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Analysis Seminar

Thursday, January 28, 2021
9:00am to 9:55am
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Nonlinear wave equations, the weak null condition, and radiation fields
Joseph Keir, Mathematical Institute, University of Oxford,

Nonlinear wave equations are ubiquitous in physics, and in three spatial dimensions they can exhibit a wide range of interesting behaviour even in the small data regime, ranging from dispersion and scattering on the one hand, through to finite-time blowup on the other. The type of behaviour exhibited depends on the kinds of nonlinearities present in the equations. In this talk I will explore the boundary between "good" nonlinearities (leading to dispersion similar to the linear waves) and "bad" nonlinearities (leading to finite-time blowup). In particular, I will give an overview of a proof of global existence (for small initial data) for a wide class of nonlinear wave equations, including some which almost fail to exist globally, but in which the singularity in some sense takes an infinite time to form. I will also show how to construct other examples of nonlinear wave equations whose solutions exhibit very unusual asymptotic behaviour, while still admitting global small data solutions.

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