On the Initial Value Problem for Hyperbolic Systems with Discontinuous Coefficients

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.