastic forcing. The equation is assumed to be one-dimensional and periodic in the space variable, and its flux function to be locally Lipschitz continuous and have at most polynomial growth. Neither the flux nor the noise need to be non-degenerate. In a first part, we show the existence and uniqueness of a global solution in a strong sense. In a second part, we establish the existence and uniqueness of an invariant measure for this strong solution.
Key words and phrases:Stochastic conservation laws, Invariant measure
2010 Mathematics Subject Classification:35A01,35R60,60H15
1.1. Stochastic viscous scalar conservation law
We are interested in the existence, uniqueness, regularity and large time behaviour of solutions of the following viscous scalar conservation law with additive and time-independent stochastic forcing
where , , is a family of independent Brownian motions. Here, denotes the one-dimensional torus , meaning that the sought solution is periodic in space. The flux function is assumed to satisfy the following set of conditions.
Assumption 1 (on the flux function).
The function is on , its first derivative has at most polynomial growth:
and its second derivative is locally Lipschitz continuous on .
The parameter is the viscosity coefficient. In order to present our assumptions on the family of functions , , which describe the spatial correlation of the stochastic forcing of (1), we first introduce some notation. For any , we denote by the subset of functions such that
The norm induced on is denoted by . For any integer , we denote by the intersection of the Sobolev space with . Equipped with the norm
and the associated scalar product , it is a separable Hilbert space. On the one-dimensional torus, the Poincaré inequality implies that and . Actually, the following stronger inequality holds: if , then and for all ,
The spaces generalise to the class of fractional Sobolev spaces , where , which will be defined in Section 2.1. We may now state:
Assumption 2 (on the noise functions).
For all , and
Let be a probability space, equipped with a normal filtration in the sense of [DZ92, Section 3.3], on which is a family of independent Brownian motions. Under Assumption 2, the series converges in , for any , towards an -valued Wiener process with respect to the filtration , defined in the sense of [DZ92, Section 4.2], with the trace class covariance operator given by
Thus, almost surely, is continuous in and for all , the process is a real-valued Wiener process with variance
1.2. Main results and previous works
First, we are interested in the well-posedness in the strong sense of Equation (1). In particular, we look for solutions that admit at least a second spatial derivative in order to give a classical meaning to the viscous term, in the sense of the following definition:
Definition 1 (Strong solution to (1)).
In the above definition, the first condition ensures that the time integral in Equation (7) is a well-defined Bochner integral in . For a careful introduction of the general concepts of random variables and stochastic processes in Hilbert spaces, the reader is referred to the third and fourth chapters of the reference book [DZ92].
Our first result is the following:
Theorem 1 (Well-posedness).
Let . Under Assumptions 1 and 2, there exists a unique strong solution to Equation (1) with initial condition . Moreover, the solution depends continuously on initial data in the following sense: if is a sequence of satisfying
then, denoting by the family of associated solutions, for any , we have almost surely
Similar results have already been established: the case where the flux is strictly convex is treated in [Bor12b, Appendix A], and the case where is globally Lipschitz continuous is treated in [Hof13]. Furthermore, the case of mild solutions (in spaces) has been looked at in [GR00]. Here, no global Lipschitz continuity assumption nor restrictions on the convexity of the flux function are made. We can also point out that the well-posedness of stochastically forced conservations laws in the inviscid case (i.e. when ) has been under a great deal of investigation in the recent years. In this "hyperbolic" framework, the appearance of shocks prevents the solutions to be smooth enough to be considered in a strong sense as in our present work. Therefore, the study of entropic solutions [FN08] or kinetic solutions [DV14] to the SPDE have been the two main approaches, both of which rely on a vanishing viscosity argument: the entropic or kinetic solution is sought as the limit of its viscous approximation as the viscosity coefficient tends to .
Let denote the set of continuous and bounded functions from to . As a consequence of Theorem 1, we can define a family of functionals on by writing
where the notation indicates that the random variable is the solution to (1) at time starting from the initial condition .
The uniqueness of a strong solution and the fact that, for all , the processes and have the same distribution, ensure that is a semigroup, and therefore that is a Markov process. The Feller property is a straightforward consequence of the result of continuous dependence on initial conditions given in Theorem 1, whereas it is a classical result that the strong Markov property of follows from the Feller property of (see for instance the proof of [Bre68, Theorem 16.21]). ∎
Let denote the Borel -algebra of the metric space , and refer to the set of Borel probability measures on . The Markov property allows us to extend the notion of strong solution to (1) by considering not only a deterministic initial condition but any -measurable random variable on . In this perspective, we define the dual semigroup of by
In particular, is the law of when is distributed according to .
Definition 2 (Invariant measure).
We say that a probability measure is an invariant measure for the semigroup (or equivalently for the process ) if and only if
Theorem 2 (Existence, uniqueness and estimates on the invariant measure).
A few similar results exist in the literature. Da Prato, Debussche and Temam [DDT94] have studied the viscous Burgers equation (which corresponds to the flux function ) perturbed by an additive space-time white noise whereas Da Prato and Gatarek [DG95] studied the same equation but with a multiplicative white noise. Both showed the well-posedness of the equation as well as the existence of an invariant measure. These results are moreover put in a much detailed context in the two reference books [DZ92, DZ96]. Boritchev [Bor12, Bor12b, Bor13] showed the existence and uniqueness of an invariant measure for the viscous generalised Burgers equation (which corresponds to the case of strictly convex flux function) perturbed by a white-in-time and spatially correlated noise. E, Khanin, Mazel and Sinai [EKMS00] showed the existence and uniqueness of an invariant measure for the inviscid Burgers equation with a white-in-time and spatially correlated noise. Debussche and Vovelle [DV15] generalised this last result by extending it to non-degenerate flux functions (roughly speaking, there is no non-negligible subset of on which is linear). Besides, the fact that these results from [EKMS00, DV15] also hold when makes them quite powerful: it shows indeed that the presence of a viscous term is not a necessary condition for the solution to be stationary.
The stochastic Burgers equation is mainly studied as a one-dimensional model for turbulence. By showing a stable behaviour at large times, this model manages, to some extent, to fit the predicitions of Kolmogorov’s "K41" theory about the universal properties of a turbulent flow [Kol41a, Kol41b]. Whether it is modelled by the Burgers equation or a by more general process such as Equation (1), turbulence is then described through the statistics of some particular small-scale quantities in the stationary state [E00, E01]. Sharp estimates were given by Boritchev for these small-scale quantities [Bor12b], which were furthermore shown to be independent of the viscosity coefficient. One of the purposes of this paper is to lay the groundwork for the numerical analysis of Equation (1). In a companion paper [BMR19], we introduce a finite-volume approximation of (1) which allows to approximate the invariant measure . Generating random variables with distribution shall eventually lead us to compute said small-scale quantities and analyse the development of turbulence in the model established by Equation (1).
1.3. Outline of the article
2. Well-posedness and regularity
This section is dedicated to the proof of Theorem 1. This proof is decomposed as follows. In Subsection 2.1, we introduce a weaker formulation of Equation (1), the so-called mild formulation. In Subsection 2.2, we show that Equation (1) is well-posed locally in time both in the mild and in the strong sense. In Subsection 2.3, we give higher bounds for the Lebesgue and Sobolev norms of this local solution. Eventually, these estimates allow us to extend the local solution to a global-in-time solution, and thus to prove Theorem 1 in Subsection 2.4.
2.1. Mild formulation of (1)
In this subsection, we collect preliminary results which shall enable us to provide a mild formulation of Equation (1), for which we prove the existence and uniqueness of a solution on a small interval. The proofs of several results are postponed to Subsection 2.5.
2.1.1. Fractional Sobolev spaces
For all , let us define , and , . The family is a complete orthogonal basis of such that, for all , is on and . With respect to this basis, we define the fractional Sobolev space , for any , as the space of functions such that
We take from [Bor12b, Appendice A] the following proposition and adapt it to our case of a flux function satisfying Assumption 1:
Under Assumption 1, for any , the mapping
is bounded on bounded subsets of . Moreover, when or , it is Lipschitz continuous on bounded subsets of .
By virtue of Proposition 1, for all , we denote by and two finite constants such that:
for all such that , ;
for all such that , .
2.1.2. Heat kernel
Let us denote by the semigroup generated by the operator :
Some of its properties are gathered in the following proposition.
Proposition 2 (Properties of the heat kernel).
The semigroup satisfies the following properties.
For any , for any , for any , and ; besides, the mapping is continuous on .
For all , there exists a constant such that
For any , and , the process belongs to .
2.1.3. Stochastic convolution and mild formulation of (1)
Let be a normal filtration on the probability space and be a -Wiener process in with respect to this filtration. Given that the orthonormal basis of the space satisfies , the family is an orthonormal basis of . We set
so that by (6), is a real-valued Brownian motion with variance . Next, we write
Under Assumption 2, for all , the series
converges in , and its sum defines an -adapted, -valued process almost surely continuous.
In the sequel, we let be a -stopping time, almost surely finite. We shall say that a process is -adapted if for all , the random variable is -measurable.
Definition 3 (Local mild solution).
We now clarify the relationship between the notions of mild and strong solutions.
Proposition 4 (Mild and strong solutions).
2.1.4. Existence and uniqueness of a mild solution on a small interval
For any integer , let us define
Notice that , almost surely.
In the spirit of [DDT94, Bor12b], we obtain the existence and uniqueness of a mild solution to (10) on the "small" interval by a fixed-point argument.
Lemma 1 (Local existence and uniqueness).
Let us introduce the random set
and notice that any satisfies Equation (11) if and only if .
We first write, for some and for any ,
furthermore, thanks to Proposition 1, if , then is bounded in uniformly in time, i.e. for all , . Thus,
By definition of , it follows that whenever .
We now take . Then, for any ,
where we have used the same arguments as above. Using now the Lipschitz continuity result in Proposition 1 and the definition of , we get for all ,
meaning that is a contraction mapping on , which is complete. Then, by the Banach fixed-point theorem, admits a unique fixed point in . To show that this solution to Equation (11) is unique among all the -valued continuous processes, let us first notice that our choice of implies
Assume that there is another solution of (11) not belonging almost surely to . Then we have with positive probability
This means that the double inequality holds on some non-negligible event. On this event, the fixed-point argument also holds in the set
which is formally a subset of . Thus, by uniqueness of the fixed point, we have and in particular , which is absurd. As a consequence, is the only -valued process with continuous trajectories satisfying Equation (11) on .
Finally, let and define the sequence of processes , by . It is clear from the definition of the operator and from Proposition 3 that each process is -adapted. On the other hand, the Banach fixed-point theorem asserts that almost surely, the sequence converges to in . As a consequence, for any , the sequence of -measurable random variables converges almost surely to , which makes this limit also -measurable. Thus, the process is -adapted. ∎
2.2. Construction of a maximal solution to (1)
Lemma 2 (Existence and uniqueness result of a maximal solution to (1)).
The random time is called the explosion time and the process is called the maximal solution to (1).