Abstract
We consider the energy supercritical defocusing nonlinear Schrödinger equation i∂tu+Δu-u|u|p-1=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} i\\partial _tu+\\Delta u-u|u|^{p-1}=0 \\end{aligned}$$\\end{document}in dimension dge 5. In a suitable range of energy supercritical parameters (d, p), we prove the existence of {mathcal {C}}^infty well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of {mathcal {C}}^infty spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.
Highlights
We consider the defocusing nonlinear Schrödinger equation (NLS)i ∂t u + u − u|u|p−1 = 0, u|t=0 = u0,(t, x) ∈ [0, T∗) × Rd, u(t, x) ∈ C. (1.1)in dimension d ≥ 3 for an integer nonlinearity p ∈ 2N∗ + 1 and address the problem of its global dynamics
Two quantities conserved along the flow (1.1) are of the utmost importance: energy: mass: M(u) = |u(t, x)|2 = |u0(x)|2, Rd
We review the main known dynamical results which rely on the scaling classification
Summary
It is a very classical statement that smooth well localized initial data u0 yield local in time, unique, smooth, strong solutions. Two quantities conserved along the flow (1.1) are of the utmost importance: energy: mass: M(u) = |u(t, x)|2 = |u0(x)|2, Rd. The problem (1.1) can be classified as energy subcritical, critical or supercritical depending on whether the critical Sobolev exponent sc lies. On blow up for the energy super critical defocusing NLS below, equal or above the energy exponent s = 1. This classification reflects the (in)/ability for the kinetic term in (1.2) to control the potential one via the Sobolev embedding H 1 → Lq
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