arXiv subjects

Zexian Zhang

Publications and source records attributed to Zexian Zhang.

6 recordsLinked to original sources

arXiv2026

Non-time-decaying global classical solutions to nonlinear wave equations in 3D under the null condition

We present an alternative proof of the global well-posedness of nonlinear wave equations in three spatial dimensions under the null condition, in the regularity regime of the classical local existence theory, assuming smallness of the angular derivatives. Unlike previous methods, which rely on decay in time, our approach is based solely on spatial decay. This provides a new perspective on the problem and offers potential applications to the study of Einstein's equations, which we intend to explore in future work.

math.AP↗

arXiv2026

Global solutions of compressible Navier-Stokes equations with small viscosity

In this paper, we study the Cauchy problem for the compressible Navier-Stokes system in $\mathbb{R}^3$. Suppose that the viscosity coefficients satisfy $0<\max\{\mu, \nu=\lambda+2\mu\}<1$, and set $\varepsilon=\min\{\mu, \nu=\lambda+2\mu\}$. We establish the global existence of classical solutions when the initial perturbations of the density and the curl-free part of the velocity are smaller than $\varepsilon^{\frac12+}$ (up to a logarithmic loss), while the divergence part of the initial velocity is smaller than $\varepsilon$. This improves the classical global existence result of Matsumura-Nishida \cite{MaN80}, which requires all the initial data to be smaller than $\varepsilon (<1)$. We expect that this result is representative of general Shizuta-Kawashima systems arising in physical applications. The improvement of the index from $1$ to $\frac12+$ relies on exploiting the hidden Kawashima-type dissipation for the density and controlling the spacetime trace norm of the solution at the scale $\sqrt{\varepsilon}$. These two ingredients are obtained through a weighted trace inequality and a Morawetz-type inequality for the perturbed sound speed and the divergence of the velocity.

math.AP↗

arXiv2026

Global well-posedness of non-integrable hyperbolic-ellptic Ishimori system in the critical Sobolev space

We consider the Cauchy problem for the hyperbolic-elliptic Ishimori system with general decoupling constant $\kappa \in \mathbb{R}$ and prove global well-posedness in the critical Sobolev space. The proof relies primarily on new bilinear estimates, which are established via a novel div-curl lemma first introduced by the second author in \cite{zhou_1+2dimensional_2022}. Our approach combines the caloric gauge technique with $U^p$-$V^p$ type Strichartz estimates to handle the hyperbolic structure of the equation. The results extend previous work on the integrable case $(\kappa = 1)$ to general $\kappa$ and provide a unified framework which also works for the hyperbolic and elliptic Schr\"odinger maps in dimensions $d \ge 2$.

math.AP↗

arXiv2025

Global well-posedness in the critical Besov space of the skew mean curvature flow in $\mathbb{R}^d: d\ge 5$

In this paper we prove small-data global well-posedness for the skew mean curvature flow of codimension-two submanifolds of $\mathbb R^{d+2}$ ($d\ge5$) in the critical Besov space. With harmonic coordinates and Coulomb gauge, the flow is formulated as a quasilinear Schr\"odinger equation for the complex mean curvature coupled to an elliptic system for the geometric and gauge variables. The main difficulty is to control the frequency interactions at critical regularity, where no derivative margin is available. Our argument combines two complementary spacetime estimates derived from the mass and momentum balance laws: a new div-curl lemma introduced by the fourth author yields a bilinear estimate with a half-derivative gain, providing the key control of low-high interactions; while a quasilinear interaction Morawetz estimate provides critical spacetime bounds for comparable and high-high frequency interactions. These estimates coupled with the Gauss-Codazzi structure of the curvature equations yield the unique global solutions to the gauge-reduced system in the critical Besov space, and improves the previous small-data global regularity results.

math.AP↗

arXiv2024

Global Well-posedness for Incompressible Hookean Elastodynamics in the Critical Besov Spaces

We identify the wave maps type nonlinearities of incompressible Hookean elastodynamics equations in Lagerangian coordinates, and iterate them in the adapted $U^2$-type spaces to prove the small data global well-posedness in the critical Besov space $\dot{B}^{\frac{n}{2}+1}_{2,1}(\mathbb{R}^n)\times \dot{B}^{\frac{n}{2}}_{2,1}(\mathbb{R}^n)\ (n\ge 2)$.

math.AP↗

arXiv2024

Data Preparation for Deep Learning based Code Smell Detection: A Systematic Literature Review

Code Smell Detection (CSD) plays a crucial role in improving software quality and maintainability. And Deep Learning (DL) techniques have emerged as a promising approach for CSD due to their superior performance. However, the effectiveness of DL-based CSD methods heavily relies on the quality of the training data. Despite its importance, little attention has been paid to analyzing the data preparation process. This systematic literature review analyzes the data preparation techniques used in DL-based CSD methods. We identify 36 relevant papers published by December 2023 and provide a thorough analysis of the critical considerations in constructing CSD datasets, including data requirements, collection, labeling, and cleaning. We also summarize seven primary challenges and corresponding solutions in the literature. Finally, we offer actionable recommendations for preparing and accessing high-quality CSD data, emphasizing the importance of data diversity, standardization, and accessibility. This survey provides valuable insights for researchers and practitioners to harness the full potential of DL techniques in CSD.

cs.SE↗