Blow up dynamics for the 3D energy-critical nonlinear Schrödinger equation

We construct a two-parameter continuum of type II blow up solutions for the energy-critical nonlinear Schrödinger equation in dimension $d = 3$ . The solutions collapse to a single energy bubble in finite time and have the form $\begin{aligned} u(t,x) = e^{i \alpha (t)}\lambda (t)^{\frac{1}{2}}W(\lambda (t) x) + \eta (t, x ),~ ~ t \in [0, T),~ x \in {{\,\mathrm{\mathbb {R}}\,}}^3, \end{aligned}$ u ( t , x ) = e i α ( t ) λ ( t ) 1 2 W ( λ ( t ) x ) + η ( t , x ) , t ∈ [ 0 , T ) , x ∈ R 3 , where $W( x) = \big ( 1 + \frac{|x|^2}{3}\big )^{-\frac{1}{2}}$ is the ground state solution, $\lambda (t) = (T -t)^{- \frac{1}{2} - \nu }$ for suitable $\nu > 0$ , $\alpha (t) = \alpha _0 \log (T - t)$ and $T= T(\nu , \alpha _0) > 0$ . Further there holds $\Vert \eta (t) - \eta _T\Vert _{\dot{H}^1 \cap \dot{H}^2} = o(1)$ as $t \rightarrow T^-$ for some $\eta _T \in \dot{H}^{1} \cap \dot{H}^2$ .