About Me
I was born in Zhejiang, China, and obtained my B.S. in Mathematics at Hangzhou Normal University (2022-2026), under the supervision of Professor Zhiyuan Xu.
Personal Awards
Curriculum Vitae
You can view and download my complete CV here: Qian Zhenye's CV (PDF)
Research
Research Interests
Partial Differential Equations and Geometric Analysis, including nonlinear elliptic PDEs, fluid dynamics, geometric measure theory and curvature flow.
Recently, I was interested in following topics:
- [January 2026 – present, joint with Tianle Wang] Extending stability analysis of travelling waves for non‑convex scalar viscous conservation laws to degenerate Lax shocks, building on the seminal work of Jones, Gardner and Kapitula. The problem reduces to a singularly perturbed ODE model: $y''+(f'(\phi)-c)y'-\epsilon y=0$, where the coefficient $f'(\phi)-c$ decays at most like $(1+z)^{-1}$ (satisfying the degenerate Lax entropy condition). Using the Levinson asymptotic theorem and Bessel estimates, we show that solutions grow at most as $O((1+z)\exp(\sqrt{\epsilon}))$.
- [December 2025 – present, second part of my thesis] Studying the De Giorgi–Nash–Moser theory (1960s) for nonlinear elliptic equations and its geometric counterpart concerning ε‑regularity in geometric measure theory. For fully nonlinear equations, we examine the Krylov–Safonov theory, further developed by Caffarelli (1987) and Caffarelli & Lihe Wang (1993). Subsequently, Savin extended the theory to small‑perturbation settings (2007), whose techniques enable establishing a crucial Harnack inequality and lead to an improved flatness theorem—a refinement central to resolving the De Giorgi conjecture for dimensions $4 \leq n \leq 8$ under necessary assumptions.
- [November 2025 – present] Studying double‑exponential growth for the 2D Euler equations, building on the small‑scale creation framework introduced in Kiselev and Šverák (2014). Recently, Andrej Zlatóš (July 6, 2025) extended this approach to unbounded domains, proving the desired growth on the half‑plane using a novel idea.
- [October 2025 – present, joint with Daoguo Zhou] Investigating nonlinear instability in fluid dynamics. This includes the linearization method in hydrodynamic stability theory developed by Yudovich, the bootstrap argument introduced by Strauss and Guo in the 1990s, and its application to the Navier–Stokes equations by Friedlander, Pavlović & Shvydkoy (2006). In collaboration with Prof. Daoguo Zhou, we established nonlinear stability and instability for Rayleigh–Bénard convection under various boundary conditions by demonstrating exponential growth.
- [September 2025 – present, first part of my thesis] Studying qualitative properties of semilinear elliptic partial differential equations in unbounded domains, including monotonicity, symmetry, Liouville‑type theorems for certain Schrödinger operators, and energy estimates. This work is motivated by Berestycki, Caffarelli and Nirenberg (1997) and Alberti, Ambrosio and Cabré (2001). The research focuses on two main directions. The first concerns monotonicity of solutions to $\Delta u + f(u) = 0$ on the half‑space with $f$ merely locally Lipschitz. The case $f(0) \geq 0$ was completely resolved by Berestycki–Caffarelli–Nirenberg, while $f(0) < 0$ in dimension $n = 2$ was proved by Farina & Sciunzi (2017). The problem remains open for $f(0) < 0$ and $n > 2$, as discussed in Cortázar, Elgueta, García‑Melián's paper.
Preprints
Publications
Selected Talks
December 2025
Phase transition, Minimal surfaces and De Giorgi conjecture
2025 Global Riemannian Geometry Workshop (Fall and Winter Semester)
Zhejiang University, Zhejiang, China
October 2025
A Short Review of Qualitative Properties for Semilinear Elliptic Equations
Elliptic Partial Differential Equations Seminar
Online
August 2024
The geometric topology of frontal surfaces
2024 International Workshop on Geometry, Topology of Singular Submanifolds and Related Topics
Northeast Normal University, Jilin, China
Study Experience
January 2026
AMSS: Geometric Partial Differential Equations Conference
July 2025
USTC Institute of Geometry and Physics: Summer School in Geometry and Analysis
- Bernstein Problem for Minimal Surfaces and Beyond (Zhihan Wang, Cornell)
June 2025
ZJU: Partial Differential Equation Advanced Workshop
- Minmax Methods in Geometric Analysis (Tristan Rivière, ETH-Zurich)
August 2024
PKU BICMR: Summer School on Differential Geometry
- Riemannian Geometry (Chao Xia, XMU)
- Complex Geometry (Yalong Shi, NJU)
- Second order Elliptic Differential Equation (Jiakun Liu, UOW)
July 2024
USTC Institute of Geometry and Physics: Summer School in Geometry
- Some aspects of Ricci flow on non-compact manifolds (Eric Chen, UCB)
- Topics in mean curvature flow (Jinze Zhu, MIT)
- An introduction to decoupling inequalities in harmonic analysis (Shenwen Gan, WSCS)
Seminars
September 2025 - now
Elliptic Partial Differential Equations Seminar
Sunday 14:00-17:00, Room 101, Building 2, Hainayuan, Zhejiang University (School of Mathematical Sciences)
with H.J. Chen (ZJU), Z.J. Hu (ZJU), M.T. Zhu (WU), T.Y. Shen (ShanghaiTech), T.L. Wang (AMSS)
Selected videos: A Short Review of Qualitative Properties for Semilinear Elliptic Equations
June 2025 - now
Minimal Surfaces Seminar (online) - selected videos
The whole talks:
- A simple survey of minimal surfaces and related aspects 6.24 (Zhenye Qian, HZNU)
- The minimal surfaces equation and minimal submanifolds 7.5 (Haoyu Pan, HZNU)
- Applications of the first variation and Bernstein problem in two dimension 7.18 (Tian Zeng, CQU)
- The second Variation formula and Stability 8.1 (Tian Zeng, CQU)
- Bernstein problem I: The fundamental objects of Geometric Measure Theory 8.8 (Mutian Zhu, WU)
- Bernstein problem II: Regularity theory in Geometric Measure theory 8.15 (Mutian Zhu, WU)
- Bernstein problem II (conti): Regularity theory in Geometric Measure theory 8.22 (Mutian Zhu, WU)
- Bernstein problem III: Bernstein problem and Simons cone 8.28 (Mutian Zhu, WU)
- Small Energy Curvature Estimates for Minimal Surfaces 10.10 (Weipeng Ruan, XMU)
- Extension of Schoen-Simon-Yau via De Giorgi iteration (Bellettini 2025) 10.17 (Weipeng Ruan, XMU)
August 2024 - March 2025
Riemannian Geometry and Geometry Analysis, Hangzhou Normal University - selected videos
Selected topics I have talked:
- Riemannian Submanifold and Basic Equations, some aspects in Minimal Submanifolds in Euclidean space
- Comparison theorem in Riemannian Geometry (Rauch, Hessian and Laplacian, Volume)
- Cheeger-Gromov Convergence and Gromov's paracompactness theorem with some applications in lower bound Ricci curvature
- Bochner Technique on Riemannian Geometry and applications (Comparison theorems, Killing fields)
- Some aspects in CMC hypersurfaces: Alexandrov's theorem, Levy-Gromov Isoperimetric inequality
- Bernstein's theorem in Minimal surfaces and Plateau problem
August 2024 - December 2024
Algebraic Topology, Hangzhou Normal University
Selected topics I have talked:
- An Introduction to Cohomology Ring: Universal coefficient theorem, Cup product and Examples
- Some Applications of Myers-Vietors Argument: Künneth formula, Leray-Hirsch theorem in fiber bundle, Poincaré's duality for manifolds
- Some aspects in homotopy theory: Freudenthal suspension and introduction to stable homotopy groups, Hurewicz-Whitehead theorem and Motivation in Poincaré's Conjecture of 3-dim manifolds
Teaching
Teaching Assistant Experience
Spring 2025
Differential Geometry (Honor), Hangzhou Normal University
Instructor: Zhiyuan Xu
References: exercises and supplementary materials
Autumn 2024
Analytic Geometry, Hangzhou Normal University
Instructor: Zhiyuan Xu
References: exercises and supplementary materials