Global Existence and Scattering of a Class of Two-Dimensional Euler--Poisson--Korteweg Systems
Abstract
For a class of two-dimensional Euler--Poisson--Korteweg systems of quantum hydrodynamic type, we establish global existence and scattering for small, smooth, spatially localized perturbations of a constant equilibrium with irrotational velocity. This extends the three-dimensional well-posedness result of Song [Math. Ann. 392 (2025), 2157--2224] to two dimensions. Our proof follows the space-time resonance approach developed by Deng, Ionescu, Pausader, and Pusateri for the three-dimensional gravity-capillary water-wave system. The new ingredients are twofold. First, to allow non-neutral initial data, we exploit the absence of space resonances and of time resonances---including those arising after the second and third iterations---and the improved approximate-finite-speed-of-propagation bounds. Second, in the energy estimates, we adapt the almost orthogonality argument used in the works of Ionescu--Pausader and Guo--Ionescu--Pausader to the regime in which the input and output frequencies are all comparable to $1$. In this regime, time resonances may occur and the relevant curvature quantity may vanish.
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BibTeX
@misc{cui2026global,
title = {{Global Existence and Scattering of a Class of Two-Dimensional Euler--Poisson--Korteweg Systems}},
author = {Han Cui and Linhao Shi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37220},
eprint = {2610.37220},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37220},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Global Existence and Scattering of a Class of Two-Dimensional Euler--Poisson--Korteweg Systems
AU - Han Cui
AU - Linhao Shi
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37220
UR - https://arxiv.org/abs/2610.37220
AB - For a class of two-dimensional Euler--Poisson--Korteweg systems of quantum hydrodynamic type, we establish global existence and scattering for small, smooth, spatially localized perturbations of a constant equilibrium with irrotational velocity. This extends the three-dimensional well-posedness result of Song [Math. Ann. 392 (2025), 2157--2224] to two dimensions. Our proof follows the space-time resonance approach developed by Deng, Ionescu, Pausader, and Pusateri for the three-dimensional gravity-capillary water-wave system. The new ingredients are twofold. First, to allow non-neutral initial data, we exploit the absence of space resonances and of time resonances---including those arising after the second and third iterations---and the improved approximate-finite-speed-of-propagation bounds. Second, in the energy estimates, we adapt the almost orthogonality argument used in the works of Ionescu--Pausader and Guo--Ionescu--Pausader to the regime in which the input and output frequencies are all comparable to $1$. In this regime, time resonances may occur and the relevant curvature quantity may vanish.
ER -