One dimensional energy cascades in a fractional quasilinear NLS
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918278393233408 |
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| author | Maspero, Alberto Murgante, Federico |
| author_facet | Maspero, Alberto Murgante, Federico |
| contents | We consider the problem of transfer of energy to high frequencies in a quasilinear Schrödinger equation with sublinear dispersion, on the one dimensional torus. We exhibit initial data undergoing finite but arbitrary large Sobolev norm explosion: their initial norm is arbitrary small in Sobolev spaces of high regularity, but at a later time becomes arbitrary large. We develop a novel mechanism producing instability, which is based on extracting, via paradifferential normal forms, an effective equation driving the dynamics whose leading term is a non-trivial transport operator with non-constant coefficients. We prove that such operator is responsible for energy cascades via a positive commutator estimate inspired by Mourre's commutator theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01097 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | One dimensional energy cascades in a fractional quasilinear NLS Maspero, Alberto Murgante, Federico Analysis of PDEs Dynamical Systems We consider the problem of transfer of energy to high frequencies in a quasilinear Schrödinger equation with sublinear dispersion, on the one dimensional torus. We exhibit initial data undergoing finite but arbitrary large Sobolev norm explosion: their initial norm is arbitrary small in Sobolev spaces of high regularity, but at a later time becomes arbitrary large. We develop a novel mechanism producing instability, which is based on extracting, via paradifferential normal forms, an effective equation driving the dynamics whose leading term is a non-trivial transport operator with non-constant coefficients. We prove that such operator is responsible for energy cascades via a positive commutator estimate inspired by Mourre's commutator theory. |
| title | One dimensional energy cascades in a fractional quasilinear NLS |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2408.01097 |