Boundary conditions for coupled quasilinear wave equations with application to isolated systems

TL;DR

This paper establishes well-posed boundary conditions for coupled quasilinear wave equations on manifolds, enabling stable formulations for physical systems like Maxwell's and Einstein's equations with potential applications in numerical simulations.

Contribution

It introduces a broad class of boundary conditions ensuring well-posedness for coupled quasilinear wave systems, including those with constraints and artificial boundaries.

Findings

Well-posedness proven for a large class of boundary conditions.

Applicable to Maxwell's equations in Lorentz gauge.

Applicable to Einstein's equations in harmonic coordinates.

Abstract

We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form $[0,T] \times \Sigma$, where $\Sigma$ is a compact manifold with smooth boundaries $\partial\Sigma$. By using an appropriate reduction to a first order symmetric hyperbolic system with maximal dissipative boundary conditions, well posedness of such problems is established for a large class of boundary conditions on $\partial\Sigma$. We show that our class of boundary conditions is sufficiently general to allow for a well posed formulation for different wave problems in the presence of constraints and artificial, nonreflecting boundaries, including Maxwell's equations in the Lorentz gauge and Einstein's gravitational equations in harmonic coordinates. Our results should also be useful for obtaining stable finite-difference discretizations for such problems.