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The aim of this paper is to study the global existence of solutions to acoupled wave-Klein-Gordon system in space dimension two when initial data aresmall, smooth and mildly decaying at infinity. Some physical models strictlyrelated to general relativity have shown the importance of studying suchsystems, but very few results are know at present in low space dimension. Westudy here a model two-dimensional system, in which the non-linearity writes interms of 'null forms', and show the global existence of small solutions. Ourgoal is to prove some energy estimates on the solution when a certain number ofKlainerman vector fields is acting on it, and some optimal uniform estimates.The former ones are obtained using systematically quasi-linear normal forms, intheir para-differential version; the latter ones are recovered by deducing anew coupled system of a transport equation and an ordinary differentialequation from the starting PDE system, by means of a semi-classical micro-localanalysis of the problem. We expect the strategy developed here to be robustenough to enable us, in the future, to treat the case of the most generalnon-linearities.

Global existence of small amplitude solutions for a model quadratic  quasi-linear coupled wave-Klein-Gordon system in two space dimension, with  mildly decaying Cauchy data