Global Existence and Scattering of a Class of Two-Dimensional Euler--Poisson--Korteweg Systems
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.