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Xueying Yu

Publications and source records attributed to Xueying Yu.

At least 19 recordsLinked to original sources
arXiv2026

Low-regularity global well-posedness for the one-dimensional, defocusing, septic nonlinear Schrödinger equation

In this paper, we show that the one dimensional septic nonlinear Schrödinger equation is globally well-posed in $H^s (\mathbb{R})$ when $s > 39/89$ with accompanying polynomial bounds in time. We utilize bilinear estimates to obtain improved bounds on the modified energy increment and prove new $L^{p+3}_{t,x}$ almost Morawetz estimates for the defocusing $p$-NLS adapted to the low-regularity setting, before specializing to the septic case $p=7$. These tools together allow us to reach Sobolev regularities $s<1/2$, showing for the first time a global existence result in the mass supercritical regime for a class of unbounded, discontinuous data.

math.AP↗
arXiv2026

Geometry-Conditioned Fourier Neural Operators for Cubic Nonlinear Schrodinger Dynamics on Periodic Domains

We consider the cubic nonlinear Schrödinger (NLS) equation on two-dimensional flat tori with varying aspect ratios. In this formulation, the choice of aspect ratio governs the Fourier resonance structure, so rational and irrational geometries can exhibit different high-frequency cascade behaviors. We present a geometry-conditioned Fourier neural operator (FNO) for the cubic defocusing NLS equation, where the input consists of the real and imaginary parts of the solution together with the aspect-ratio parameter \(ω^2\). The model is trained to approximate the one-step solution operator and is evaluated on unseen trajectories generated from random-phase initial data using Fourier pseudospectral method. Our numerical experiments show that the learned operator captures the main solution dynamics on both tori and reproduces the distinct Sobolev norm behavior of the two geometries, with stronger \(H^2\)-growth on the rational torus and more constrained behavior on the irrational torus, consistent with the findings of \cite{hrabski2021energy}. We perform ablation studies to examine the roles of retained Fourier modes, activation functions, Fourier-layer depth, and explicit geometry conditioning. The results indicate that including $ω^2$ improves long-time predictive accuracy, especially for the rational geometry, and supports the use of geometry-aware neural operators for learning spectral-transfer phenomena in nonlinear dispersive partial differential equations.

cs.LG↗
arXiv2026

Satellite-based emissions estimate indicates progress toward China's methane mitigation goals

China emits the most methane of any country worldwide, but there are large uncertainties in recent emissions trends, sources, and the potential impacts of policy actions. This study focuses on a period when the government initiated ambitious methane control efforts, linking sectoral policies with atmospheric evidence on sectoral, sub-national, and seasonal emissions during 2019--2024. We quantify daily methane emissions from China using a regional atmospheric inverse model with TROPOMI satellite observations. Our results reveal an average methane emissions increase rate of 0.3 Tg yr$^{-2}$ in Eastern & Central China, likely a milder trend than in the 2010s. Coal industry methane emissions intensity declined for the first time (-3.2% yr$^{-1}$) despite rising production, possibly associated with diverse policy instruments, mandates, and incentives. We further highlight two emerging challenges for future mitigation: leaks from expanding urban gas use amid the energy transition and rising agricultural emission yet with substantial uncertainty in estimates. Lastly, declining emissions intensity of coal mines points to the future role of targeted mandates and incentives in encouraging methane reduction for other sectors.

physics.ao-ph↗
arXiv2025

Nonlinear Schrödinger equation with Ornstein-Uhlenbeck operator

In this work, we introduce and study nonlinear Schrödinger equations (NLS) with anisotropic dispersion, where the standard Laplacian acts on the Euclidean variable \(x \in \mathbb{R}^d\), and an Ornstein-Uhlenbeck ($\mathcal{OU}$) operator governs the confined direction \(α\in \mathbb{R}\). We consider models with two natural variants of $\mathcal{OU}$-induced confinement: (Model Div) based on the divergence form \(\nabla_α\cdot (e^{-\frac{α^2}{2}} \nabla_α)\), and (Model Non-Div) based on the non-divergence form \(Δ_α- α\cdot \nabla_α\). For both models, we establish the Strichartz estimates and Gaussian-weighted Morawetz estimates. In addition, for (Model Div), we prove a virial-type finite-time blow-up result; for (Model Non-Div), we establish global well-posedness and small data scattering in the 2D quintic and 3D cubic cases. The primary motivation of this work is to capture waveguide-type dispersive behavior in a Euclidean setting. To the best of our knowledge, this is the first rigorous analysis of NLS with $\mathcal{OU}$ operators in both divergence and non-divergence forms.

math.AP↗
arXiv2025

Uniform periodic counterexamples to Carleson's convergence problem with polynomial symbols

In Carleson's convergence problem for dispersive equations $i\, \partial_t u + P(D)u=0$ in the periodic setting $\mathbb T^d$, we prove that the Sobolev exponent $d/(2(d+1))$ is necessary for any non-singular polynomial symbol $P$, including the natural powers of the Laplacian $Δ^k$. This is in contrast with the results known in the Euclidean case, in which for symbols $P(ξ) = |ξ|^a$ with $a > 1$ the exponent $d/(2(d+1))$ is sufficient, but we do not know if it is necessary.

math.AP↗
arXiv2024

On uniqueness properties of solutions of the generalized fourth-order Schrödinger equations

In this paper, we study uniqueness properties of solutions to the generalized fourth-order Schrödinger equations in any dimension $d$ of the following forms, $$i \partial_t u + \sum_{j=1}^d \partial_{x_j}^{\, 4} u = V(t, x) u, \quad \text{and} \quad i \partial_t u + \sum_{j=1}^d \partial_{x_j}^{\, 4} u + F (u, \bar{u}) = 0.$$ We show that a linear solution $u$ with fast enough decay in certain Sobolev spaces at two different times has to be trivial. Consequently, if the difference between two nonlinear solutions $u_1$ and $u_2$ decays sufficiently fast at two different times, it implies that $u_1 \equiv u_2$.

math.AP↗
arXiv2024

On Strichartz estimates for many-body Schrödinger equation in the periodic setting

In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori $\mathbb{T}^d$, where $d\geq 3$. The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in Bourgain-Demeter \cite{BD}. As a comparison, this result can be regarded as a periodic analogue of Hong \cite{hong2017strichartz} though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate.

math.AP↗
arXiv2023

On recovering the nonlinearity for generalized higher-order Schrödinger equations

In this note, we generalize the nonlinearity-recovery result in [7] for classical cubic nonlinear Schrödinger equations to higher-order Schrödinger equations with a more general nonlinearity. More precisely, we consider a spatially-localized nonlinear higher-order Schrödinger equation and recover the spatially-localized coefficient by the solutions with data given by small-amplitude wave packets.

math.AP↗
arXiv2023

On the uniqueness of variable coefficient Schrödinger equations

We prove unique continuation properties for linear variable coefficient Schrödinger equations with bounded real potentials. Under certain smallness conditions on the leading coefficients, we prove that solutions decaying faster than any cubic exponential rate at two different times must be identically zero. Assuming a transversally anisotropic type condition, we recover the sharp Gaussian (quadratic exponential) rate in the series of works by Escauriaza-Kenig-Ponce-Vega [14, 17, 18].

math.AP↗
arXiv2023

Singular Levy processes and dispersive effects of generalized Schrödinger equations

We introduce new models for Schrödinger-type equations, which generalize standard NLS and for which different dispersion occurs depending on the directions. Our purpose is to understand dispersive properties depending on the directions of propagation, in the spirit of waveguide manifolds, but where the diffusion is of different types. We mainly consider the standard Euclidean space and the waveguide case but our arguments extend easily to other types of manifolds (like product spaces). Our approach unifies in a natural way several previous results. Those models are also generalizations of some appearing in seminal works in mathematical physics, such as relativistic strings. In particular, we prove the large data scattering on waveguide manifolds $\mathbb{R}^d \times \mathbb{T}$, $d \geq 3$. This result can be regarded as the analogue of \cite{TV2, YYZ2} in our setting and the waveguide analogue investigated in \cite{GSWZ}. A key ingredient of the proof is a Morawetz-type estimate for the setting of this model.

math.AP↗
arXiv2023

On well-posedness results for the cubic-quintic NLS on $\mathbb{T}^3$

We consider the periodic cubic-quintic nonlinear Schrödinger equation \begin{align}\label{cqnls_abstract} (i\partial_t +Δ)u=μ_1 |u|^2 u+μ_2 |u|^4 u\tag{CQNLS} \end{align} on the three-dimensional torus $\mathbb{T}^3$ with $μ_1,μ_2\in \mathbb{R} \setminus\{0\}$. As a first result, we establish the small data well-posedness of \eqref{cqnls_abstract} for arbitrarily given $μ_1$ and $μ_2$. By adapting the crucial perturbation arguments in \cite{zhang2006cauchy} to the periodic setting, we also prove that \eqref{cqnls_abstract} is always globally well-posed in $H^1(\mathbb{T}^3)$ in the case $μ_2>0$.

math.AP↗
arXiv2022

Global well-posedness for the cubic fractional NLS on the unit disk

In this paper, we prove that the cubic nonlinear Schrödinger equation with the fractional Laplacian on the unit disk is globally well-posed for certain radial initial data below the energy space. The result is proved by extending the I-method in the fractional nonlinear Schrödinger equation setting.

math.AP↗
arXiv2022

On the decay property of the cubic fourth-order Schrödinger equation

In this short paper, we prove that the solution of the cubic fourth-order Schrödinger equation (4NLS) on $\mathbb{R}^d$ ($5 \leq d \leq 8$) enjoys the same (pointwise) decay property as its linear solution does. This result is proved via a bootstrap argument based on the corresponding global result Pausader \cite{Pau1}. This result can be extended to more general dispersive equations (including some more 4NLS models) with scattering asymptotics.

math.AP↗
arXiv2021

Global Well-posedness and scattering for fourth-order Schrödinger equations on waveguide manifolds

In this paper, we study the well-posedness theory and the scattering asymptotics for fourth-order Schrödinger equations (4NLS) on waveguide manifolds (semiperiodic spaces) $\mathbb{R}^d\times \mathbb{T}^n$, $d \geq 5$, $n=1,2,3$. The tori component $\mathbb{T}^n$ can be generalized to $n$-dimensional compact manifolds $\mathcal{M}^n$. First, we modify Strichartz estimates for 4NLS on waveguide manifolds, with which we establish the well-posedness theory in proper function spaces via the standard contraction mapping method. Moreover, we prove the scattering asymptotics based on an interaction Morawetz-type estimate established for 4NLS on waveguides. At last, we discuss the higher dimensional analogue, the focusing scenario and give some further remarks on this research line. This result can be regarded as the waveguide analogue of Pausader \cite{Pau2,Pau1,Pau3} and the 4NLS analogue of Tzvetkov-Visciglia \cite{TV2}.

math.AP↗
arXiv2021

Almost sure global well-posedness for the energy supercritical NLS on the unit ball of $\mathbb{R}^3$

In this paper, we present two almost sure global well-posedness (GWP) results for the energy supercritical nonlinear Schrödinger equations (NLS) on the unit ball of $\Bbb R^3$ using two different approaches. First, for the NLS with algebraic nonlinearities with the subcritical initial data, we show the almost sure global well-posedness and the invariance of the underlying measures, and establish controls on the growth of Sobolev norms of the solutions. This global result is based on a deterministic local theory and a probabilistic globalization. Second, for the NLS with generic power nonlinearities with critical and supercritical initial conditions, we prove the almost sure global well-posedness, and the invariance of the measure under the solution flows. This global result is built on a compactness argument and the Skorokhod representation theorem.

math.AP↗