arXiv subjects

Weiren Zhao

Publications and source records attributed to Weiren Zhao.

At least 19 recordsLinked to original sources

arXiv2026

Couette Flow with Robin Boundary Condition (I): the viscosity-independent friction

This article is the first paper in the series. In this series of articles, we will examine the influence of the friction factor $\alpha$ at the solid--fluid boundary on the stability of Couette flow. Specifically, we consider the stability of Couette flow in a bounded periodic channel $\mathbb{T} \times [-1,1]$ under Robin-type boundary conditions ($u^2|_{y=\pm 1} = 0$, $[\alpha \partial_n u^1 + u^1]|_{y=\pm1} = f $), where $\alpha$ is the friction factor and $n$ is the unit outer normal vector. In this article, we prove that for a given friction factor $\alpha$, as long as the fluid viscosity coefficient $\nu\ll \alpha$ is sufficiently small, the system is asymptotically stable if the initial perturbation satisfies $\|\omega_{\rm in}\| \leq \epsilon \nu^{1/3}$. Moreover, inviscid damping and enhanced dissipation hold.

math.AP↗

arXiv2026

Long-time behaviour of two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane

In this paper, we study the long-time behavior of solutions to the two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane with Navier-slip boundary conditions. We prove that the total vorticity will approach \begin{align*} -1+\frac{M_2(\omega_{0})}{\nu^{3/2}(1+t)^{5/2}} \bar{\Omega}\left( \frac{x}{\sqrt{\nu(1+t)^3}}, \frac{y}{\sqrt{\nu(1+t)}} \right), \end{align*} where $-1$ is the vorticity of the Couette flow and $\bar{\Omega}$ is the kernel of a Fokker-Planck type operator $\mathcal{L}=\partial_Y^2+\frac32 X\partial_X+\frac12 Y\partial_Y+\frac52-Y\partial_X$. In the proof, we introduce a new idea of studying the spectrum of such type operators with boundary.

math.AP↗

arXiv2026

SynerMedGen: Synergizing Medical Multimodal Understanding with Generation via Task Alignment

Unifying multimodal understanding and generation is a compelling frontier that is beginning to emerge in the medical field. However, the limited existing unified medical models typically treat understanding and generation as disjoint objectives, lacking a meaningful functional synergy. In this work, we identify and address a critical question in unified medical modeling: what form of understanding truly benefits generation. We present SynerMedGen, a unified framework built on the proposed principle of generation-aligned understanding, which synergizes understanding objectives with generation tasks via task alignment. SynerMedGen introduces three generation-aligned understanding tasks and a two-stage training strategy that transfers generation-beneficial representations learned during understanding training to medical image synthesis. Remarkably, even with understanding training alone, our SynerMedGen achieves strong zero-shot performance across 22 medical image synthesis tasks and demonstrates robust generalization to unseen datasets. When combined with generation training, SynerMedGen consistently outperforms state-of-the-art specialized medical image synthesis models as well as recent unified medical models. We also release a large-scale dataset named SynerMed consisting of 1M paired synthesis samples and 2M generation-derived understanding instances to support further research on understanding-generation synergy. Our project can be accessed at https://github.com/piooip/SynerMedGen.

cs.CV↗

arXiv2026

Optimal stability threshold in lower regularity spaces for the Vlasov-Poisson-Fokker-Planck equations

In this paper, we study the optimal stability threshold for the Vlasov-Poisson equation with weak Fokker-Planck collision. We prove that if the initial perturbation is of size $\nu^{\frac{1}{2}}$ in the critical weighted space $H_x^{\log}L^2_{v}(\langle v\rangle^m)$, then the solution remains the same size in the same space. Moreover, a space-time type Landau damping holds, namely, $\|E\|_{L^2_tL^2_x}\lesssim \nu^{\frac{1}{2}}$; and a point-wise type Landau damping holds, namely, $\|E(t)\|_{L^2}\lesssim \nu^{1/2}\langle t\rangle^{-N}$ for any $N>0$ for $t\geq \nu^{-1}$. We also prove that there exists an initial perturbation in $H^{1}_xL^2_v(\langle v\rangle^m)$ with size $\nu^{\frac12-\frac32\epsilon_0}$ for any ${\epsilon_0>0}$, such that the enhanced dissipation fails to hold in the following sense: there is $0 0$. The paper solves the open problem raised in [Bedrossian; arXiv: 2211.13707] about the sharp stability threshold in lower regularity spaces.

math.AP↗

arXiv2026

Traveling waves near shear flows for the inhomogeneous Euler equations with non-constant density

We investigate the existence and nonexistence of traveling wave solutions near monotonic shear flows with non-constant background density for the two-dimensional inhomogeneous Euler equations in a finite channel. For any small $\tau>0$, first, we construct nontrivial traveling waves with velocity and density in $H^{5/2-\tau}$ and $H^{3/2-\tau}$, respectively, showing that inviscid damping fails at these regularities. Second, when the distorted Rayleigh operator has no eigenvalues, we prove that such traveling wave solutions cannot exist in higher regularity spaces ($H^{5/2+\tau}$ for velocity and $H^{3/2+\tau}$ for density).

math.AP↗

arXiv2025

MetaSSL: A General Heterogeneous Loss for Semi-Supervised Medical Image Segmentation

Semi-Supervised Learning (SSL) is important for reducing the annotation cost for medical image segmentation models. State-of-the-art SSL methods such as Mean Teacher, FixMatch and Cross Pseudo Supervision (CPS) are mainly based on consistency regularization or pseudo-label supervision between a reference prediction and a supervised prediction. Despite the effectiveness, they have overlooked the potential noise in the labeled data, and mainly focus on strategies to generate the reference prediction, while ignoring the heterogeneous values of different unlabeled pixels. We argue that effectively mining the rich information contained by the two predictions in the loss function, instead of the specific strategy to obtain a reference prediction, is more essential for SSL, and propose a universal framework MetaSSL based on a spatially heterogeneous loss that assigns different weights to pixels by simultaneously leveraging the uncertainty and consistency information between the reference and supervised predictions. Specifically, we split the predictions on unlabeled data into four regions with decreasing weights in the loss: Unanimous and Confident (UC), Unanimous and Suspicious (US), Discrepant and Confident (DC), and Discrepant and Suspicious (DS), where an adaptive threshold is proposed to distinguish confident predictions from suspicious ones. The heterogeneous loss is also applied to labeled images for robust learning considering the potential annotation noise. Our method is plug-and-play and general to most existing SSL methods. The experimental results showed that it improved the segmentation performance significantly when integrated with existing SSL frameworks on different datasets. Code is available at https://github.com/HiLab-git/MetaSSL.

cs.CV↗

arXiv2025

Landau damping and the long-time collisionless limit of the Vlasov-Poisson-Landau Equation

In this paper, we study the Vlasov-Poisson-Landau Equations on $\mathbb{T}^3\times \mathbb{R}^3$ with small collision frequency $\nu\ll 1$. We prove that for $\nu$-independent perturbations of the global Maxwellians in Gevrey-$2_-$, solutions display uniform-in-$\nu$ Landau damping and enhanced dissipation. Moreover, the collisionless limit holds, that is, as $\nu\to 0_+$ for $0 0$ solutions converge uniformly (and in much stronger norms) to the solution of the Vlasov-Poisson equation with the same initial data. To our knowledge, this work is hence the first justification that the collisionless prediction matches those of collisional plasmas in the nonlinear equations. The interaction between Landau damping and collisions requires several new ideas: (1) an infinite-regularity commuting vector field method, merged with Guo's weighted energy methods for the Landau operator and hypocoercivity to extract the enhanced dissipation; (2) A novel nearly-physical side treatment of the collisionless Vlasov echoes; (3) A new set of decomposition methods to treat the effects of the nonlinear collisions in the Volterra equation for the density (i.e., the ``collisional echoes'') (4) A new quasi-linearization method for treating the effect of the slowly evolving homogeneous modes over long times. As a side result, we also prove Landau damping and enhanced dissipation of $O(\epsilon\nu^{1/3})$ Sobolev-space perturbations of homogeneous distributions that are only $O(\epsilon)$ perturbations of global Maxwellians, generalizing the recent results of Chaturvedi, Luk, and Nguyen. As another side result, our methods also provide a nearly-completely physical-side proof of Mouhot and Villani's theorem in the full range of Gevrey-$3_-$.

math.AP↗

arXiv2025

Nonlinear Evolution Toward the Linear Diffusive Profile in the Presence of Couette Flow

In this paper, we investigate the long-time behavior of solutions to the two-dimensional Navier-Stokes equations with initial data evolving under the influence of the planar Couette flow. We focus on general perturbations, which may be large and of low regularity, including singular configurations such as point vortices, and show that the vorticity asymptotically approaches a constant multiple of the fundamental solution of the corresponding linearized vorticity equation after a long-time evolution determined by the relative Reynolds number.

math.AP↗

arXiv2025

UD-Mamba: A pixel-level uncertainty-driven Mamba model for medical image segmentation

Recent advancements have highlighted the Mamba framework, a state-space model known for its efficiency in capturing long-range dependencies with linear computational complexity. While Mamba has shown competitive performance in medical image segmentation, it encounters difficulties in modeling local features due to the sporadic nature of traditional location-based scanning methods and the complex, ambiguous boundaries often present in medical images. To overcome these challenges, we propose Uncertainty-Driven Mamba (UD-Mamba), which redefines the pixel-order scanning process by incorporating channel uncertainty into the scanning mechanism. UD-Mamba introduces two key scanning techniques: 1) sequential scanning, which prioritizes regions with high uncertainty by scanning in a row-by-row fashion, and 2) skip scanning, which processes columns vertically, moving from high-to-low or low-to-high uncertainty at fixed intervals. Sequential scanning efficiently clusters high-uncertainty regions, such as boundaries and foreground objects, to improve segmentation precision, while skip scanning enhances the interaction between background and foreground regions, allowing for timely integration of background information to support more accurate foreground inference. Recognizing the advantages of scanning from certain to uncertain areas, we introduce four learnable parameters to balance the importance of features extracted from different scanning methods. Additionally, a cosine consistency loss is employed to mitigate the drawbacks of transitioning between uncertain and certain regions during the scanning process. Our method demonstrates robust segmentation performance, validated across three distinct medical imaging datasets involving pathology, dermatological lesions, and cardiac tasks.

eess.IV↗

arXiv2025

Stability threshold of the two-dimensional Couette flow in the whole plane

In this paper, we study the stability threshold for the two-dimensional Couette flow in the whole plane. Our main result establishes that the asymptotic stability threshold is at most $\frac{1}{3}+$ for Sobolev perturbations with additional control over low horizontal frequencies, aligning with the threshold results in periodic domains. As a secondary outcome of our approach, we also prove the asymptotic stability for perturbations in weak Sobolev regularity with size $\nu^{\frac{1}{2}}$.

math.AP↗

arXiv2024

On The Hydrostatic Approximation of Navier-Stokes-Maxwell System with 2D Electronic Fields

In this paper, we prove the local well-posedness of a scaled anisotropic Navier-Stokes-Maxwell system in a two-dimensional striped domain with a transverse magnetic field around $ (0,0,1)$ in Gevrey-2 class. We also justify the limit from the scaled anisotropic equations to the associated hydrostatic system and obtain the precise convergence rate. Then, we prove the global well-posedness for the system and show that small perturbations near $(0,0,1)$ decay exponentially in time. Finally, we show the optimality of the Gevrey-2 regularity by proving the solution to linearized hydrostatic system around shear flows $(V(y),0,0)=(y(1-y),0,0)$ with some initial data $(\zeta, \zeta ^1)$ grows exponentially. More precisely, for some large parameter $ \lvert k \rvert>M\gg 1 $ corresponding to the frequency in $x$, there exists a solution $ h_k(t,x,y)$ of the system \begin{equation*} \begin{cases} \partial_{tt}h_k+\partial_th_k-\partial_{yy}h_k+V(y) \partial_x h_k =0,\\ h_k(0,x,y)=\zeta,\quad \partial_th_k(0,x,y)= \zeta ^1,\\ h_k(t, x,0)=h_k(t, x, 1)=0, \end{cases} \end{equation*} such that for any $s\in [0,\frac{1}{2})$ and $t\in [T_k,T_0)$ with $T_{k}\approx |k|^{s-\frac{1}{2}}\to 0$ as $|k|\to \infty$ and some $T_0$ small and independent of $k$, it satisfies \begin{align*} \lVert h_k(t) \rVert_{L^2 }\geq C \, e^{\sqrt{|k|}t}( \lVert \zeta \rVert_{L^2} + \lVert \zeta ^1 \rVert_{L^2}), \end{align*} for some $C > 0$ independent of $k$.

math.AP↗

arXiv2024

Viscosity driven instability of shear flows without boundaries

In this paper, we study the instability effect of viscous dissipation in a domain without boundaries. We construct a shear flow that is initially spectrally stable but evolves into a spectrally unstable state under the influence of viscous dissipation. To the best of our knowledge, this is the first result of viscosity driven instability that is not caused by boundaries.

math.AP↗

arXiv2024

Improved stability threshold of the Two-Dimensional Couette flow for Navier-Stokes-Boussinesq Systems via quasi-linearization

In this paper, we improve the size requirement of the perturbations for the asymptotic stability of the Couette flow in stratified fluids governed by the two-dimensional Navier-Stokes-Boussinesq system. More precisely, the size of perturbed temperature is improved to $\nu^{2/3}$ from $\nu^{5/6}$ in the paper of Zhang and Zi [J. Math. Pure. Anal. 179:123-182 (2023)]. The idea is the quasi-linearization. The main system is decomposed into two or more equations: a good equation (might be linear) that carries the regularity and size of the initial data and some quasi-linear and nonlinear equations that contain the nonlinear part, which start from zero initial data.

math.AP↗

arXiv2024

Boundary driven instabilities of Couette flows

In this article, we prove that the threshold of instability of the classical Couette flow in $H^s$ for large $s$ is $ν^{1/2}$. The instability is completely driven by the boundary. The dynamic of the flow creates a Prandtl type boundary layer of width $ν^{1/2}$ which is itself linearly unstable. This leads to a secondary instability which in turn creates a sub-layer.

math.AP↗

arXiv2024

Asymptotic Stability of the two-dimensional Couette flow for the Stokes-transport equation in a finite channel

We study the Stokes-transport system in a two-dimensional channel with horizontally moving boundaries, which serves as a reduced model for oceanography and sedimentation. The density is transported by the velocity field, satisfying the momentum balance between viscosity, pressure, and gravity effects, described by the Stokes equation at any given time. Due to the presence of moving boundaries, stratified densities with the Couette flow constitute one class of steady states. In this paper, we investigate the asymptotic stability of these steady states. We prove that if the stratified density is close to a constant density and the perturbation belongs to the Gevrey-3 class with compact support away from the boundary, then the velocity will converge to the Couette flow as time approaches infinity. More precisely, we prove that the horizontal perturbed velocity decays as $\frac{1}{\langle t\rangle^3}$ and the vertical perturbed velocity decays as $\frac{1}{\langle t\rangle^4}$.

math.AP↗

arXiv2024

Asymptotic stability of the three-dimensional Couette flow for the Stokes-transport equation

In this paper, we investigate the asymptotic stability of the three-dimensional Couette flow in a stratified fluid governed by the Stokes-transport equation. We observe that a similar lift-up effect to the three-dimensional Navier-Stokes equation near Couette flow destabilizes the system. We find that the inviscid damping type decay due to the Couette flow together with the damping structure caused by the decreasing background density stabilizes the system. More precisely, we prove that if the initial density is close to a linearly decreasing function in the Gevrey-$\frac{1}{s}$ class with $\frac{1}{2}< s\leq 1$, namely, $\|\varrho_{\mathrm{in}}(X,Y,Z)-(-Y)\|_{\mathcal{G}^{s}}\leq ε$, then the perturbed density remains close to $-Y$. Moreover, the associated velocity field converges to Couette flow $(Y, 0, 0)^{\top}$ with a convergence rate of $\frac{1}{\langle t\rangle^3}$.

math.AP↗

arXiv2024

Landau damping, collisionless limit, and stability threshold for the Vlasov-Poisson equation with nonlinear Fokker-Planck collisions

In this paper, we study the Vlasov-Poisson-Fokker-Planck (VPFP) equation with a small collision frequency $0 < ν\ll 1$, exploring the interplay between the regularity and size of perturbations in the context of the asymptotic stability of the global Maxwellian. Our main result establishes the Landau damping and enhanced dissipation phenomena under the condition that the perturbation of the global Maxwellian falls within the Gevrey-$\frac{1}{s}$ class and obtain that the stability threshold for the Gevrey-$\frac{1}{s}$ class with $s>s_{\mathrm{k}}$ can not be larger than $γ=\frac{1-3s_{\mathrm{k}}}{3-3s_{\mathrm{k}}}$ for $s_{\mathrm{k}}\in [0,\frac{1}{3}]$. Moreover, we show that for Gevrey-$\frac{1}{s}$ with $s>3$, and for $t\ll ν^{\frac13}$, the solution to VPFP converges to the solution to Vlasov-Poisson equation without collision.

math.AP↗

arXiv2023

The transition to instability for stable shear flows in inviscid fluids

In this paper, we study the generation of eigenvalues of a stable monotonic shear flow under perturbations in $C^s$ with $s<2$. More precisely, we study the Rayleigh operator $\mathcal{L}_{U_{m,γ}}= U_{m,γ}\partial_x-U''_{m,γ}\partial_xΔ^{-1}$ associated with perturbed shear flow $(U_{m,γ}(y),0)$ in a finite channel $\mathbb{T}_{2π}\times [-1,1]$ where $U_{m,γ}(y)=U(y)+mγ^2\widetildeΓ(y/γ)$ with $U(y)$ being a stable monotonic shear flow and $\big\{mγ^2\widetildeΓ(y/γ)\big\}_{m\geq 0}$ being a family of perturbations parameterized by $m$. We prove that there exists $m_*$ such that for $0\leq m m_*$ which also leads to instability.

math.AP↗