Vanishing viscosity solution to a 2 x 2 system of conservation laws with linear damping

TL;DR

This paper investigates a 2x2 non-strictly hyperbolic conservation law system with linear damping, focusing on vanishing viscosity limits, weak solutions, and large-time behavior for general initial data, extending previous Riemann data results.

Contribution

It introduces a weak asymptotic solution framework for the system with general initial data and analyzes the vanishing viscosity limit and asymptotic behavior, which were previously studied only for Riemann data.

Findings

Established weak asymptotic solutions for the system.

Proved the vanishing viscosity limit yields a distribution solution.

Analyzed large time asymptotic behavior of the viscous system.

Abstract

Systems of the first order partial differential equations with singular solutions appear in many multiphysics problems and the weak formulation of solutions involve in many cases product of distributions. In this paper we study such a system derived from Eulerian droplet model for air particle flow. This is a 2 x 2 non - strictly hyperbolic system of conservation laws with linear damping. We first study a regularized viscous system with variable viscosity term and obtain a weak asymptotic solution with general initial data and also get solution in the Colombeau algebra. We also study the vanishing viscosity limit and show that this limit is a distribution solution. Further we study the large time asymptotic behaviour of the viscous system. This important system, is not very well studied due to complexities in the analysis. As far as we know the only work done on this system is for Riemann type of initial data. The significance of this paper is that we work on the system having general initial data and not just initial data of the Riemann type.