Differential Equations Seminar

In this talk we discuss the almost-global well-posedness of a wide class of coupled Wave-Klein-Gordon equations in 2+1 space-time dimensions, when initial data are assumed to be small and localized. The Wave-Klein-Gordon systems arise from several physical models especially related to General Relativity, but few results are know at present in lower space-time dimensions. Compared with prior related results, our novel contributions include a strong quadratic quasilinear coupling between the wave and the Klein-Gordon equation, and no restriction is made on the support of the initial data which are supposed to only have a mild decay at infinity and very limited regularity. Our proof relies on a combination of energy estimates localized to dyadic space-time regions, and pointwise interpolation type estimates within the same regions. This is akin to ideas previously used by Metcalfe-Tataru-Tohaneanu in a liner setting, and is also related to Alinhac's ghost weight method. A refinement of these estimates through different techniques will allow us to pass, in a future work, from almost global existence to global existence of solutions under the same hypothesis on the initial data. This is a joint work with M. Ifrim. Speaker(s): Annalaura Stingo (UC Davis)