Generalized solution to a system of conservation law which is not strictly hyperbolic

TL;DR

This paper addresses the existence and uniqueness of solutions for non-strictly hyperbolic conservation laws with and without viscosity, employing Colombeau generalized functions, shadow wave, and Volpert product methods.

Contribution

It introduces a novel framework for solving non-strictly hyperbolic systems using Colombeau generalized functions and constructs solutions via shadow wave and Volpert product approaches.

Findings

Existence and uniqueness of solutions in Colombeau algebra

Construction of inviscid solutions as viscosity approaches zero

Application of shadow wave and Volpert product methods

Abstract

In this paper we study a non strictly system of conservation law when viscosity is present and viscosity is zero, which is studied in [10]. We show the existence and uniqueness of the solution in the space of generalized functions of Colombeau for the viscous problem and construct a solution to the inviscid system in the sense of association. Also we construct a solution using shadow wave approach [5] and Volpert product which was partly determined as vanishing viscosity limit in [10].