PDE Coffee Chat ☕

Organized by Alvis Zahl


PDE Coffee Chat is an informal online discussion series centered on partial differential equations and related areas of analysis.
It is intended as a relaxed but mathematically serious space for students, early career researchers, and other participants interested in PDEs to share ideas, discuss papers and techniques, present ongoing work, and explore open problems in a collegial setting.

The format is intentionally flexible. Sessions may range from introductory expository talks to discussions of current research, technical methods, or broader questions connected to the study of PDEs. While the atmosphere is informal, the goal is to encourage thoughtful mathematical exchange, foster new connections, and create a consistent community for conversation across institutions and research backgrounds.

Meetings are typically held online on Saturdays at 10:00 AM Eastern Time.
If you are interested in participation, please sign up Here .

Upcoming Meetings

Well-posedness and Threshold Analysis for a Climate-Explicit Size–Condition Structured Population Model. Notes.
Time: Saturday, Apr 11, 2026, 10:00–11:00 AM ET
Presenter: Louis Shuo Wang, PhD Candidate. Northeastern University.
Abstract: Climate variability can reshape population dynamics not only through size-dependent growth and mortality, but also through changes in energetic condition, recruitment, and resource limitation. Motivated by this mechanism, we study a climate-explicit structured population model in which individuals are described by size and condition, recruitment is endogenous through a renewal-type inflow boundary condition, and harvesting acts through effort and selectivity. Our main result is global well-posedness for the full coupled PDE--ODE system: we prove existence, uniqueness, nonnegativity, and continuous dependence of weak solutions on finite time horizons. We then turn to a reduced autonomous single-zone setting and construct a next-generation operator for the linearisation at the extinction equilibrium. Its spectral radius defines a reduced basic reproduction number \[ \mathcal R_0(y^\ast,\bar q)=r(\mathcal K), \] which determines the sign of the spectral bound of the associated linearised transport--renewal problem. In the same reduced autonomous regime, we prove existence of a nontrivial stationary state under an explicit nonlinear operator hypothesis. Finally, for a finite-horizon harvesting problem, we prove existence of an optimal control on a compact regular admissible class. The paper provides a rigorous analytical foundation for climate-explicit size--condition structured population models with endogenous renewal, resource feedback, and selective harvesting.

Past Meetings

Some studies on the regularity of solutions to degenerate/singular equations with gradient terms. Notes.
Time: Saturday, Apr 4, 2026, 10:00–11:00 AM ET
Presenter: Jiangwen Wang, PhD Candidate. Southeast University.
Abstract: In this talk, we consider $C^{1,\alpha}$ regularity of solutions to degenerate normalized p-Laplacian equations and degenerate/singular fully nonlinear integro-differential equations. If time allows, we will also introduce related applications，such as dead-core problems. This talk is based on joint works with Prof. João Vitor da Silva, Prof. Feida Jiang and Dr. Yunwen Yin.

The Fractional Laplacian and Fractional Sobolev Spaces. Notes.
Time: Saturday, March 28, 2026, 10:00–11:00 AM ET
Presenter: Alvis Zahl, PhD. Rutgers University
Abstract: The Fractional Laplacian has attracted many interest in recent years. A major pheonomen of the fractional Laplacian is its nonlocality. This talk gives an introduction to the fractional Laplacian (-\Delta)^s and the fractional Sobolev Spaces W^{s,p}. Topics will include Cacciopolli inequality, the Caffarelli-Silvestre extension, local maximum principle, and fractional Sobolev inequalities.

Introduction to Elliptic PDE, Part II. Notes.
Time: Saturday, March 21, 2026, 10:00–11:00 AM ET
Presenter: Mark Ma, PhD Candidate. Columbia University
Abstract: In the second part, we conclude Dirichlet Problem to Poisson’s equation on Balls. Then we focus on Schauder theory for uniformly elliptic operators for both divergence form and non-divergence form operators, and (time-permitting) solve the corresponding Dirichlet problems.

Introduction to Elliptic PDE, Part I. Notes.
Time: Saturday, March 14, 2026, 10:00–11:00 AM ET
Presenter: Mark Ma, PhD Candidate. Columbia University
Abstract: This introductory session will discuss several classical topics in elliptic partial differential equations, including the Laplace equation, the Poisson equation, and basic Schauder theory. The talk is intended as the first part of a two-session introduction to elliptic PDE.