Smooth singularity formation for the energy critical wave equation without quantization I: Approximate solutions

Abstract: 
In this talk, I present ongoing work on singularity formation for the energy-critical nonlinear wave equation (NW) in three dimensions. In their foundational work, Krieger-Schlag-Tataru constructed non-smooth solutions to (NW) that develop singularities with arbitrarily fast, polynomial blow-up rates exceeding the self-similar rate. For comparison, their analogous results for the wave map equation stand in sharp contrast to the quantized blow-up rates for smooth solutions obtained by Raphaël-Rodnianski. In this talk, I discuss a new result establishing the existence of smooth approximate solutions to (NW) that develop singularities at a continuum of polynomial rates, achieved through the collapse of multiple solitons.