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

2023/08/03 by Schmid, Tobias · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.01883

Abstract

We construct a two-parameter continuum of type II blow up solutions for the energy-critical focusing NLS in dimension d = 3. The solutions collapse to a single energy bubble in finite time, precisely they have the form u(t,x) = e^{i α(t)}λ(t)^(1)/(2)W(λ(t) x) + η(t, x ), t ∈[0, T), x ∈ ℝ³, where W( x) = ( 1 + (|x|²)/(3))^-(1)/(2) is the ground state solution, λ(t) = (T-t)^{- \frac12 - ν} for suitable ν> 0 , α(t) = α₀ log(T - t) and T= T(ν, α₀) > 0 . Further ‖η(t) - η_T‖_H¹ ∩ H² = o(1) as t → T^- for some η_T ∈ H¹ ∩~ H².

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