Xinyu Cheng

Xinyu Cheng

PDE/Analysis/Numerics

Research Interest

My interest is on the mathematical analysis of PDEs and applied mathematics, including fluid models, interfacial change and free boundary problems, low-regularity structure-preserving numerical methods, mathematical biology, and machine-learning based PDE solvers. This is my research statement.

Research highlights

Four representative lines of work. The full publication and preprint list below is unchanged.

1. Well-posedness

\[\partial_t\theta + u\cdot\nabla\theta=0 \quad\text{on }\mathbb{T}^2,\qquad u=\nabla^\perp\Lambda^{-1}\theta=(-\mathcal{R}_2\theta,\mathcal{R}_1\theta)\]

The inviscid surface quasi-geostrophic (SQG) equation on the torus.

  • Non-uniqueness of weak solutions. Stationary weak solutions of SQG are not unique at low regularity, which gives a route toward a stationary analogue of Onsager’s conjecture (Comm. Math. Phys. 2021).
  • Global strong solutions and scattering. Global well-posedness for wave and Klein–Gordon–wave systems, including low-regularity global solutions, and global effective approximation and scattering for non-compactly supported data (Calc. Var. PDE 2024; Dynam. PDE 2022; arXiv:2312.00821; scattering manuscript submitted).
  • Vanishing viscosity. For dissipative SQG, the inviscid limit holds in borderline Besov spaces, with a quantitative rate, and the convergence takes place in the same topology as the initial data (to appear in Proc. Amer. Math. Soc. 2026).
  • Oldroyd-B vanishing damping. Global well-posedness of the inviscid Oldroyd-B model (with damping) in low-regularity spaces, optimal temporal decay, and the uniform-in-time vanishing damping limit in two and three dimensions (arXiv:2410.09340).
  • Related analysis also includes equivalent formulations of Helmholtz equations and localization (Comm. Contemp. Math. 2023; J. Differ. Equ. 2024).

Klein-Gordon-wave field at t=20 Klein-Gordon-wave field at t=9

Wave–Klein–Gordon fields at two times (t = 20 and t = 9).

2. Interface dynamics

Phase-field coalescence: two interfaces merge into one Interface coalescence in a phase-field model.

  • Capturing the dynamics. Stable schemes give a quantitative description of Allen–Cahn / Cahn–Hilliard interface evolution, including splitting, merging, and other topological changes (J. Sci. Comput. 2021).
  • Crystal growth. Related interface work includes a Sinc-type molecular beam epitaxy model (arXiv:2106.16193).

Allen-Cahn evolution at t=0.1, 1, 5, 10 Cahn-Hilliard evolution at t=0.1, 5, 10, 20

Allen–Cahn (left) and Cahn–Hilliard (right): small components disappear or merge as time evolves.

3. Low-regularity structure-preserving schemes

Comparison of a low-regularity scheme at N=128 and N=1024

Same initial data, two spatial resolutions (N = 128 vs N = 1024). The scheme remains stable while resolving the singular structure.

  • Structure preservation. Energy-stable semi-implicit schemes for phase-field models (Allen–Cahn, Cahn–Hilliard) that capture singularities and keep structural stability over long times (IMA J. Numer. Anal. 2026; BIT Numer. Math. 2025).
  • Low-regularity convergence. Convergence can be proved in the same low-regularity space as the initial data, for SQG, Cahn–Hilliard, and related fluid models. The analysis and the schemes are designed to check each other (J. Differ. Equ. 2026).
  • SQG at endpoint regularity. A low-regularity integrator for SQG admits optimal error estimates when the solution has only endpoint regularity (to appear in IMA J. Numer. Anal. 2026).
  • Fluid models in borderline spaces. Unconditionally stable semi-implicit schemes for the incompressible Euler equations via the vanishing viscosity limit, and a Besov-based integration-by-parts framework with stability and convergence for the Navier–Stokes equations in borderline Besov spaces (arXiv:2406.12320; arXiv:2509.23192).

4. Mathematical biology and AI

\[\partial_t u=\Delta\bigl(u\,\phi(w)\bigr),\qquad \partial_t w=D\Delta w-uw\]

A chemotaxis–consumption system with nutrient-dependent motility.

  • Oxygen consumption. For the oxygen-consumption free-boundary problem we work with gradient-flow and variational-inequality formulations, and prove that the different models are equivalent (SIAM J. Appl. Math. 2023).
  • Chemotaxis. Nonlinear stability of positive nonconstant steady states in chemotaxis–consumption systems with general motility on bounded domains in two and three dimensions, with exponential convergence (arXiv:2609.34202).
  • Machine learning for PDEs. A deep truncated FBSDE method that decouples forward and backward processes via gradient truncation and fictitious-play averaging, giving a robust solver for high-dimensional nonlinear PDEs and fully coupled FBSDEs with residual-based error estimates (arXiv:2609.39616).

Population perturbation relaxing to a nonconstant steady state

Chemotaxis–consumption: a population perturbation on an ellipse relaxes to a nonconstant steady state.

Relative error of the deep truncated FBSDE solver versus training iterations

Relative error of the deep truncated FBSDE solver versus training iterations.

Publications and preprints (by year)

published

  1. On the Spectral Gap of a Square Distance Matrix, joint with D. Li, D. Shirokoff and B. Wetton, J. Stat. Phys., 2017, 166(3-4), 1029–1035. pdf
  2. Asymptotic Behaviour of Time Stepping Methods for Phase Field Models, joint with D. Li, K. Promislow and B. Wetton, J. Sci. Comput., 2021, 86(3), 1–34. pdf
  3. On a parabolic Sine-Gordon model, joint with D. Li, C. Quan and W. Yang, Numerical Mathematics: Theory, Methods and Applications., 2021, 14(4), 1068–1084. pdf
  4. Non-uniqueness of stationary weak solutions to the surface quasi-geostrophic equations, joint with H. Kwon and D. Li, Comm. Math. Phys., 2021, 388 (3), 1281-1295. pdf
  5. Global wellposedness for 2D quasilinear wave without Lorentz, joint with D. Li, J. Xu and D. Zha, Dynam. Part. Differ. Eq., 2022, 19(2), 123-140. pdf
  6. On the equivalence of classical Helmholtz equation and fractional Helmholtz equation with arbitrary order, joint with D. Li and W. Yang, Comm. Contemp. Math., 25, 2250036, 2023. pdf
  7. Equivalent formulations of the oxygen diffusion problem and other implicit free boundary value problems and implications for numerical approximation, joint with Z. Fu and B. Wetton, SIAM J. Appl. Math., 2023, 83(1), 52-78. pdf
  8. On the global well-posedness and scattering of the 3D Klein-Gordon-Zakharov system, joint with J. Xu, Calc. Var. Part. Differ. Eqn., 63(17), 2024. pdf
  9. Localization for general Helmholtz, joint with D. Li and W. Yang, J. Diff. Eqn., 393: 139-154, 2024. pdf
  10. Energy stable semi-implicit schemes for the 2D Allen-Cahn and fractional Cahn-Hilliard equations, IMA. J. Numer. Anal, 46(2):758–794, 2026. pdf
  11. Second order energy stable semi-implicit schemes for the 2D Allen-Cahn equation, BIT Numer. Math., 65(26): 26, 2025. pdf
  12. Stability and convergence of an iterative low-regularity method for the Cahn-Hilliard equation, joint with Z. Luo and S. Wang, to appear in J. Diff. Eq. pdf
  13. Optimal error estimates of a low-regularity integrator for the SQG equations under endpoint regularity, joint with Z. Luo and S. Wang, to appear in IMA. J. Numer. Anal. pdf
  14. The vanishing viscosity limit of the surface quasi-geostrophic equations in borderline Besov spaces: Rates and same topology convergence, joint with Z. Luo and S. Wang, to appear in Proc. Amer. Math. Soc. pdf

preprints

  1. On a Sinc-type MBE model, joint with D. Li, C. Quan and W. Yang, submitted, arXiv:2106.16193.
  2. Uniform Boundedness of Highest Norm for 2D Quasilinear Wave, joint with D. Li and J. Xu, submitted, arXiv:2104.10019.
  3. Energy stability and convergence of Strang splitting method for Cahn-Hilliard equation, joint with D. Li, preprint.
  4. Global well-posedness for 2D quasilinear wave equations with non-compactly supported initial data, joint with D. Li and J. Xu, preprint.
  5. Global well-posedness of a two dimensional wave-Klein-Gordon system with small non-compactly supported data, arXiv:2312.00821.
  6. Viscosity-uniform error estimates for a semi-implicit Fourier method in Besov spaces, joint with Z. Luo and S. Wang, preprint arXiv:2406.12320.
  7. Global well-posedness and uniform-in-time vanishing damping limit for the inviscid Oldroyd-B model, joint with Z. Luo, Z. Yang and C. Yuan, preprint arXiv:2410.09340.
  8. Nonlinear Stability of Nonconstant Steady States in Chemotaxis–Consumption Systems with General Motility, joint with X. Lai and X. Song, preprint arXiv:2609.34202.
  9. Deep Truncated FBSDE Method: A Robust Solver for High-Dimensional Nonlinear PDEs and Fully Coupled FBSDEs, joint with Y. Li and W. Xiong, preprint arXiv:2609.39616.
  10. Global Effective Approximation and Scattering for a Wave–Klein–Gordon System, submitted.

Academic activities

conferences and talks

  • SNSA 2026 （分组召集人） @ Guilin, Guangxi, China 2026/07

  • AIMS 2026 （分组报告人，分组召集人）@ Athens, Greece 2026/07

  • 2026年兰州大学计算数学前沿论坛 (报告）@ Lanzhou, Gansu, China 2026/04

  • 2026年中山大学理学院五周年院庆暨偏微分方程、动力系统与计算数学学术会议 （分组报告）@ Shenzhen, Guangdong, China 2026/03

  • 2025 HKU Workshop on PDEs and Analysis （报告）@ Hong Kong, China 2025/11

  • 中国工业与应用数学学会第二十三届年会（CSIAM 2025） (分组报告）@ Changsha, Hunan, China 2025/10

  • 第30届差分方程和应用国际会议(ICDEA 2025) （分组报告） @ Guangzhou, Guangdong, China 2025/07

  • 第二届“面向未来的复杂系统科学–交叉与融合”学者讲习班 （组织）@ IICS, Fudan University 2025/07

  • 2025微分方程与动力系统研讨会 （报告） @ SUSTech University 2025/06

  • 现代偏微分方程分析与计算学术研讨会 （报告）@ Chongqing Normal University 2025/05

  • 第一届微分方程与非线性分析青年学术研讨会 （组织） @ Fudan University 2024/12

  • 第四届国际生物数学建模、分析与应用研讨会 （报告） @ Harbin Engineering University 2024/06

  • 非线性分析青年学术研讨会（报告） @ Wuhan University of Technology 2024/06

  • 三亚波国际前沿论坛@清华三亚国际数学论坛 2024/01

  • 第一届“面向未来的复杂系统科学–交叉与融合”学者讲习班 （组织，报告）@ IICS, Fudan University 2023/11

  • Seminar talk （报告）@ Institute of Mathematical Sciences, ShangTech University 2023/11

  • Seminar talk（报告） @ School of Mathematics, South China University of Technology 2023/07

  • CSIAM 2022 （报告）@ Guangzhou, Guangdong, China 2022/09

  • Workshop on Analysis and PDE @ Tianjin Center for Applied Mathematics (TCAM) 2019/08

  • SIAM Conference on Applications of Dynamical Systems （报告） @ Snowbird, UT, US 2019/05

visiting

  • South China University of Technology, Guangzhou, China 2023/06-2023/07

  • SUSTech, Shenzhen, China 2021/10-2022/01

  • SUSTech, Shenzhen, China 2021/02-2021/04

  • Tianjin University, Tianjin, China 2019/06-2019/07

  • SUSTech, Shenzhen, China 2018/07

  • Michigan State University, East Lansing, US 2018/04

Academic Service

I have refereed articles for publications in

Numerical Analysis and Scientific Computing

  • Mathematics of Computation, IMA Journal of Numerical Analysis, Advances in Computational Mathematics, Applied Mathematics Letters, BIT Numerical Mathematics

PDE and Analysis

  • Acta Mathematica Scientia, Zeitschrift für Angewandte Mathematik und Mechanik, Communications in Analysis and Mechanics, Fractional Calculus and Applied Analysis, Dynamics of Partial Differential Equations, Applied Mathematics-A Journal of Chinese Universities

Mathematical Physics

  • Classical and Quantum Gravity, Physica Scripta

Mathematical Biology

  • Mathematical Biosciences and Engineering

Academic advising

Current

  • Haoran Wu (Phd student, since Fall 2024)
  • Weiran Xiong (Phd student, since Fall 2026)
  • Xuewen Lai (Phd student, since Fall 2026)
  • Yanqiao He (Phd student, since Fall 2027)
  • Jia Xiao (Phd student, since Fall 2027)

Academic seminars

Past and current

Opportunities for graduate students and postdoc fellows

I am looking to recruit graduate students for three projects

  • numerical analysis of methods for capturing phase change
  • analysis and computation of fluid models
  • some aspects of machine learning using neural ODEs, deepBSDE and others.