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I am interested in mathematical physics, scientific computing, and differential geometry, where the central language is PDE. The following are some of my finished projects.
Slowly converging mean curvature flow
Slowly converging mean curvature flow
Paper: Yunhong Deng, Max Engelstein. "Phase Transition and Mean Curvature Flow near Stable Degenerate Minimal Surfaces." in preparation. [link to the preliminary mean curvature flow part]
Key words: Mean curvature flow, Minimal surfaces, Allen-Cahn equation
Date: May 2025 - Aug 2026
Abstract: We investigate the quantitative convergence of mean curvature flow around a degenerate minimal surface on a warped product manifold, where the manifold was considered by M. Engelstein with O. Chodosh and L. Spolaor [J. Eur. Math. Soc. 25.5 (2022): 1711-1741] to illustrate their sharp quantitative Riemannian isoperimetric inequality.
In our work, we prove that the convergence rate for a general initial datum is t to the power of 1/(2-p), where p is the leading order of theTaylor expansion of the warping function w at the minimal surface. The main technical difficulty is the degeneracy, which makes the principal linearization non-applicable.
The following figure is an illustration I draw.
Homogenization of Fokker-Planck equations
Homogenization of Fokker-Planck equations
Paper: Yunhong Deng, Yuan Gao, Li Wang. "Homogenization of Time-discrete Gradient Flows." arXiv preprint arXiv: 2607.25206 (2026). [link]
Key words: Periodic homogenization, Fokker-Planck equation, Optimal transport, JKO scheme
Date: Sep 2024 - May 2026
Abstract: Inspired by Y. Gao's previous work on homogenization of Wasserstein gradient flows with N. K. Yip [link], we proceed to investigate the homogenization at a time discrete level. One important time discrete scheme for Wasserstein gradient flows is the JKO scheme, which is the minimizing movement scheme in the Wasserstein space with an in-depth optimal-transport geometric structure.
In our work, we show that, however, the JKO scheme fails to capture the homogenization limit of the flow, while the backward Euler scheme (which we call the weighted L2 scheme because it has a special gradient flow structure via a change of variable) is consistent.
Efficient Primal-Dual methods for optimal-transport related gradient flows
Efficient Primal-Dual methods for optimal-transport related gradient flows
Paper Part II: Yunhong Deng and Chaozhen Wei. “A Primal-dual Forward-backward Splitting Method for Cross-diffusion Gradient Flows with General Mobility Matrices.” arXiv preprint arXiv: 2510.27660 (2025). [link]
Paper Part I: Yunhong Deng and Chaozhen Wei. “Efficient Primal-dual Forward-backward Splitting Method for Wasserstein-like Gradient Flows with General Nonlinear Mobilities.” arXiv preprint arXiv: 2504.12713 (2025). [link]
Key words: Computational optimal transport, Structure-preserving numerical scheme
Date: Sep 2023 - Oct 2025
Abstract: From practical models of physics and mathematical biology, transport equations with various forms of nonlinear mobility are taken into consideration. Efficient (in particular, stable) numerical method for solving these models remains partially open. In the previous work, Carrillo-Wang-Wei[link] considered using the JKO scheme with nonlinear mobility to address these problems. However, the considered optimization approach lacks certain flexibility and applications are limited to quadratic mobility functions.
In Part I, we constructed an alternative saddle point form which decouples the mobility within the minimization problem, and proposed an efficient numerical method that has the ability to address general mobility functions. In Part II, we later pushed this idea to solve cross-diffusion models involving matrix mobility (which is the first method based on the JKO scheme).
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