# The Fuchsian approach to global existence for hyperbolic equations

#### General Relativity, Geometry and Analysis: beyond the first 100 years after Einstein

#### 31 October 11:00 - 12:00

#### Todd Oliynyk - Monash University

Systems of first order hyperbolic equations that can be expressed in the form

are said to be Fuchsian. Traditionally, these systems have been viewed as singular initial value problems (SIVP), where asymptotic data is prescribed at the singular time *t *= 0 and then the Fuchsian equation is used to evolve the asymptotic data away from the singular time to construct solutions on time intervals *t *∈ (0, *T*]. In this talk, I will not consider the SIVP, but instead I will focus on the standard initial value problem where initial data is specified at some *T > *0 and the Fuchsian equation is used to evolve the initial to obtain solutions on time intervals *t *∈ (*T*_{*}, *T*], *T >*

*T*0. I will describe recent small initial data existence results for these systems that guarantee the existence of solutions all the way to

_{* }*>*_{ }*t*= 0, that is, on time intervals

*t*∈ (0,

*T*]. I will then discuss how this existence theory for Fuchsian systems can be used to obtain global existence results for a variety of hyperbolic equations including the relativistic Euler equations, the Einstein-Euler equations, and non-linear systems of wave equations.