Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.

Very few results are known for the two-dimensional problem and they only concern the case of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity. The proof of the main theorem is based on the propagation of some energy and $L^\infty$ estimates on the solution. We derive the wished energy estimates by performing several normal form arguments and using an adapted version of Klaineman vector fields’ method.

The $L^\infty$ estimates are instead obtained by rewriting the starting problem in a semi-classical setting and deducing from it a new coupled system, made of a transport equation and of an ODE. This will require a new normal form argument together with a semi-classical micro-local analysis of the problem.