# 401-4767-66L Partial Differential Equations (Hyperbolic PDEs)

Semester | Autumn Semester 2016 |

Lecturers | D. Christodoulou |

Periodicity | non-recurring course |

Language of instruction | English |

Abstract | The course begins with characteristics, the definition of hyperbolicity, causal structure and the domain of dependence theorem. The course then focuses on nonlinear systems of equations in two independent variables, in particular the Euler equations of compressible fluids with plane symmetry and the Einstein equations of general relativity with spherical symmetry. |

Objective | The objective is to introduce students in mathematics and physics to an area of mathematical analysis involving differential geometry which is of fundamental importance for the development of classical macroscopic continuum physics. |

Content | The course shall begin with the basic structure associated to hyperbolic partial differential equations, characteristic hypersurfaces and bicharacteristics, causal structure, and the domain of dependence theorem. The course shall then focus on nonlinear systems of equations in two independent variables. The first topic shall be the Euler equations of compressible fluids under plane symmetry where we shall study the formation of shocks, and second topic shall be the Einstein equations of general relativity under spherical symmetry where we shall study the formation of black holes and spacetime singularities. |

Prerequisites / Notice | Basic real analysis and differential geometry. |