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Almost global existence for quadratic Schrödinger equations in two dimensions
Abstract
We study the two-dimensional quadratic Schrödinger equation (i∂t+Δ)u=λ1u2+λ2u̅ 2 with initial data of size ε in a weighted H2 space. We prove almost global existence, with a lifespan of at least exp(c/ε) and decay at the optimal linear dispersive rate t−1. Our proof is based on the space--time resonance method introduced by Germain, Masmoudi, and Shatah. The main new ingredient is a Besov-type profile norm 𝒵, which allows us to close the bootstrap argument without controlling the global second spatial momentof the free profile, thereby avoiding the substantial time-dependent losses arising from direct second-moment estimates.
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BibTeX
@misc{liu2026almost,
title = {{Almost global existence for quadratic Schrödinger equations in two dimensions}},
author = {Xuan Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34237},
eprint = {2610.34237},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34237},
}
RIS
TY - UNPB
TI - Almost global existence for quadratic Schrödinger equations in two dimensions
AU - Xuan Liu
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34237
UR - https://arxiv.org/abs/2610.34237
AB - We study the two-dimensional quadratic Schrödinger equation (i∂_t+Δ)u=λ_1u^2+λ_2u̅^ 2 with initial data of size ε in a weighted H^2 space. We prove almost global existence, with a lifespan of at least exp(c/ε) and decay at the optimal linear dispersive rate t^−1. Our proof is based on the space--time resonance method introduced by Germain, Masmoudi, and Shatah. The main new ingredient is a Besov-type profile norm 𝒵, which allows us to close the bootstrap argument without controlling the global second spatial momentof the free profile, thereby avoiding the substantial time-dependent losses arising from direct second-moment estimates.
ER -