Stochastic Entropy Solutions for Stochastic Nonlinear Transport Equations.

Rongrong Tian,Yanbin Tang

doi:10.3390/e20060395

Abstract

This paper considers the existence and uniqueness of stochastic entropy solution for a nonlinear transport equation with a stochastic perturbation. The uniqueness is based on the doubling variable method. For the existence, we develop a new scheme of parabolic approximation motivated by the method of vanishing viscosity given by Feng and Nualart (J. Funct. Anal. 2008, 255, 313–373). Furthermore, we prove the continuous dependence of stochastic strong entropy solutions on the coefficient b and the nonlinear function f.

Highlights

IntroductionWe consider the existence and uniqueness of the solutions to the nonlinear transport equation with a stochastic forcing:
To overcome the first difficulty, we develop a new scheme of parabolic approximation, which sheds some new light on the method of vanishing viscosity
Proof of Theorem 2 (i) We prove the existence of stochastic strong entropy solutions for (1) by the method of vanishing viscosity, that is, we regard (1) as the ε ↓ 0 limit of the viscosity equation dρε (t, x ) + b( x ) · ∇ f (ρε (t, x ))dt = ε∆ρε (t, x )dt + A(ρε (t, x ))dWt, t > 0, x ∈ Rd, ρε (t, x )|t=0 = ρ0ε ( x ), x ∈ Rd, (22)

Summary

Introduction

We consider the existence and uniqueness of the solutions to the nonlinear transport equation with a stochastic forcing:. Let ρ be the unique stochastic strong entropy solution of (1) and ρbe the unique stochastic strong entropy solution of dρ(t, x ) + b( x ) · ∇ x f(ρ(t, x ))dt = A(ρ(t, x ))dWt , t > 0, x ∈ Rd , ρ(t, x )|t=0 = ρ0 ( x ), x ∈ Rd. For every T > 0, there exists a constant C > 0, which depends only on kbkW 1,∞ (Rd ) , k f 0 k L∞ (R) , k f0 k L∞ (R) , kdivbk L∞ (Rd ) , kbk L∞ (Rd ) and T, such that sup E. Some results on the existence and uniqueness of solutions as well as continuous dependence on b and f have been obtained in [33]. The letter C will mean a positive constant, whose values may change in different places

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 3

Conclusions