AN INTRODUCTION TO WELL-POSEDNESS AND FREE-EVOLUTION

作者：Hilditch David^*

来源：International Journal of Modern Physics A, 2013, 28(22-23): UNSP 1340015.

DOI：10.1142/S0217751X13400150

• Initial value problem
• initial boundary value problem
• strong hyperbolicity
• symmetric hyperbolicity
• Laplace-Fourier method
• Maxwell equations
• gauge freedom

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.

• 出版日期2013-9-20