3/17/2008

Jared Speck
Rutgers University

Well-Posedness for the Euler-Nordström System via the Method of Energy Currents

In this talk, I will discuss the well-posedness of the Cauchy problem for the Euler-Nordström (EN) system, a first-order quasilinear hyperbolic system of PDEs modeling the motion of a relativistic perfect fluid with self-interaction mediated by Nordström's scalar theory of gravity. It is well-known that for first-order symmetric hyperbolic (FOSH) and strictly hyperbolic (STRHYP) systems, an energy principle is available that implies well-posedness (local existence, uniqueness, and continuous dependence on initial data) for initial data belonging to an appropriate Sobolev space. Because the EN system is neither symmetric hyperbolic nor strictly hyperbolic, I will discuss alternate techniques recently developed by Christodoulou that provide energy currents for equations derivable from a Lagrangian. After providing energy currents for the EN system, I will discuss their key properties and show that they can be used to derive Sobolev estimates that are similar to the ones implied by the energy principle that is available in the FOSH and STRYHYP cases. If time permits, I will discuss the Newtonian limit (allowing the speed of light to go to infinity) of the EN system. It turns out that the method of energy currents is also a useful tool for analyzing this singular limit.