Tracey Balehowsky : An inverse problem for the relativistic Boltzmann equation

Jeudi 11 mars 14:00-15:00 - Tracey Balehowsky - Université d'Helsinki

Résumé : In this talk, we consider the following problem : Given the source-to-solution map for a relativistic Boltzmann equation on a neighbourhood V of an observer in a Lorentzian spacetime (M,g) and knowledge of the metric g restricted to V, can we determine (up to diffeomorphism) the spacetime metric g on the domain of causal influence for the set V ?
We will show that the answer is yes. The problem we consider is a so-called inverse problem. We will briefly motivate the mathematical area of inverse problems. We will also introduce the relativistic Boltzmann equation and comment on the existence of solutions to this nonlinear PDE given some initial data. Then, we present a broad sketch of the key ideas in the proof of our result. One such key point is that the nonlinear term in the relativistic Boltzmann equation which describes the behaviour of particle collisions captures information about a source-to-solution map for a related linear problem. We use this relationship together with an analysis of the behaviour of particle collisions by classical microlocal techniques to determine the set of locations in V where we first receive light particle signals from collisions in the unknown domain. From this data we are able to parametrize the unknown region and determine the metric.
The new results presented in this talk are joint work with Antti Kujanpää, Matti Lassas, and Tony Liimatainen, (University of Helsinki).
A preprint can be found here :

Tracey Balehowsky : An inverse problem for the relativistic Boltzmann equation  Version PDF