# Mathematics Colloquia and Seminars

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### Global existence of small solutions to a model wave-Klein-Gordon system in 2D

**PDE and Applied Math Seminar**

Speaker: | Annalaura Stingo, UC Davis |

Location: | 2112 MSB |

Start time: | Thu, Oct 18 2018, 3:10PM |

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 fieldsâ€™ 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.