Construction of Global Weak Entropy Solution of Initial-Boundary Value Problem for Scalar Conservation Laws with Weak Discontinuous Flux

Jing Zhang,Yihong Dai

doi:10.4236/ajcm.2017.74033

Abstract

This paper is concerned with the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a constant function. Under the condition that the flux function has a finite number of weak discontinuous points, by using the structure of weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux-Nedelec, we give a construction method to the global weak entropy solution for this initial-boundary value problem, and by investigating the interaction of elementary waves and the boundary, we clarify the geometric structure and the behavior of boundary for the weak entropy solution.

Highlights

The purpose of our present paper is devoted to constructing the global weak entropy solution of the initial-boundary value problem (1) for scalar conservation laws with two pieces of constant initial data and constant boundary data under the condition that the flux function has a finite number of weak discontinuous points, and clarifying the geometric structure and the behavior of boundary for the weak entropy solution
We will construct the global weak entropy solution of (1) with two pieces of constant initial data and constant boundary data under the condition that the flux function has a finite number of weak discontinuous points by employing Lemma 2 and the structure of weak entropy solution to the corresponding initial value problem
This paper is mainly concerned about the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a constant function

Summary

Introduction

Consider the following initial-boundary value problem for scalar conservation laws:. Y. The purpose of our present paper is devoted to constructing the global weak entropy solution of the initial-boundary value problem (1) for scalar conservation laws with two pieces of constant initial data and constant boundary data under the condition that the flux function has a finite number of weak discontinuous points, and clarifying the geometric structure and the behavior of boundary for the weak entropy solution.

Definition of Weak Entropy Solution and Related Lemmas

Construction of Global Weak Entropy Solutions

The Case That f Has Only One Weak Discontinuous Point

Riemann Initial-Boundary Problem

The Case That f Has Finitely Many Weak Discontinuous Points

Conclusion