Smooth radial quantized Type II blow-up for the three-dimensional energy-critical nonlinear Schrödinger equation

Abstract: We consider the focusing energy-critical nonlinear Schrödinger equation in three dimensions. In \cite{Schmid2026CVPDE}, Schmid constructed finite-time type II blow-up solutions with a continuum of blow-up rates, while the existence of smooth finite-time type II blow-up solutions remained open. We resolve this problem by constructing, for each integer $L\ge1$, smooth radial solutions concentrating a single ground state with the blow-up rate $λ(t)=(\ell+o(1))(T-t)<sup>{2L}$ for some $\ell&gt;0$. Our proof follows a forward construction based on modulation analysis with the energy method. The main obstacle is the lack of a repulsive structure that can be directly exploited to control the remainder. To overcome this obstacle, we adapt Kato's smoothing theory from asymptotic stability analysis to the forward construction. This yields a global spacetime Morawetz estimate for the linearized flow, providing the spacetime control needed to bound the higher-order energy in place of the monotonicity derived from repulsive structures in previous blow-up constructions.