An Introduction to Well-posedness and Free-evolution

TL;DR

This paper introduces fundamental concepts for analyzing well-posedness of evolution PDEs, emphasizing hyperbolicity, boundary conditions, and gauge effects, with applications to electromagnetism and system stability.

Contribution

It provides an overview of hyperbolicity conditions, boundary treatments, and gauge considerations for ensuring well-posedness in evolution PDE systems, especially in electromagnetism.

Findings

Strong hyperbolicity guarantees well-posedness.

Symmetric hyperbolic systems enable well-posed boundary value problems.

Gauge choices influence hyperbolicity and boundary stability.

Abstract

These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations. I show how strong hyperbolicity guarantees well-posedness of the initial value problem. Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions. I discuss the Laplace-Fourier method for analyzing the initial boundary value problem. Finally I state how these notions extend to systems that are first order in time and second order in space. In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem. I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions. I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable.