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Jiaxi Huang

Publications and source records attributed to Jiaxi Huang.

13 recordsLinked to original sources
arXiv2025

Local well-posedness of the skew mean curvature flow for large data

The skew mean curvature flow is an evolution equation for $d$ dimensional ma\-nifolds embedded in $\mathbb{R}^{d+2}$ (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove large data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $d\geq 2$. This is achieved by introducing several new ideas: (i) a time discretization method to establish the existence of smooth solutions, (ii) constructing the orthonormal frame by a parallel transport method and a lifting criterion, (iii) introducing intrinsic fractional function spaces $X^s\subset H^s$ on a noncompact manifold for any $s>\frac{d}{2}$, such that the $X^s$-norm of the second fundamental form can be propagated well along the quasilinear Schrödinger flow, (iv) deriving a difference equation to prove the uniqueness result for solutions $F\in C^2$, which is independent in the choices of gauge. Our method turns out to be more robust for large data problem.

math.AP↗
arXiv2025

Q-Doc: Benchmarking Document Image Quality Assessment Capabilities in Multi-modal Large Language Models

The rapid advancement of Multi-modal Large Language Models (MLLMs) has expanded their capabilities beyond high-level vision tasks. Nevertheless, their potential for Document Image Quality Assessment (DIQA) remains underexplored. To bridge this gap, we propose Q-Doc, a three-tiered evaluation framework for systematically probing DIQA capabilities of MLLMs at coarse, middle, and fine granularity levels. a) At the coarse level, we instruct MLLMs to assign quality scores to document images and analyze their correlation with Quality Annotations. b) At the middle level, we design distortion-type identification tasks, including single-choice and multi-choice tests for multi-distortion scenarios. c) At the fine level, we introduce distortion-severity assessment where MLLMs classify distortion intensity against human-annotated references. Our evaluation demonstrates that while MLLMs possess nascent DIQA abilities, they exhibit critical limitations: inconsistent scoring, distortion misidentification, and severity misjudgment. Significantly, we show that Chain-of-Thought (CoT) prompting substantially enhances performance across all levels. Our work provides a benchmark for DIQA capabilities in MLLMs, revealing pronounced deficiencies in their quality perception and promising pathways for enhancement. The benchmark and code are publicly available at: https://github.com/cydxf/Q-Doc.

cs.CV↗
arXiv2025

CoCoL: A Communication Efficient Decentralized Collaborative Method for Multi-Robot Systems

Collaborative learning enhances the performance and adaptability of multi-robot systems in complex tasks but faces significant challenges due to high communication overhead and data heterogeneity inherent in multi-robot tasks. To this end, we propose CoCoL, a Communication efficient decentralized Collaborative Learning method tailored for multi-robot systems with heterogeneous local datasets. Leveraging a mirror descent framework, CoCoL achieves remarkable communication efficiency with approximate Newton-type updates by capturing the similarity between objective functions of robots, and reduces computational costs through inexact sub-problem solutions. Furthermore, the integration of a gradient tracking scheme ensures its robustness against data heterogeneity. Experimental results on three representative multi robot collaborative learning tasks show the superiority of the proposed CoCoL in significantly reducing both the number of communication rounds and total bandwidth consumption while maintaining state-of-the-art accuracy. These benefits are particularly evident in challenging scenarios involving non-IID (non-independent and identically distributed) data distribution, streaming data, and time-varying network topologies.

cs.RO↗
arXiv2024

Asymptotic behavior of 3-D evolutionary model of Magnetoelasticity for small data

In this article, we consider the evolutionary model for magnetoelasticity with vanishing viscosity/damping, which is a nonlinear dispersive system. The global regularity and scattering of the evolutionary model for magnetoelasticity under small size of initial data is proved. Our proof relies on the idea of vector-field method due to the quasilinearity and the presence of convective term. A key observation is that we construct a suitable energy functional including the mass quantity, which enable us to provide a good decay estimates for Schrödinger flow. In particular, we establish the asymptotic behavior in both mass and energy spaces for Schrödinger map, not only for gauged equation.

math.AP↗
arXiv2023

Almost global well-posedness of 2-D Ericksen-Leslie's hyperbolic liquid crystal model for small data

This article is devoted to the two dimensional simplified Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model, where the direction $d$ of liquid crystal molecules satisfies a wave map equation with an acoustical metric. We established the almost global well-posedness for small and smooth initial data near the constant equilibrium. Our proof relies on the idea of vector-field method and ghost weight method. There are two key ingredients in our proof: (i) Inspired by the gauge theory in Tataru \cite{Tataru,Tataru05}, we reformulate the wave map equation into a free wave equation with acoustical metric, where the nonlinearity is annihilated due to the geometry of $\mathbb S^1$; (ii) Motivated by the ghost weight method in Alinhac \cite{A01}, we introduce a new and important ``good unknown", the velocity $u$, which provides the additional dissipation $u/\langle t-r\rangle\in L^2_tL^2_x$. These new observations turn out to be extremely crucial in resolving the system in low dimensions.

math.AP↗
arXiv2023

On the hydrodynamics of hyperbolic liquid crystal elastomers

Liquid crystal elastomers are special cross-linked polymer materials combining the large elastic deformability of elastomers with the orientational orders of liquid crystals. This model exhibits markedly different phenomena than the liquid crystal model due to the strong coupling between mechanical elastic deformation and orientation vector. Our results are threefold. (i) First we derive the hydrodynamics of liquid crystal elastomers with inertial effect by energetic variational approach inspired by Chun Liu [33]. (ii) Then we study the hyperbolic liquid crystal elastomers from mathematical point, which is a fully quasilinear hyperbolic system. The local well-posedness for large data is proved by zero-viscosity limit method. (iii) Finally, we show the global regularity for small and smooth initial data near the constant equilibrium in three dimensions, which is achieved by carefully investigating the structure of second-order material derivatives and the cancellations in the system. This is the first global result on hyperbolic liquid crystal elastomers.

math.AP↗
arXiv2023

Asymptotic behaviors of incompressible Schrödinger flow for small data in three dimensions

The incompressible Schrödinger flow is a Madelung's hydrodynamical form of quantum mechanics, which can simulate classical fluids with particular advantage in its simplicity and its ability of capturing thin vortex dynamics. This model enables robust simulation of intricate phenomena such as vortical wakes and interacting vortex filaments. In this article, we prove the global regularity and asymptotic behaviors for incompressible Schrödinger flow with small and localized data in three dimensions. We choose a suitable gauge to rewrite the system, and then use Fourier analysis and vector fields method to prove global existence and asymptotic behaviors.

math.AP↗
arXiv2022

Global regularity of Skew mean curvature flow for small data in $d\geq 4$ dimensions

The skew mean curvature flow is an evolution equation for a $d$ dimensional manifold immersed into $\mathbb{R}^{d+2}$, and which moves along the binormal direction with a speed proportional to its mean curvature. In this article, we prove small data global regularity in low-regularity Sobolev spaces for the skew mean curvature flow in dimensions $d\geq 4$. This extends the local well-posedness result in \cite{HT}.

math.AP↗
arXiv2022

Local well-posedness of the Skew mean curvature flow for small data in $d\geq 2$ dimensions

The skew mean curvature flow is an evolution equation for $d$ dimensional manifolds embedded in $\mathbb{R}^{d+2}$ (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In an earlier paper, the authors introduced a harmonic/Coulomb gauge formulation of the problem, and used it to prove small data local well-posedness in dimensions $d \geq 4$. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $d\geq 2$. This is achieved by introducing a new, heat gauge formulation of the equations, which turns out to be more robust in low dimensions.

math.AP↗
arXiv2021

Local well-posedness of skew mean curvature flow for small data in $d\geq 4$ dimensions

The skew mean curvature flow is an evolution equation for $d$ dimensional manifolds embedded in $\mathbb{R}^{d+2}$ (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $d\geq 4$.

math.AP↗
arXiv2020

Small data global regularity for 3-D Ericksen-Leslie's hyperbolic liquid crystal model without kinematic transport

In this article, we consider the Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model without kinematic transport in three spatial dimensions, which is a nonlinear coupling of incompressible Navier-Stokes equations with wave map to $\mathbb{S}^2$. Global regularity for small and smooth initial data near the equilibrium is proved. The proof relies on the idea of space-time resonance.

math.AP↗
arXiv2019

Small data global regularity for simplified 3-D Ericksen-Leslie's compressible hyperbolic liquid crystal model

In this article, we consider the Ericksen-Leslie's hyperbolic system for compressible liquid crystal model in three spatial dimensions. Global regularity for small and smooth initial data near equilibrium is proved for the case that the system is a nonlinear coupling of compressible Navier-Stokes equations with wave map to $\mathbb{S}^2$. Our argument is a combination of vector field method and Fourier analysis. The main strategy to prove global regularity relies on an interplay between the control of high order energies and decay estimates, which is based on the idea inspired by the method of space-time resonances. In particular the different behaviors of the decay properties of the density and velocity field for compressible fluids at different frequencies play a key role.

math.AP↗
arXiv2017

Equivariant Schrödinger maps from two dimensional hyperbolic space

In this article, we consider the equivariant Schrödinger map from $\Bbb H^2$ to $\Bbb S^2$ which converges to the north pole of $\Bbb S^2$ at the origin and spatial infinity of the hyperbolic space. If the energy of the data is less than $4π$, we show that the local existence of Schrödinger map. Furthermore, if the energy of the data sufficiently small, we prove the solutions are global in time.

math.AP↗