Global Solution of a Nonlinear Conservation Law with Weak Discontinuous Flux in the Half Space

Jing Zhang,Xiaoqian Li

doi:10.4236/ajcm.2018.84026

Abstract

This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space R+= {x |x > 0} where a>0 , u(x,t) is an unknown function of x ∈ R+ and t>0 , u ± , um are three given constants satisfying um=u+≠u- or um=u-≠u+ , and the flux function f is a given continuous function with a weak discontinuous point ud. The main purpose of our present manuscript is devoted to studying the structure of the global weak entropy solution for the above initial-boundary value problem under the condition of f '-(ud) > f '+(ud). By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial-boundary value problem, and investigate the interaction of elementary waves with the boundary and the boundary behavior of the weak entropy solution.

Highlights

In order to clarify the structure of the global weak entropy solution for the initial-boundary value problem (2)-(4) under the assumptions (A1) and (A2), we need the following lemma 2
When u− = u2* or u2* < u− < ud, if we set u ( x, t ) = v ( x, t ) x,t>0, similar to the case of u− < u2*, we can give the expression for u (0+,t ), and by which and Lemma 2, it is easy to be verified that u ( x,t ) is the global weak entropy solution of the problem (2)-(4)
We state the interaction of the unique elementary wave ( ) E u−,u+ x,t>0 in u ( x,t ) with the boundary for this case. ( ) E u−,u+ x,t>0 will be far away from the boundary as f ′(u− ) ≥ 0, or one part R (u−,u5*;(a, 0)) of rarefaction wave in ( ) E u−,u+ x,t>0 will be absorbed by the boundary as f ′(u− ) < 0 and f ′(u4* ) > 0 or the whole of ( ) E u−,u+ x,t>0 will be completely absorbed by the boundary f ′(u4* ) ≤ 0

Summary

Introduction

Liu-Pan [13] [14] [15] gave a construction method to the global weak entropy solution of the initial-boundary value problem with piecewise smooth initial dada and constant boundary data for scalar nonlinear hyperbolic conservation laws, and clarified the structure and boundary behavior of the weak entropy solution.

Definition of Weak Entropy Solution and Related Lemma

Solution Structures

Conclusion