Qian Zhenye

About Me

I was born in Zhejiang, China. I am currently a Ph.D. student in Mathematics at The Chinese University of Hong Kong, under the supervision of Professor Juncheng Wei.

Education

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:

Elliptic PDEs

• [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.
• [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.

Geometric Analysis

• [February 2026 – present, third part of my thesis] Studying the evolution of level sets under generalized mean curvature flow (MCF), focusing on well‑posedness, singularity formation, and applications to dynamic phase transition models such as the parabolic Allen–Cahn equation. The theory of generalized MCF (the classical formulation was pioneered by Hamilton [Ricci flow] in 1982 and Huisken in 1984) splits into two main approaches. The first, introduced by Brakke in 1978, uses geometric measure theory (motivated by Allard 1972) to construct varifold solutions and establish partial regularity. The second is the theory of viscosity solutions for the level‑set formulation, developed by many authors in the 1980s–1990s, including Evans, Crandall–Lions, Crandall–Evans–Lions, Lions, Jensen, and Ishii. Our work focuses on the seminal contributions of Evans–Spruck (“Motion of level sets by mean curvature” I–IV) and their applications to the Ginzburg–Landau model (see Evans–Soner–Souganidis and Chen–Giga–Goto in the 1990s).
• [January 2026 – present] Studying harmonic maps in geometric variational problems and elliptic PDE theory, with an emphasis on regularity and the structure of singularity sets. Key contributions include the foundational work of Schoen–Uhlenbeck, followed by the regularity theories developed by Evans, Hélein, Rivière, Fanghua Lin, Lin–Rivière, and more recently the quantitative stratification results of Naber–Valtorta.
• [September 2025 – present, lectures in the Minimal Surfaces Seminar] Studying the min‑max construction for minimal surfaces and more general geometric variational problems. The min‑max method traces back to Schwarz in the 1940s (for harmonic functions) and Birkhoff in the 1910s (for closed geodesics on 2‑spheres). Our focus is on its modern refinements, including Croke's work and the two seminal articles by Colding–Minicozzi that connect width to Ricci flow and mean curvature flow. For a comprehensive overview of these developments, we refer to the survey by Colding–De Lellis.

Fluid equations

• [January 2026 – present, joint with Tianle Wang] Extending the stability analysis of travelling waves for non‑convex scalar viscous conservation laws to degenerate Lax shocks in a weighted $L^\infty$ space, 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 $(1+z)^{-1}$, consistent with the degenerate Lax entropy condition. The analysis relies on sharp pointwise Green's function estimates for such ODE profiles, drawing on a series of seminal works by Howard.
• [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.

Preprints

Publications

Selected Talks

Study Experience

Seminars

Teaching

Teaching Assistant Experience

Notes & Materials

Lecture Notes

Miscellaneous

• Vlogs about studying: Bilibili
• Ideas and Discussions: Zhihu