Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure
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Nonlinear Differential Equations and Applications NoDEA
Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure Sofiane Martel
and Julien Reygner
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. Mathematics Subject Classification. 35A01, 35R60, 60H15. Keywords. Stochastic conservation laws, Invariant measure.
1. Introduction 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 gk dW k (t), x ∈ T, t ≥ 0, (1) du = −∂x A(u)dt + ν∂xx udt + k≥1 k
where (W (t))t≥0 , k ≥ 1, is a family of independent Brownian motions. Here, T denotes the one-dimensional torus R/Z, meaning that the sought solution is periodic in space. The flux function A is assumed to satisfy the following set of conditions. This work is partially supported by the French National Research Agency (ANR) under the programs ANR-17-CE40-0030—EFI—Entropy, flows, inequalities and QuAMProcs.
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S. Martel and J. Reygner
NoDEA
Assumption 1. (on the flux function) The function A : R → R is C 2 on R, its first derivative has at most polynomial growth: ∃C1 > 0,
∃pA ∈ N∗ ,
|A (v)| ≤ C1 (1 + |v|pA ) ,
∀v ∈ R,
(2)
and its second derivative A is locally Lipschitz continuous on R. The parameter ν > 0 is the viscosity coefficient. In order to present our assumptions on the family of functions gk : T → R, k ≥ 1, which describe the spatial correlation of the stochastic forcing of (1), we first introduce some notation. For any p ∈ [1, +∞], we denote by Lp0 (T) the subset of functions v ∈ Lp (T) such that vdx = 0. T
The Lp norm induced on Lp0 (T) is denoted by · Lp0 (T) . For any integer m ≥ 0, we denote by H0m (T) the intersection of the Sobolev space H m (T) with L20 (T). Equipped with the norm 1/2 m 2 m v H0 (T) := |∂x v| dx , T
and the associated scalar product ·, · H0m (T) , it is a separable Hilbert space. On the one-dimensional torus, the Poincar´e inequality implies that H0m+1 (T) ⊂ H0m (T) and · H0m (T) ≤ · H m+1 (T) . Actually, the following stronger inequality 0 holds: if v ∈ H01 (T), then v ∈ L∞ 0 (T) and for all p ∈ [1, +∞), ≤ v H01 (T) . v Lp0 (T) ≤ v L∞ 0 (T)
(3)
H0m (T), m
≥ 0, generalise to the class of fractional Sobolev The spaces spaces H0s (T), where s ∈ [0, +∞), which will be defined in Sect. 2.1. We may now state: Assumption 2. (on the noise functions) For all k ≥ 1, gk ∈ H02 (T) and
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