Stochastic Entropy Solutions for Stochastic Scalar Balance Laws

Jinlong Wei,Rongrong Tian,Bin Liu,Liang Ding

doi:10.3390/e21121142

Abstract

We are concerned with the initial value problem for a multidimensional balance law with multiplicative stochastic perturbations of Brownian type. Using the stochastic kinetic formulation and the Bhatnagar-Gross-Krook approximation, we prove the uniqueness and existence of stochastic entropy solutions. Furthermore, as applications, we derive the uniqueness and existence of the stochastic entropy solution for stochastic Buckley-Leverett equations and generalized stochastic Burgers type equations.

Highlights

We are interested in the uniqueness and existence of the stochastic entropy solution for the following stochastic scalar balance law: d dρ(t, x ) + divx ( F (ρ))dt + ∑
The uniqueness and existence of stochastic entropy solutions are proved in Sections 4 and 5
In order to prove the uniqueness of the stochastic entropy solution, we need another two lemmas below, the first one follows from DiPerna and Lions [34], and the proof is analogue, we only give the details for the second one

Summary

Introduction

We are interested in the uniqueness and existence of the stochastic entropy solution for the following stochastic scalar balance law:. We will prove the uniqueness and existence of the stochastic entropy solution to (1), (2). We define the stochastic entropy solution by the inequality (9), and the source or motivation for this definition comes from the ε → 0 limit of the following equation d dρε (t, x ) + divx ( F (ρε ))dt + ∑. We should assume the growth rates on the coefficients Bi,j , i.e., Bi,j (t, ρ) is at most linear growth in ρ, and regularity property of A on spatial variables (e.g., Lipschitz continuous) In this case, we will establish the existence for stochastic entropy solutions. The uniqueness and existence of stochastic entropy solutions are proved in Sections 4 and 5.

Preliminaries

Proof of Theorem 1

Proofs of Theorem 2 and Corollary 1

Proof of Theorem 3

Conclusions