arXiv subjects

Wenting Huang

Publications and source records attributed to Wenting Huang.

5 recordsLinked to original sources

arXiv2026

Stability threshold for 3D Boussinesq equations with rotation near the Couette flow and stratified temperature

This paper examines the stability threshold at high Reynolds numbers $\textbf{Re}$ for the three-dimensional Boussinesq equations with rotation on the domain $\Omega=\{(x,\,y,\,z)\in \mathbb{T} \times \mathbb{R} \times \mathbb{T}\}$ around the Couette flow $(y,0,0)$ and the vertically stratified temperature $\Theta_s=1+\alpha^2 z$. For the linear system without rotation, stratification not only suppresses the lift-up effect but also exhibits certain dispersion effects, except for some points where degradation occurs, which will bring essential difficulties to nonlinear estimates. In contrast, when rotation is taken into account, we observe that this degeneracy in dispersion effects disappears; furthermore, we can derive dispersive estimates for the second and third components of the simple-zero mode within the velocity field. Additionally, we develop three good unknowns to minimize linear coupling terms as much as possible while mitigating growth induced by linear stretching terms; through constructing a series of multipliers, we achieve enhanced dissipation and inviscid damping effects. In our analysis of the nonlinear system aimed at establishing an improved stability threshold, we utilize quasi-linearization methods to rectify deficiencies in dispersive estimates related to both the first component of velocity and temperature, as well as address regularity issues along vertical directions caused by buoyancy forces and stratification. Consequently, we demonstrate that if initial perturbations in velocity and temperature satisfy $\left\|u_{\mathrm{in}}\right\|_{H^{N+2}\cap W^{N+3,1}}+\left\|\theta_{\mathrm{in}}\right\|_{H^{N+1}\cap W^{N+3,1}}<\delta \mathbf{Re}^{-\frac{14}{15}}$, for any $N\geq 11$ and some $\delta>0$ independent of $\mathbf{Re}$, then the solution to the 3D Boussinesq equations with rotation is nonlinearly stable without transitioning away from the steady state.

math.AP↗

arXiv2024

Stability of the Couette flow for 3D Navier-Stokes equations with rotation

Rotation significantly influences the stability characteristics of both laminar and turbulent shear flows. This study examines the stability threshold of the three-dimensional Navier-Stokes equations with rotation, in the vicinity of the Couette flow at high Reynolds numbers ($\mathbf{Re}$) in the periodical domain $\mathbb{T} \times \mathbb{R} \times \mathbb{T}$, where the rotational strength is equivalent to the Couette flow. Compared to the classical Navier-Stokes equations, rotation term brings us more two primary difficulties: the linear coupling term involving in the equation of $u^2$ and the lift-up effect in two directions. To address these difficulties, we introduce two new good unknowns that effectively capture the phenomena of enhanced dissipation and inviscid damping to suppress the lift-up effect. Moreover, we establish the stability threshold for initial perturbation $\left\|u_{\mathrm{in}}\right\|_{H^{\sigma}} < \delta \mathbf{Re}^{-2}$ for any $\sigma > \frac{9}{2}$ and some $\delta=\delta(\sigma)>0$ depending only on $\sigma$.

math.AP↗

arXiv2024

On Temporal Decay of Compressible Hookean Viscoelastic Fluids with Relatively Large Elasticity Coefficient

Recently, Jiang--Jiang (J. Differential Equations 282, 2021) showed the existence of unique strong solutions in spatial periodic domain (denoted by $\mathbb{T}^3$), whenever the elasticity coefficient is larger than the initial velocity perturbation of the rest state. Motivated by Jiang--Jiang's result, we revisit the Cauchy problem of the compressible viscoelastic fluids in Lagrangian coordinates. Employing an energy method with temporal weights and an additional asymptotic stability condition of initial density in Lagrangian coordinates, we extend the Jiang--Jiang's result with exponential decay-in-time in $\mathbb{T}^3$ to the one with algebraic decay-in-time in the whole space $\mathbb{R}^3$. Thanks to the algebraic decay of solutions established by the energy method with temporal weights, we can further use the spectral analysis to improve the temporal decay rate of solutions. In particular, we find that the $k$-th order spatial derivatives of both the density and deformation perturbations converge to zero in $L^2(\mathbb{R}^3)$ at a rate of $(1+t)^{-\frac{3}{4}-\frac{k+1}{2}}$, which is faster than the decay rate $(1 +t)^{-\frac{3}{4}-\frac{k}{2}}$ obtained by Hu--Wu (SIAM J. Math. Anal. 45, 2013) for $k=0$ and $ 1$. In addition, it's well-known that the decay rate $(1+t)^{-\frac{3}{4}-\frac{k}{2}}$ of the density perturbation is optimal in the compressible Navier--Stokes equations (A.~Matsumura, T.~Nishida, Proc. Jpn. Acad. Ser-A. 55, 1979). Therefore, our faster temporal decay rates indicate that the elasticity accelerates the decay of the density perturbation after the rest state of a compressible viscoelastic fluid being perturbed.

math.AP↗

arXiv2024

Towards a Comprehensive Benchmark for Pathological Lymph Node Metastasis in Breast Cancer Sections

Advances in optical microscopy scanning have significantly contributed to computational pathology (CPath) by converting traditional histopathological slides into whole slide images (WSIs). This development enables comprehensive digital reviews by pathologists and accelerates AI-driven diagnostic support for WSI analysis. Recent advances in foundational pathology models have increased the need for benchmarking tasks. The Camelyon series is one of the most widely used open-source datasets in computational pathology. However, the quality, accessibility, and clinical relevance of the labels have not been comprehensively evaluated. In this study, we reprocessed 1,399 WSIs and labels from the Camelyon-16 and Camelyon-17 datasets, removing low-quality slides, correcting erroneous labels, and providing expert pixel annotations for tumor regions in the previously unreleased test set. Based on the sizes of re-annotated tumor regions, we upgraded the binary cancer screening task to a four-class task: negative, micro-metastasis, macro-metastasis, and Isolated Tumor Cells (ITC). We reevaluated pre-trained pathology feature extractors and multiple instance learning (MIL) methods using the cleaned dataset, providing a benchmark that advances AI development in histopathology.

eess.IV↗

arXiv2024

On the Sobolev stability threshold for 3D Navier-Stokes equations with rotation near the Couette flow

Rotation is a crucial characteristic of fluid flow in the atmosphere and oceans, which is present in nearly all meteorological and geophysical models. The global existence of solutions to the 3D Navier-Stokes equations with large rotation has been established through the dispersion effect resulting from Coriolis force (i.e., rotation). In this paper, we investigate the dynamic stability of periodic, plane Couette flow in the three-dimensional Navier-Stokes equations with rotation at high Reynolds number $\mathbf{Re}$. Our aim is to determine the stability threshold index on $\mathbf{Re}$: the maximum range of perturbations within which the solution remains stable. Initially, we examine the linear stability effects of a linearized perturbed system. Comparing our results with those obtained by Bedrossian, Germain, and Masmoudi [Ann. Math. 185(2): 541--608 (2017)], we observe that mixing effects (which correspond to enhanced dissipation and inviscid damping) arise from Couette flow while Coriolis force acts as a restoring force inducing a dispersion mechanism for inertial waves that cancels out lift-up effects occurred at zero frequency velocity. This dispersion mechanism exhibits favorable algebraic decay properties distinct from those observed in classical 3D Navier-Stokes equations. Consequently, we demonstrate that if initial data satisfies $\left\|u_{\mathrm{in}}\right\|_{H^{\sigma}}<\delta \mathbf{Re}^{-1}$ for any $\sigma>\frac{9}{2}$ and some $\delta=\delta(\sigma)>0$ depending only on $\sigma$, then the solution to the 3D Navier-Stokes equations with rotation is global in time without transitioning away from Couette flow. In this sense, Coriolis force contributes as a factor enhancing fluid stability by improving its threshold from $\frac{3}{2}$ to 1.

math.AP↗