Pseudo-differential noise and nonlocal singularity formation in the stochastic Córdoba--Córdoba--Fontelos equation
Abstract
We study the stochastic Córdoba--Córdoba--Fontelos equation driven by multiplicative Stratonovich noise. The noise amplitude is allowed to be a pseudo-differential operator whose leading part is nearly skew-adjoint. This class contains classical transport noise and also permits genuinely nonlocal perturbations. We first develop a local-in-time theory for maximal classical solutions in Sobolev spaces, proving existence, uniqueness, and a blow-up criterion. We then consider the special case of Stratonovich transport. For sufficiently large initial nonlocal steepness at a global maximum, we prove finite-time blow-up with arbitrarily high prescribed probability and obtain an explicit upper bound on the lifespan. Finally, on the event that the nonlocal steepness at the transported maximum diverges, we establish a conditional Type-I upper bound. When the terminal Cesàro average of the normalised nonlocal energy converges, we further identify the exact leading-order blow-up rate in terms of its limiting value.
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BibTeX
@misc{alonsooran2026pseudodifferential,
title = {{Pseudo-differential noise and nonlocal singularity formation in the stochastic Córdoba--Córdoba--Fontelos equation}},
author = {Diego Alonso-Orán and Rafael Granero-Belinchón and Yingting Miao and Hao Tang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02019},
eprint = {2609.02019},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02019},
}
RIS
TY - UNPB
TI - Pseudo-differential noise and nonlocal singularity formation in the stochastic Córdoba--Córdoba--Fontelos equation
AU - Diego Alonso-Orán
AU - Rafael Granero-Belinchón
AU - Yingting Miao
AU - Hao Tang
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02019
UR - https://arxiv.org/abs/2609.02019
AB - We study the stochastic Córdoba--Córdoba--Fontelos equation driven by multiplicative Stratonovich noise. The noise amplitude is allowed to be a pseudo-differential operator whose leading part is nearly skew-adjoint. This class contains classical transport noise and also permits genuinely nonlocal perturbations. We first develop a local-in-time theory for maximal classical solutions in Sobolev spaces, proving existence, uniqueness, and a blow-up criterion. We then consider the special case of Stratonovich transport. For sufficiently large initial nonlocal steepness at a global maximum, we prove finite-time blow-up with arbitrarily high prescribed probability and obtain an explicit upper bound on the lifespan. Finally, on the event that the nonlocal steepness at the transported maximum diverges, we establish a conditional Type-I upper bound. When the terminal Cesàro average of the normalised nonlocal energy converges, we further identify the exact leading-order blow-up rate in terms of its limiting value.
ER -