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Special Colloquium

The Radiative Uniqueness Conjecture for Bubbling Wave Maps

Casey Rodriguez, MIT

Location:  Hill 705
Date & time: Friday, 24 January 2020 at 2:00PM - 3:00PM

Abstract.   One of the most fundamental questions in partial dif-
ferential equations is that of regularity and the possible breakdown of
solutions. We will discuss this question for solutions to a canonical ex-
ample of a geometric wave equation; energy critical wave maps. Break-
through works of Krieger-Schlag-Tataru, Rodnianski-Sterbenz and
Rapha ?el-Rodnianski produced examples of wave maps that develop sin-
gularities in finite time. These solutions break down by concentrating
energy at a point in space (via bubbling a harmonic map) but have a
regular limit, away from the singular point, as time approaches the final
time of existence. The regular limit is referred to as the radiation. This
mechanism of breakdown occurs in many other PDE including energy
critical wave equations, Schr ?odinger maps and Yang-Mills equations.
A basic question is the following:


• Can we give a precise description of all bubbling singularities
for wave maps with the goal of finding the natural unique con-
tinuation of such solutions past the singularity?


In this talk, we will discuss recent work (joint with J. Jendrej and A.
Lawrie) which is the first to directly and explicitly connect the radiative
component to the bubbling dynamics by constructing and classifying
bubbling solutions with a simple form of prescribed radiation. Our
results serve as an important first step in formulating and proving the
following Radiative Uniqueness Conjecture for a large class of wave
maps: every bubbling solution is uniquely characterized by it’s radia-
tion, and thus, every bubbling solution can be uniquely continued past
blow-up time while conserving energy.