paper

An Introduction to Well-posedness and Free-evolution

arXiv:1309.2012 · doi:10.1142/S0217751X13400150

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.

Lecture notes from the NRHEP spring school held at IST-Lisbon, March 2013. To be published by IJMPA (V. Cardoso, L. Gualtieri, C. Herdeiro and U. Sperhake, Eds., 2013)

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