Construction of two-bubble solutions for some energy-critical wave equations | Zendy

Jacek Jendrej | Zendy

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Construction of two-bubble solutions for some energy-critical wave equations

Author(s) -

Jacek Jendrej

Publication year - 2016

Publication title -

séminaire laurent schwartz — edp et applications

Language(s) - English

Resource type - Journals

ISSN - 2266-0607

DOI - 10.5802/slsedp.90

Subject(s) - superposition principle , bubble , exponential function , equivariant map , mathematical analysis , mathematics , space (punctuation) , energy (signal processing) , scale (ratio) , nonlinear system , wave equation , dimension (graph theory) , power (physics) , function (biology) , physics , classical mechanics , mechanics , pure mathematics , quantum mechanics , computer science , statistics , evolutionary biology , biology , operating system

We present a construction of pure two-bubbles for some energy-critical wave equations, that is solutions which in one time direction approach a superposition of two stationary states both centered at the origin, but asymptotically decoupled in scale. Our solution exists globally, with one bubble at a fixed scale and the other concentrating in infinite time, with an error tending to 0 in the energy space. We treat the cases of the power nonlinearity in space dimension 6, the radial Yang-Mills equation and the equivariant wave maps equation with equivariance class k ≥ 3. The concentration speed of the second bubble is exponential for the first two models and a power function in the last case.

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