Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure

Abstract

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.

Finite-Volume approximation of the invariant measure of a viscous stochastic scalar conservation law

“…The viscosity coefficient ν is assumed to be positive. In the companion paper [27], we have shown the well-posedness in a strong sense of Equation (1), as well as the existence and uniqueness of an invariant measure for its solution. These results are recalled in Proposition 1.2 below.…”

“…In this work, we aim to provide a numerical scheme, based on the finite-volume method, that allows to approximate this invariant measure. In this perspective, we place ourselves in the setting of [27] and recall our main notations and assumptions. Notations.…”

“…The assumptions (4) and (5) will be needed in the arguments contained in this paper while the local Lipschitz continuity of A is only necessary for Proposition 1.2, whose proof is done in [27].…”

“…The proofs for these two statements are given separately in Section 2. The structure of the proof is the same as for [27,Theorem 2] where we derived the existence and uniqueness of an invariant measure for the solution of (1) from two important properties: respectively the dissipativity of the solution and an L 1 -contraction property. In Lemma 2.4 below, we show that both of these properties are preserved at the discrete level.…”

“…In particular, the case of the Burgers' equation is covered. However, Equation (1) satisfies an L 1 -contraction property [27,Proposition 5] which may be viewed as a one-sided Lipschitz condition in the Banach space L 1 0 (T). 1.6.…”