HOMOGENIZATION OF A STOCHASTIC VISCOUS TRANSPORT EQUATION

. In the present paper we prove an homogenisation result for a locally perturbed transport stochastic equation. The model is similar to the stochastic Burgers’ equation and it is inspired by the LWR model. Therefore, the interest in studying this equation comes from it’s application for traﬃc ﬂow modelling. In the ﬁrst part of paper we study the inhomogeneous equation. More precisely we give an existence and uniqueness result for the solution. The technical diﬃculties of this part come from the presence of the function ϕ under assumptions coherent for the model, which is giving the inhomogeneity with respect to the space variable, not present in the classical results. The second part of the paper is the homogenisation result in space.


Introduction.
1.1. General model. The goal of this paper is to present a homogenization result for a stochastic Burgers' type equation perturbed by white noise. We consider the following perturbed transport equation with a viscosity term and with a random force which is a white noise in space and time, du δ (t, x) = µ ∂ 2 u δ (t, x) ∂x 2 + ϕ x δ ∂ ∂x g(u δ (t, x)) dt + dW, x ∈ [0, 1], t > 0. (1.1) Here u represents a certain density in space and time and g : R → [0, +∞) denotes the flux. Finally, µ is a postive constant, the function ϕ : R → [0, +∞) is a local perturbation (non-periodic) and W is a Q-Wiener process in L 2 (0, 1). More precisely, we consider an operator Q which is a trace class non-negative operator on L 2 (0, 1) and we define where {e k } is an orthonormal basis of L 2 (0, 1), {λ k } is the family of eigenvalues of the operator Q (i.e. Qe k = λ k e k , k ∈ N), and {β k } k is a sequence of mutually independent real Brownian motions in a probability space (Ω, F, P), adapted to a filtration {F t } t≥0 .
The model is similar from the stochastic Burgers' equation which is used to study turbulent flows in the presence of random forces. Note that, even if the deterministic model is not realistic because it does not display any chaos, the situation is different when the force is a random one, as in our case.
Several authors studied the stochastic Burgers equation in one dimension, driven by additive or multiplicative noise as model describing turbulences (see [4], [2], [7], [8]). A result concerning the homogenization was proved for the integro-differential deterministic Burgers equation in [1].
Equation (1.1) is also inspired to the LWR model (see for instance [9,10]) for traffic flow. In fact, the LWR model is a transport equation on the density of vehicles ρ: where v is called the average speed. Notice that equation (1.1) contains an extra viscosity term µρ xx . In the deterministic case, the LWR model with a viscosity term is not used since under certain conditions the vehicles can have negative velocities (see [6]). However, the stochastic case is different and we can consider µ very small.
To the best of our knowledge, this is the first paper which treats the stochastic equation (1.1) and its homogenization.
The problem (1.1) shall be studied with the following boundary conditions, 2) and the initial condition, where the self-adjoint operator A on L p (0, 1) is defined by 5) and the operator B δ on L p (0, 1), for p ≥ 3, is defined, for all δ > 0, by In fact, since A is a self-adjoint negative operator in L 2 (0, 1), we have where {α k } is the sequence of positive eigenvalues.
(A3) (Asymptotic behaviour). There exits a constant ϕ ∈ R such that (A4) (Regularity of the noise). The eigenvalues of the operators Q and A satisfy for some γ ∈ (0, 1). Figure 1 we give a schematic representation of a function ϕ which satisfies assumptions (A1), (A2) and (A3). Function ϕ can oscillate near 0 but there must be a constant ϕ such that ϕ = lim x→+∞ ϕ(x). In this framework we shall prove existence and uniqueness of a solution to the equation describing the inhomogeneous model (1.4) for each δ fixed, followed by the convergence of this solution, for δ → 0, to the one of the following homogeneous model.

Remark 1.1 (Local perturbation). In
where Bu = ϕ ∂ ∂x u. The organisation of the article is the following: Section 2 contains the definition of a mild solution and some preliminary results. Section 3 contains the proof of the result of existence and uniqueness of the solution while Section 4 contains the proof of the homogenization result.  is given by the stochastic convolution, Keeping in mind that, for a constant C > 0, we have for the eigenvalues of A that and also considering the properties of the semi-group generated by the Laplace operator (see Lemma 2.2 below), we can apply [3,Theorem 5.24] and get that p ≥ 3. Furthermore, by using the previous assumption (A4) we get by [3,Theorem 5.21] that W A has a version which is an α−Hölder continuous function for all α ∈ (0, γ/2) with respect to the variables t and x.
We can now give the definition of the solution.

Preliminary results.
The idea which shall be used for both existence and homogenization results consists in rewriting the stochastic equations as random differential equations. In fact, for each fixed ω ∈ Ω, we shall rewrite equation (1.4) in the form The mild solution corresponding to (2.3) will be then given by Using this form of the equation we shall prove that v δ (and implicitly u δ ) exists and is unique. Like before, for the homogeneous problem (1.
The idea used to prove homogenization is then to obtain the convergence result of v δ to v, which will imply the same for u δ and u.
In order to see the properties of the stochastic convolution, we recall the following result concerning contraction semi-group on L p (0, 1) (see for instance [11, Lemma 3, Part I]).

Lemma 2.2.
For any s 1 ≤ s 2 ∈ R, and r ≥ 1, e At : W s1,r (0, 1) → W s2,r (0, 1), for all t > 0 and we have that there exists a constant C 1 depending only on s 1 , s 2 and r such that Finally, we present a useful (deterministic) result that we will use several times in the rest of this paper. Lemma 2.3. Assume ϕ continuous and derivable such that ϕ ∈ L p (0, 1) and that g : R → R is bounded and Lipschitz continuous. Then, there exists two constants Proof of Lemma 2.3. Using the Sobolev embedding theorem (the reader is referred to the Hitchhiker's guide of Nezza, Palatucci and Valdinoci [5] for results concerning fractional Sobolev spaces), we have that there exists a constant C 2 > 0 such that where we used Lemma 2.2 for the second inequality, with s 1 = −1, s 2 = 1/p and r = p/2. By definition, we have that where we have used for the fourth and fifth line the Hölder inequality with coefficients p/2 and p/(p−2), for the sixth line the fact that |u|    Proof of Lemma 3.4. This proof is done by a fixed point argument in Σ p (m, T * ). Therefore, we want to prove that for all v ∈ Σ p (m, T * ), the transformation Gv = z defined by is a contraction from Σ p (m, T * ) into Σ p (m, T * ). We define, for all v ∈ Σ p (m, T * ), the norm || · || Σp(m,T * ) by |v| L p (0,1) .
Step 1: G is stable in Σ p (m, T * ). Let us first prove that z = Gv is in Σ p (m, T * ). First we have, Let us now treat the term inside the integral, using Lemma 2.3, we obtain that Since e At is a contraction on L p (0, 1), we have that |e tA u 0 | L p (0,1) ≤ |u 0 | L p (0,1) . Injecting this result and (3.2) into (3.1) we obtain We can see that if m > |u 0 | L p (0,1) , given that p ≥ 3, there exists a time T * such that this implies that for all t ∈ [0, T * ], |z(t)| L p (0,1) ≤ m and therefore z ∈ Σ p (m, T * ).
Step 2: G is a contraction on Σ p (m, T * ). Let us consider v 1 , v 2 ∈ Σ p (m, T * ). We Using Lemma 2.3, we get Using the regularity of g, we obtain If we choose T * small enough such that we obtain |z(t)| L p (0,1) < ||v 1 − v 2 || Σp(m,T * ) . By taking the supremum in time on the left hand side of the inequality we obtain the desired result.
) be a sequence of regularised processes such that provided by Lemma 3.4, with g n ∈ C 2 (R) a sequence of regularised functions such that Given ∂t Let us now consider the last term of the previous equation. We notice that it can be separated into two parts, Let us begin by considering (3.7), where we have used for the first line the fact that ϕ and g are bounded, for the second line we simply have used Young inequality ((ab ≤ (1/p)a p + b p p−1 (p − 1)/p) and finally for the third line we have used the Sobolev embedding theorem. Let us now consider (3.8), where we have used for the second line the fact that ϕ and g are bounded, for the third line we have used Hölder inequality with coefficients 2, for the fourth line Young inequality (ab ≤ a 2 /4 + b 2 ), for the fifth line we have used Young inequality (ab ≤ (2/p)a p 2 +b p p−2 (p−2)/p) and for the sixth line the Sobolev embedding theorem. Injecting (3.9) and (3.10) into (3.6), we obtain Thanks to Gronwall's lemma we obtain Passing to the limit as n goes to infinity we obtain the desired result.
Combining Lemma 3.4 and 3.5 with a continuation argument, we obtain the global existence of the mild solution (Lemma 3.3) and it also implies the following result. If ϕ is defined as in (A3), then Theorem 3.1 implies also the existence of a unique mild solution of the equation (1.8), which describes the homogeneous model.   Combining (4.1) and (4.2), we obtain that ds.