Quantitative Unique Continuation on Simplex and $\mathbb{Z}^n$
In this work, we establish the quantitative unique continuation on both $n$-dimensional simplex lattice $Δ^{(n)}_{N}$ and $\mathbb{Z}^n$ for arbitrary $n\geq 3$.
math.AP↗
arXiv subjects
Zhifei Zhang
Publications and source records attributed to Zhifei Zhang.
Sharp $H_{x}^{s}$ Ill-posedness of the Hard-sphere Boltzmann Equation
We investigate the ill-posedness mechanism of the hard-sphere Boltzmann equation in $H_{x}^{s}$ Sobolev space. Via a direct construction, we prove a strong-weak type ill-posedness result in the low-regularity regime $s<1$, establishing a sharp threshold in connection to the local $s>1$ well-posedness result [11]. Instead of originating from the large-velocity growth of the collision kernel, this illposedness is generated by the loss term and dispersive effects. Consequently, we prove a dispersion-driven nonlinear instability mechanism for the hard-sphere Boltzmann equation, and provide a capstone of the ill-posedness series [18,20].
math.AP↗
SupGRPO: Enhancing GRPO with Matching-based Online SFT for Text Spotting
Text spotting requires both accurate text recognition and precise spatial localization. Current specialised spotters excel at predicting tight bounding boxes in natural scenes, but falter on complex or artistic text, whereas multimodal large language models (MLLMs) possess strong recognition capabilities yet remain weak at localisation. To equip the text spotter with general and powerful recognition capabilities and to maximize its localization ability, we explore two MLLM-based fine-tuning methods: Supervised Fine-Tuning (SFT) and reinforcement learning fine-tuning based on Group Relative Policy Optimisation (GRPO). An interesting finding is that SFT is less effective than GRPO at enhancing recognition, while GRPO is less effective than SFT at enhancing detection. To compensate for each other's shortcomings, we introduce a joint training strategy, SupGRPO, which simultaneously optimizes the model using both SFT and GRPO. SupGRPO employs the specially designed reward functions and develops a matching-based online SFT applied solely to coordinate tokens. It both mitigates the reward sparsity problem of GRPO and avoids the instance order dependency problem of SFT. To evaluate particularly challenging cases, we curate ATS, a dataset for artistic text spotting. Experiments demonstrate that SupGRPO improves both text recognition and detection, and attains superior performance. Our code and dataset will be released at https://github.com/Psycho-9/SupGRPO.
cs.CV↗
Exponential mixing for the stochastic Navier--Stokes equation with localized noise
This paper studies the 2D Navier--Stokes equation on a bounded domain with Navier-slip boundary conditions, driven by spatially localized stationary forcing generated by a stochastic heat equation. We prove exponential mixing for the associated Navier--Stokes--heat system, and consequently exponential convergence of the velocity law under stationary forcing. The main difficulty is that the white noise is confined to a subdomain and reaches the velocity indirectly through the heat component. To address the degeneracy, the proof combines Malliavin calculus with PDE control theory. The key ingredient is a stabilization scheme that converts localized Navier--Stokes controls into time-regular controls compatible with the heat dynamics.
math.PR↗
Benchmarking Clinical Decision Pathway Adherence in Large Language Models
Following clinical decision pathways (CDPs) defined by clinical practice guidelines is essential for safe and reliable medical decision-making. However, existing medical large language model (LLM) benchmarks mainly evaluate final-answer accuracy, providing limited evaluation of models' ability to adhere to guidelines. To address this gap, we introduce MEGA-CDP, a benchmark for evaluating whether medical LLMs can generate guideline-adherent CDPs using provided guidelines as references. MEGA-CDP is constructed from 2,274 English and Chinese clinical practice guidelines through a guideline-to-case pipeline, yielding 42,353 clinical cases with explicit reference CDPs. It supports both single-turn vignette and multi-turn interactive settings, and introduces a CDP-oriented evaluation framework for measuring pathway consistency. Experiments on 16 representative LLMs show that reliable clinical decision support remains challenging for current models, demonstrating the need for CDP-oriented evaluation and the value of MEGA-CDP for advancing guideline adherence in medical LLMs.
cs.CL↗
Asymptotic limit for the isotropic-nematic problem with anisotropic elasticity
We consider the isotropic-nematic interface problem for liquid crystals based on the Landau-de Gennes model. We show that when the anisotropic elastic constant $L_2$ is negative, then the surface tension strength of the isotropic-nematic interface is proportional to $\sqrt{L_1+\frac{2}{3}{L}_2}$, and the homeotropic alignment is more favored near the interface. This result gives a rigorous confirmation for de Gennes' formal derivation in \cite{deGen1971} on the isotropic-nematic tension strength and the anchoring alignment condition in the case of $L_2<0$.
math.AP↗
PixelControl: Fine-Grained Condition Fidelity in Text-to-Image Diffusion
Controllable text-to-image diffusion models can often follow the global layout of spatial conditions, yet still violate fine-grained structures such as object boundaries, thin contours, and medium/small conditioned regions. This limitation is especially problematic for VAE-based latent diffusion, where spatial compression can weaken high-frequency and low-area condition signals. We propose PixelControl, a pixel-space controllable diffusion framework for fine-grained condition fidelity. Built on a PixelDiT-style backbone, PixelControl avoids the latent bottleneck and introduces two complementary designs. First, Structure-Aware Control Injection derives a condition structure map and uses it to strengthen injected control residuals around spatially sensitive regions. Second, Multi-Scale Pyramid Cycle Loss verifies generated images against condition-derived structures across multiple resolutions, balancing global layout consistency with local boundary and detail accuracy. PixelControl supports depth, segmentation, edge, and their combinations through modality-specific control branches with lightweight gated fusion. Experiments across depth, segmentation, and edge control show that PixelControl improves structural fidelity and visual quality over existing controllable generation methods, with especially strong gains on boundaries and medium/small conditioned regions. The project page can be found at: https://linxin0.github.io/pixelcontrol_homepage/pixelcontrol-site/
cs.CV↗
Long-Wave Spectral Instability of Shear Layers for the Compressible Euler Equations
We study the long-wave spectral instability of the two-dimensional compressible Euler equations around smooth monotone shear layers. We construct decaying half-line solutions of the compressible Rayleigh equation through a long-wave expansion and derive a second-order expansion of the matching Wronskian. For every fixed Mach number $m>0$, we prove the existence of unstable modes for sufficiently small wavenumbers. For $m<\sqrt2$, this holds for a class of profiles, with $c_i$ tending to a positive constant as $α\to0$. At $m=\sqrt{2}$, the profile $U_s(Y)=\tanh Y$ admits an unstable mode with $c\to0$ and $c_i$ of order $α^{1/3}$. For $m>\sqrt2$, the same profile remains unstable, with $c\to c_*(m)\in(0,1)$ and $c_i>0$ of order $α$. The corresponding temporal growth rates are of order $α$, $α^{4/3}$ and $α^2$, respectively, showing a change in the long-wave instability scaling at $m=\sqrt2$. In the zero-thickness limit, the supercritical unstable eigenvalue approaches the real axis, consistently with the stability results for supersonic compressible vortex sheets in \cite{CS1,CS2}.
math.AP↗
On the absence of point defects in biaxial Landau--de Gennes models
We study local minimizers of a sextic-potential Landau--de Gennes energy for nematic liquid crystals in the small-elastic-constant limit. Under the uniform energy and $L^\infty$ bounds, these minimizers converge to a locally energy-minimizing harmonic map $\mathbf{Q}_0$ into a biaxial vacuum manifold. The main result of this paper is that such a limiting map $\mathbf{Q}_0$ has no interior point singularities. The proof relies on a geometric identification of the lifted Frobenius metric on the universal cover $\mathbb{S}^3$ with a rescaled Berger metric. For a hypothetical tangent cone at a point singularity, its link is a nonconstant harmonic two-sphere into the Berger sphere. We construct a smooth variation field adapted to the Hopf direction and show that it induces a quantitative instability estimate, in contradiction with the shifted stability inequality inherited from local minimality. This replaces the usual round-sphere test fields by a target-specific construction and rules out interior point defects in the biaxial setting. The result is sharp in view of the well-known existence of point defects in the uniaxial theory.
math.AP↗
Exponential mixing for the randomly forced NLS equation
This paper investigates exponential mixing of the invariant measure for randomly forced nonlinear Schrödinger equation, with damping and random noise localized in space. Our study emphasizes the crucial role of exponential asymptotic compactness and control properties in establishing the ergodic properties of random dynamical systems. This work extends the series [16, 47] on the statistical behavior of randomly forced dispersive equations.
math.AP↗
Exponential mixing for Korteweg-de Vries equation with localized noise
We establish exponential mixing for the randomly forced and weakly damped KdV equation in $L^2(\mathbb{T})$. The noise is bounded, localized, and degenerate in high frequencies. Our proof relies on a general probabilistic framework in [11,33], nonlinear smoothing for KdV and its linearization via normal form transformation, and stabilization of the system by localized force. This paper continues a series of works connecting asymptotic compactness, control theory, and ergodicity and mixing for randomly forced dispersive PDEs.
math.AP↗
Neutral curves and traveling waves in plane Poiseuille flow
We study the spectrum of the Orr--Sommerfeld operator associated with the incompressible plane Poiseuille flow in the high-Reynolds-number regime. In the Tollmien--Schlichting eigenvalue region, we prove the existence and uniqueness of the lower and upper branches of the neutral curve. For each sufficiently small fixed wavenumber $α$, the viscosities corresponding to the lower and upper neutral branches satisfy $ν\sim |α|^7$ and $ν\sim |α|^{11}$, respectively. Equivalently, for each sufficiently small viscosity $ν$, the corresponding lower and upper neutral wavenumbers satisfy $α^2\sim ν^{2/7}$ and $α^2\sim ν^{2/11}$, respectively. We also establish the simplicity of the neutral eigenvalues and verify the transversal crossing condition on both neutral branches. The proof is based on a boundary-adapted version of the Rayleigh--Airy iteration scheme, together with precise expansions and refined estimates for the correction terms and their parameter derivatives. These spectral results verify the assumptions required in the classical Hopf bifurcation framework of Joseph--Sattinger \cite{JS1972} and Iooss \cite{Iooss1972}, and hence yield traveling-wave solutions bifurcating from the plane Poiseuille flow at the neutral points.
math.AP↗
Self-similar blow-up solutions of $d$-dimensional incompressible Euler equations with $C^{1,\left(1-2/d\right)-}$ velocity
We investigate self-similar blow-up solutions to the $d$-dimensional axisymmetric incompressible Euler equations without swirl for $d\ge 3$. For any $α\in(0, α_d)$ with $α_d=1-2/d$, we construct a self-similar blow-up solution whose initial velocity field satisfies $u_0\in C^{1,α}_{\rm loc}(\mathbb R^d)\cap C^\infty(\mathbb R^d\setminus\{0\})$. Our construction relies on a fixed-point argument formulated for the self-similar profile equations, which form a coupled elliptic-transport system. Specifically, the transport equation recovers the vorticity profile from given data along characteristic curves, while the elliptic equation reconstructs the velocity field via Newtonian potentials defined in an auxiliary $(d+4)$-dimensional space. The main challenge consists in choosing appropriate function spaces that remain invariant under such nonlinear compositions and that simultaneously capture the exact singular behavior near the origin and the symmetry axis. Furthermore, we establish a finite-codimensional stability result for the self-similar profiles obtained above. As a consequence, after suitable truncation and correction of finitely many unstable modes, we obtain finite-energy blow-up solutions with initial velocity in $C^{1,α}(\mathbb R^d)\cap C^\infty(\mathbb R^d\setminus\{0\})\cap L^2(\mathbb R^d)$ and compactly supported initial vorticity. These solutions are asymptotically self-similar near the blow-up time.
math.AP↗
Existence of weak solutions for two-phase matrix-valued harmonic map flows
We study the existence of weak solutions to a two-phase matrix-valued harmonic map flow with lifespan determined by that of the prescribed interface. This system arises as the limiting equation for the matrix-valued Keller--Rubinstein--Sternberg problem studied in (Invent. Math., 233(1):1--80, 2023). Our construction is based on a modified minimizing movement scheme. Precisely, we discretize time and build approximate solutions by interpolating minimizers of the associated variational problem on each time step.
math.AP↗
Localization and unique continuation for the Anderson-Bernoulli model with long-range hopping on $\mathbb{Z}$
In this paper, we study Anderson localization near the spectral edge for the Anderson-Bernoulli model on $\mathbb{Z}$ with long-range hopping. When the hopping has a rational Laurent symbol, a quantitative version of the unique continuation principle can be proved, and localization occurs. For the unique continuation in the general case, we give some counterexamples and prove a weaker result for hopping that decays faster than exponential rate. To the best of our knowledge, this is the first localization result for the long-range Anderson model with pure Bernoulli potentials.
math.SP↗
Stratification and rectifiability of harmonic map flows via tangent measures
In this paper, we investigate the stratification theory for ``suitable solutions" of harmonic map flows based on the spatial symmetry of tangent measures. Building on the quantitative stratifications and Reifenberg-rectifiable theory developed by Naber and Valtorta in breakthrough research of harmonic maps (\emph{Ann. Math.} 185 (2017), 131-227), we prove that each time slice of the singular set in our model is rectifiable. By making some additional assumptions about the target manifolds to exclude specific tangent flows and measures, we can also obtain a sharp regularity of suitable solutions for harmonic map flows.
math.AP↗
Solutions of the 3D inhomogeneous incompressible Navier-Stokes system with initial velocity in $VMO^{-1}$
In this paper, we establish local existence of strong solutions for the three-dimensional inhomogeneous incompressible Navier-Stokes equations with initial data $(ρ_0,u_0)$ lying in $C^1 \times (L^2 \cap VMO^{-1})$, where $ρ_0$ has a positive lower bound. Furthermore, if $ρ_0 \in C^2$ and $||ρ_0-1||_{L^\infty}+||u_0||_{BMO^{-1}}$ is sufficiently small, we prove global existence of the solution. To achieve this, we employ an estimate for the transport equation to obtain regularity for the density and apply a new freezing-coefficient method for the momentum equation.
math.AP↗
Critical mass threshold for the 2D Patlak-Keller-Segel-Navier-Stokes system
In this paper, we investigate critical mass threshold for the Patlak-Keller-Segel-Navier-Stokes system on the two-dimensional whole space and obtain global existence of strong solutions if the initial mass is less than or equal to $8π$, regardless of the initial norm of the velocity. One new observation is that the local mass of the density function rearrangement satisfies a good inequality that is independent of velocity; and then an improved maximum principle is applied by choosing a nice auxiliary function.
math.AP↗