Heterogeneous stochastic scalar conservation laws with non-homogeneous Dirichlet boundary conditions

Abstract

We introduce a notion of stochastic entropy solutions for heterogeneous scalar conservation laws with multiplicative noise on a bounded domain with non-homogeneous boundary condition. Using the concept of measure-valued solutions and Kruzhkov’s semi-entropy formulations, we show the existence and uniqueness of stochastic entropy solutions. Moreover, we establish an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux function and the random source function.

Renormalized entropy solutions of stochastic scalar conservation laws with boundary condition

“…In paper [21], the authors gave the following definition of stochastic entropy solution of (1.1)-(1.3). For convenience, for any function u of N 2 w (0, T ; L 2 (D)), any real number k and any regular function η ∈ E + , denote dP-a.s. in Ω by μ η,k , the distribution in D defined by…”

“…(H 2 ): h : R → R is a Lipschitz-continuous function with h(0 [21] obtained the existence and uniqueness of stochastic entropy solutions of (1.1)-(1.3) in sense of Definition 2.1.…”

“…where p ≥ 2 and C does not depend on ε. By using Young measure theory, we prove u ε converges an "entropy process" denoted by u in [21].…”

“…The initial data u 0 : D ⊂ R N → R will be specified later and the boundary data a : Σ → R is supposed to be measurable. Problem (1.1)- (1.3) was studied recently by Kobayasi-Noboriguchi [16] and Lv-Wu [21] via kinetic solution approach and Kruzhkov's semi-entropy method, respectively. By introducing a notion of kinetic formulations in which the kinetic defect measures on the boundary of domain are turncated, Kobayasi-Noboriguchi [16] obtained the well-posedness of (1.1)- (1.3).…”

“…By introducing a notion of kinetic formulations in which the kinetic defect measures on the boundary of domain are turncated, Kobayasi-Noboriguchi [16] obtained the well-posedness of (1.1)- (1.3). Motivated by the deterministic case [1,24], Lv-Wu [21] (see also [22]) introduced a notion of stochastic entropy solutions of (1.1)- (1.3) and obtained the existence and uniqueness of stochastic entropy solutions by utilising vanishing viscosity method and Kruzhkov's technique of doubling variables.…”