On the Initial Value Problem for Hyperbolic Systems with Discontinuous Coefficients

TL;DR

The paper addresses the initial value problem for hyperbolic systems with discontinuous coefficients across interfaces in piecewise-homogeneous media. It first derives explicit 1D solutions via the method of characteristics for piecewise constant coefficient matrices with one or two discontinuities and proves existence for the general 1D case. It then extends to multi-dimensional symmetric hyperbolic systems, formulating a weak solution with a sharp interface condition and establishing energy estimates to obtain existence and uniqueness. The work clarifies how interface conditions arise from both characteristic and energy methods and has potential implications for multiphysics problems with layered media.

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.

Figures (6)

• Figure 1: Characteristics for $\lambda_j>0$
• Figure 2: Characteristics for $\lambda_j<0$
• Figure 3: $\lambda_{1j}, \lambda_{2j}, \lambda_{3j}$ are all positive
• Figure 4: $\lambda_{1j}, \lambda_{2j}, \lambda_{3j}$ are all negative.
• Figure 5: When $\alpha(s,z,t)$ does not touch the line $z=z_m$

Theorems & Definitions (1)

• Remark 1.1