The Schrödinger equation with fluctuating nonlinearity in the energy space
Abstract
We study nonlinear Schrödinger equations with nonlinear Stratonovich noise d u = i[ Δu + λ|u|p−1u] d t + i|u|(q−1)/2u∘ d W, in their energy space H1(ℝd;ℂ). By combining the stochastic Strichartz estimates derived in [Potential Anal. 41 (2014), pp.\ 269--315] with the approach from [Ann.\ Inst.\ H.\ Poincaré Phys.\ Théor.\ 46 (1987), pp.\ 113--129] we obtain local well-posedness for all energy-subcritical nonlinearities p,q∈ [1, 1+4/(d−2)+) together with a corresponding blow-up alternative. For a linear multiplicative noise q=1, a real-valued noise W and a defocusing nonlinearity λ≤ 0, we check this blow-up condition using a bound on the energy, resulting in the global well-posedness of the equation. If both nonlinearities are mass-subcritical, i.e., p,q∈ [1, 1+4/d), we provide an improved blow-up criterion involving the L2(ℝd;ℂ)-norm. Using the conservation of mass for real-valued W, we obtain global well-posedness also in this case. Compared to previous results on stochastic nonlinear Schrödinger equations, we thereby improve the range of exponents p and q and the spatial regularity assumption on the noise.
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BibTeX
@misc{sauerbrey2026schrödinger,
title = {{The Schrödinger equation with fluctuating nonlinearity in the energy space}},
author = {Max Sauerbrey and Joris van Winden},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10417},
eprint = {2609.10417},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10417},
}
RIS
TY - UNPB
TI - The Schrödinger equation with fluctuating nonlinearity in the energy space
AU - Max Sauerbrey
AU - Joris van Winden
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10417
UR - https://arxiv.org/abs/2609.10417
AB - We study nonlinear Schrödinger equations with nonlinear Stratonovich noise d u = i[ Δu + λ|u|^p−1u] d t + i|u|^(q−1)/2u∘ d W, in their energy space H^1(ℝ^d;ℂ). By combining the stochastic Strichartz estimates derived in [Potential Anal. 41 (2014), pp.\ 269--315] with the approach from [Ann.\ Inst.\ H.\ Poincaré Phys.\ Théor.\ 46 (1987), pp.\ 113--129] we obtain local well-posedness for all energy-subcritical nonlinearities p,q∈ [1, 1+4/(d−2)_+) together with a corresponding blow-up alternative. For a linear multiplicative noise q=1, a real-valued noise W and a defocusing nonlinearity λ≤ 0, we check this blow-up condition using a bound on the energy, resulting in the global well-posedness of the equation. If both nonlinearities are mass-subcritical, i.e., p,q∈ [1, 1+4/d), we provide an improved blow-up criterion involving the L^2(ℝ^d;ℂ)-norm. Using the conservation of mass for real-valued W, we obtain global well-posedness also in this case. Compared to previous results on stochastic nonlinear Schrödinger equations, we thereby improve the range of exponents p and q and the spatial regularity assumption on the noise.
ER -