Initial Boundary Value Problem For A Non-conservative System In Elastodynamics

TL;DR

This paper investigates the initial boundary value problem for a nonlinear, nonconservative hyperbolic system in elastodynamics, employing vanishing viscosity to construct solutions for specific initial and boundary data related to Riemann invariants.

Contribution

It introduces a method to solve a class of nonlinear elastodynamic boundary value problems using vanishing viscosity and Riemann invariants.

Findings

Constructed solutions for data on Riemann invariant level sets.

Demonstrated the applicability of vanishing viscosity in nonconservative systems.

Analyzed boundary conditions based on characteristic speeds.

Abstract

This paper is concerned with the initial boundary value problem for a nonconservative system of hyperbolic equation appearing in elastodynamics in the space time domain $x > 0, t > 0$. The number of boundary conditions to be prescribed at the boundary $x = 0$, depend on the number of characteristics entering the domain. Since our system is nonlinear the characteristic speeds depends on the unknown and the direction of the characteristics curves are known apriori. As it is well known, the boundary condition has to be understood in a generalised way. One of the standard way is using vanishing viscosity method. We use this method to construct solution for a particular class of initial and boundary data, namely the initial and boundary datas that lie on the level sets of one of the Riemann invariants.