arXiv subjects

Qingtang Su

Publications and source records attributed to Qingtang Su.

11 recordsLinked to original sources

arXiv2026

Full-radius dimension-free maximal inequalities for discrete Euclidean balls

For every $1<p\le\infty$, we prove dimension-free maximal inequalities over all radii for normalized averages over Euclidean balls in $\mathbb Z^d$. In particular, this settles the $\ell^2$ question attributed to Stein. The proof uses a two-saddle expansion at integer squared radii to compare ball multipliers with normalized discrete Gaussians at the zero and parity frequencies. First-order estimates give the full-radius $\ell^2$ bound. For $1<p<2$, we combine higher-order residual estimates with dimension-uniform $\ell^1$ bounds for the residuals and levelwise interpolation to obtain the full range.

math.CA↗

arXiv2026

Global regularity for 2D gravity water waves with two retreating point vortices

We prove global well-posedness for the two-dimensional infinite-depth gravity water wave system coupled to a pair of point vortices of opposite strengths. The vortex dipole is initially placed deep below the free surface and oriented so that its leading motion is away from the interface. We show that the vortices retreat almost linearly, and that the velocity induced on the free boundary is therefore integrable in time. Combining this decay with Wu's cubic formulation, together with a transition-of-derivatives and localization argument, we obtain global bounds for small localized perturbations of the flat surface.

math.AP↗

arXiv2025

A New Structure for the 2D water wave equation: Energy stability and Global well-posedness

We study the two-dimensional gravity water waves with a one-dimensional interface with small initial data. Our main contributions include the development of two novel localization lemmas and a Transition-of-Derivatives method, which enable us to reformulate the water wave system into the following simplified structure: $$(D_t^2-iA\partial_{\alpha})\theta=i\frac{t}{\alpha}|D_t^2\zeta|^2D_t\theta+R$$ where $R$ behaves well in the energy estimate. As a key consequence, we derive the uniform bound $$ \sup_{t\geq 0}\Big(\norm{D_t\zeta(\cdot,t)}_{H^{s+1/2}}+\norm{\zeta_{\alpha}(\cdot,t)-1}_{H^s}\Big)\leq C\epsilon, $$ which enhances existing global uniform energy estimates for 2D water waves by imposing less restrictive constraints on the low-frequency components of the initial data.

math.AP↗

arXiv2020

Nonlinear modulational instabililty of the Stokes waves in 2d full water waves

The well-known Stokes waves refer to periodic traveling waves under the gravity at the free surface of a two dimensional full water wave system. In this paper, we prove that small-amplitude Stokes waves with infinite depth are nonlinearly unstable under long-wave perturbations. Our approach is based on the modulational approximation of the water wave system and the instability mechanism of the focusing cubic nonlinear Schrödinger equation.

math.AP↗

arXiv2020

Developing Univariate Neurodegeneration Biomarkers with Low-Rank and Sparse Subspace Decomposition

Cognitive decline due to Alzheimer's disease (AD) is closely associated with brain structure alterations captured by structural magnetic resonance imaging (sMRI). It supports the validity to develop sMRI-based univariate neurodegeneration biomarkers (UNB). However, existing UNB work either fails to model large group variances or does not capture AD dementia (ADD) induced changes. We propose a novel low-rank and sparse subspace decomposition method capable of stably quantifying the morphological changes induced by ADD. Specifically, we propose a numerically efficient rank minimization mechanism to extract group common structure and impose regularization constraints to encode the original 3D morphometry connectivity. Further, we generate regions-of-interest (ROI) with group difference study between common subspaces of $Aβ+$ AD and $Aβ-$ cognitively unimpaired (CU) groups. A univariate morphometry index (UMI) is constructed from these ROIs by summarizing individual morphological characteristics weighted by normalized difference between $Aβ+$ AD and $Aβ-$ CU groups. We use hippocampal surface radial distance feature to compute the UMIs and validate our work in the Alzheimer's Disease Neuroimaging Initiative (ADNI) cohort. With hippocampal UMIs, the estimated minimum sample sizes needed to detect a 25$\%$ reduction in the mean annual change with 80$\%$ power and two-tailed $P=0.05$ are 116, 279 and 387 for the longitudinal $Aβ+$ AD, $Aβ+$ mild cognitive impairment (MCI) and $Aβ+$ CU groups, respectively. Additionally, for MCI patients, UMIs well correlate with hazard ratio of conversion to AD ($4.3$, $95\%$ CI=$2.3-8.2$) within 18 months. Our experimental results outperform traditional hippocampal volume measures and suggest the application of UMI as a potential UNB.

cs.CV↗

arXiv2020

On the transition of the Rayleigh-Taylor instability in 2d water waves

In this paper we prove the existence of water waves with sign-changing Taylor sign coefficients, that is, the strong Taylor sign holds initially, while breaks down at a later time, and vice versa. Such a phenomenon can be regarded as the transition between the stable and unstable regime in the sense of Rayleigh-Taylor of water waves. As a byproduct, we prove the sharp wellposedness of 2d water waves in Gevrey-2 spaces.

math.AP↗

arXiv2019

Justification of Peregrine soliton from full water waves

The Peregrine soliton $Q(x,t)=e^{it}(1-\frac{4(1+2it)}{1+4x^2+4t^2})$ is an exact solution of the 1d focusing nonlinear schrödinger equation (NLS) $iB_t+B_{xx}=-2|B|^2B$, having the feature that it decays to $e^{it}$ at the spatial and time infinities, and with a peak and troughs in a local region. It is considered as a prototype of the rogue waves by the ocean waves community. The 1D NLS is related to the full water wave system in the sense that asymptotically it is the envelope equation for the full water waves. In this paper, working in the framework of water waves which decay non-tangentially, we give a rigorous justification of the NLS from the full water waves equation in a regime that allows for the Peregrine soliton. As a byproduct, we prove long time existence of solutions for the full water waves equation with small initial data in space of the form $H^s(\mathbb{R})+H^{s'}(\mathbb{T})$, where $s\geq 4, s'>s+\frac{3}{2}$.

math.AP↗

arXiv2018

Long time behavior of 2d water waves with point vortices

In this paper, we study the motion of the two dimensional inviscid incompressible, infinite depth water waves with point vortices in the fluid. We show that Taylor sign condition $-\frac{\partial P}{\partial \boldmath{n}}\geq 0$ can fail if the point vortices are sufficient close to the free boundary, so the water waves could be subject to the Taylor instability. Assuming the Taylor sign condition, we prove that the water wave system is locally wellposed in Sobolev spaces. Moreover, we show that if the water waves is symmetric with a certain symmetric vortex pair traveling downward initially, then the free interface remains smooth for a long time, and for initial data of size $ε\ll 1$, the lifespan is at least $O(ε^{-2})$.

math.AP↗

arXiv2018

Dynamics of subcritical threshold solutions for energy-critical NLS

In this paper, we study the dynamics of subcritical threshold solutions for focusing energy critical NLS on $\mathbb{R}^d$ ($d\geq 5$) with nonradial data. This problem with radial assumption was studied by T. Duyckaerts and F. Merle in \cite{DM} for $d=3,4,5$ and later by D. Li and X. Zhang in \cite{LZ} for $d \geq 6$. We generalize the conclusion for the subcritical threshold solutions by removing the radial assumption for $d\geq 5$. A key step is to show exponential convergence to the ground state $W(x)$ up to symmetries if the scattering phenomenon does not occur. Remarkably, an interaction Morawetz-type estimate are applied.

math.AP↗

arXiv2012

Global well-posedness and scattering for defocusing, cubic NLS in $\mathbb{R}^3$

We prove global well-posedness and scattering for the defocusing, cubic NLS on $\mathbb{R}^3$ with initial data in $H^s(\mathbb{R}^3)$ for $s>49/74$. The proof combines the ideas of resonance decomposition in \cite{CKSTT4} and linear-nonlinear decomposition in \cite{ben1}\cite{roy} together with the idea of large time iteration.

math.AP↗

arXiv2012

Sharp global well-posedness for 1D NLS with derivatives

We show that the 1d derivative nonlinear Schrödinger equation (\ref{equ}) is globally well-posed in $H^s(\mathbb{R})$ for $s\geq 1/2$. We use the linear-nonlinear decomposition method to take advantage of the local smoothing effect of the nonlinearity, which enables us to establish a refined version of the almost conservation law. Note that $H^{1/2}$ is the endpoint that we have uniform continuous for the solution map and hence our result is sharp.

math.AP↗