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On stability of self-similar blowup for mass supercritical NLS

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arxiv 2304.02078 v2 pith:EDRN3QEF submitted 2023-04-04 math.AP

Zexing Li This is my paper · ORCID
classification math.AP
keywords deltaself-similarstrichartzbeceanublowdeformedestimateslaplacian
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abstract

We consider the mass supercritical (NLS) in dimension $d\ge 1$ in the mass-supercritical range. The existence of self-similar blow up dyamics is known [Merle-Rapha\"el-Szeftel, 2010], and suitable self-similar blow up profiles were constructed [Bahri-Martel-Rapha\"el, 2021]. In this work, we prove the finite codimensional nonlinear asymptotic stability of a large class of self-similar profiles. The heart of the proof is, following the approach of Beceanu [Beceanu, 2011], the derivation of Strichartz dispersive estimates for matrix operators with a deformed Laplacian $\Delta_b = \Delta + ib\left(\frac d2 + x\cdot\nabla\right)$ in homogeneous Sobolev spaces and energy space $H^1$. The deformed Laplacian $\Delta_b$ arises from the renormalization and its operator group $e^{it\Delta_b}$ exhibits self-similar dispersion, which not only recovers the free Strichartz but also enables an extension of resolvent families. Compared with Strichartz estimates based on $\Delta$, this one has a larger admissible region, works for arbitrarily small polynomial decaying potential, and requires no spectral assumption.

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Cited by 2 Pith papers

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  1. Mode stability for self-similar blowup of slightly supercritical NLS: II. high-energy spectrum

    math.AP 2025-07 conditional novelty 7.0 of 10

    For slightly mass-supercritical NLS in 1 to 10 dimensions, no unstable high-energy eigenmodes exist for the linearized self-similar profile operator, which settles its asymptotic stability.

  2. Mode stability for self-similar blowup of slightly supercritical NLS: I. low-energy spectrum

    math.AP 2025-07 accept novelty 7.0 of 10

    For slightly mass-supercritical NLS in any dimension, the low-energy unstable spectrum of the self-similar linearized operator consists exactly of the symmetry modes 0, -bi, and -2bi.

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