On blow up for the energy super critical defocusing nonlinear Schrödinger equations

Abstract

We consider the energy supercritical defocusing nonlinear Schrödinger equation $$\begin{aligned} i\partial _tu+\Delta u-u|u|^{p-1}=0 \end{aligned}$$ i ∂ t u + Δ u - u | u | p - 1 = 0 in dimension $$d\ge 5$$ d ≥ 5 . In a suitable range of energy supercritical parameters (d, p), we prove the existence of $${\mathcal {C}}^\infty $$ C ∞ 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 $${\mathcal {C}}^\infty $$ C ∞ 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.

On smooth self similar solutions to the compressible Euler equations

“…The case of higher dimensions will be relevant for the study of the energy super critical defocusing (NLS) equation in [8], for which the power nonlinearity involves the real number p given by p = 1 + 4 .…”

“…This is contrary to the Lyapunov analysis which would suggest that it might never be possible. In the companion papers [8,9], we will use these solutions as the leading order blow up profiles for respectively the energy super-critical defocusing nonlinear Schrödinger equations and the compressible Navier-Stokes equation (as well as its inviscid Euler limit) to produce blowing up solutions arising from smooth initial data. The C ∞ regularity of the profile is needed not only for the regularity of initial data but much more importantly, in fact, crucially, for the stability analysis.…”

“…The above gluing procedure produces a solution with limited C k(r) regularity 3 at the degenerate point P 2 , see Remark 3.7. As it turns out, this limited regularity has a dramatic effect on the spectral structure of the linearized operator for (1.6) close to this self-similar profile, and we do not know how to use these non C ∞ solutions to produce finite energy imploding solutions to the structural perturbations of (1.1) studied in [8,9].…”

“…The exceptional solution is C ∞ but it does not go to P 6 and is thus inadmissible as a global profile. The C λ − λ + regularity is sharply insufficient for our purposes (linear stability analysis of these solutions as solutions of the Euler equations misses exactly one derivative, see [8]). Instead, we consider the regime where λ + ↑ 0 (and λ − stays uniformly bounded away from 0 and −∞) which corresponds to a limiting degeneracy of the phase portrait for the critical speed r ↑ r o (d, ) given by (1.13) where a vanishing "o" structure appears to the left of P 2 .…”

Random tensors, propagation of randomness, and nonlinear dispersive equations

“…This is also important from the PDE point of view, as it would mean that blowup for defocusing H 1 supercritical nonlinear Schrödinger equations is non-generic and unstable 50 . Note that the blowup example in R d , recently constructed in [60], is indeed non-generic.…”