Todd Oliynyk: The Fuchsian approach to global existence for hyperbolic equations

Time: Thu 2019-10-31 11.00 - 12.00

Lecturer: Todd Oliynyk, Monash University

Location: Seminar Hall Kuskvillan, Institut Mittag-Leffler

Abstract

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

\(B^0(t,u)\partial_t u + B^i(t,u)\nabla_i u = \frac1t B(t,u)u + F(t,u)\)

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 \(t \in (0, T]\). 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 \(t \in (T^\ast, T], T > T^\ast > 0\). 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 \(t \in (0, T]\). 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.

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