Annalaura Stingo, UC Davis – Analysis & PDE Seminar

Title: Almost global well-posedness for quadratic quasilinear 2D wave-Klein-Gordon systems with small and localized initial data.

Abstract: In this talk we will discuss our recent result on the almost global existence of small solutions of strongly coupled wave-Klein-Gordon systems (WKG) in 2+1 space-time dimensions. The coupling we consider is quadratic, quasilinear, and satisfies the so-called null condition. No restriction is made on the support of the initial data, that are small and only mildly decaying at infinity.

Wave-Klein-Gordon systems arise from physical models especially related to General Relativity, but only few results on the long time/global existence of solutions are known at present. In 2+1 space-time dimensions, the only contributions to the subject are due to Y. Ma, who studies some particular examples of weakly coupled WKG systems with small compactly supported initial data, and to Stingo, who proved the global existence of solutions for a model strongly coupled WKG with small and localized data (no restriction on their support).

The work we present here, in collaboration with M. Ifrim, addresses a much wider class of strongly coupled WKG systems than the one treated by Stingo. We prove that small solutions are almost global, using 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 linear setting, and is also related to Alinhac’s ghost weight method. A refinement of these estimates will lead us to pass, in a future work, from almost global existence to global existence of solutions under the same hypothesis on the initial data.