On the Initial Value Problem for Hyperbolic Systems with Discontinuous Coefficients

TL;DR

This paper investigates the initial value problem for hyperbolic systems with discontinuous coefficients, establishing existence, regularity, and explicit solutions for wave propagation in piecewise homogeneous media with interface conditions.

Contribution

It provides explicit solutions and existence proofs for hyperbolic systems with discontinuous coefficients across multiple lines and extends results to higher dimensions with interface conditions.

Findings

Explicit solutions for piecewise constant coefficient matrices with discontinuities along one or two lines.

Proof of existence and regularity of solutions for hyperbolic systems with discontinuous coefficients.

Development of energy estimates and weak solutions for symmetric hyperbolic systems in multiple dimensions.

Abstract

Hyperbolic systems of the first and higher-order partial differential equations appear in many multiphysics problems. We will be dealing with a wave propagation problem in a piece-wise homogeneous medium. Mathematically, the problem is reduced to analyzing two systems of partial differential equations posed on two domains with a common boundary. The differential equations may not be satisfied on the boundary (or part of the boundary), but some interface conditions are satisfied. These interface conditions depend on a specific physical problem. We aim to prove the existence and regularity of the solution for the case of hyperbolic systems of first-order equations with different domains separated by a hyperplane, where we need to formulate the interface conditions. We do this for the initial value problem in 1D-space variable when the coefficient matrix has discontinuity on $m$ lines. More specifically, we find explicit solutions to the case when the coefficient matrix is piecewise constant with a discontinuity along $1$ line or $2$ lines. We also prove the existence of solution to the general initial value problem. We then formulate the weak solution of initial value problem for the corresponding symmetric hyperbolic system in $n $D-space variables with interface conditions, get the energy estimates for this system, and prove the existence of solution to the system.