Spring 2026
Yanyan Li
Subtitle:
The Incompressible Navier-Stokes Equations, Focusing on Stationary Solutions
Course Description:
This course presents selected topics concerning the incompressible Navier–Stokes equations, with particular emphasis on stationary (steady-state) problems. We will develop the mathematical tools needed to understand classical and modern results in the area. Foundational material from partial differential equations—including Sobolev spaces, embedding theorems, and elements of the Leray–Schauder degree theory—will be reviewed as needed.
1. Overview and exposition of the existence theory for nonhomogeneous stationary Navier–Stokes equations in two dimensions.
This includes Leray’s pioneering work and several significant developments from the last few years.
2. Overview and exposition of Leray’s problem for steady Navier–Stokes flow past a body in the plane.
3. Regularity of solutions of stationary Navier-Stokes in R^n for n \le 15.
Text:
[MB] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, Vol. 27, Cambridge University Press, 2002. [G] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Steady-State Problems, Second Edition, Springer Monographs in Mathematics, Springer, 2011. [KPR] M. Korobkov, K. Pileckas, and R. Russo, The Steady Navier–Stokes System: Basics of the Theory and the Leray Problem, Advances in Mathematical Fluid Mechanics, Birkhäuser, 2024.
Prerequisites:
640:517 or permission of the instructor
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Fall 2024
Paul Feehan
Subtitle:
Nonlinear Evolutionary Equations
Course Description:
This course is intended for students in who are currently pursuing or would like to pursue research in geometric analysis (especially geometric flows, such as Ricci flow, mean curvature flow, etc.) or applied mathematics/partial differential equations (Navier—Stokes, nonlinear wave equations, reaction-diffusion equations, etc.).
The course is intended to complement existing graduate courses in geometric analysis that focus specifically on Ricci or mean curvature flows and courses in analysis or applied mathematics that focus on nonlinear hyperbolic or parabolic partial differential equations and qualitative properties of their solutions (such as stability, behavior near equilibria, bifurcation, etc.)
The framework of nonlinear evolutionary equations (developed, for example, in the text by Sell and You) allows one to view the examples listed above, and many others, as nonlinear ordinary differential equations on Banach spaces and use this convenient common framework for all of these equations to develop local existence, uniqueness, and regularity properties of solutions, continuity and smoothness of solutions with respect to initial data and parameters, global existence of solutions, and qualitative properties of solutions.
Text:
- G. Sell and Y. You, Dynamics of evolutionary equations (2002), Springer.
- Lecture notes developed by the instructor
Prerequisites:
Permission of Instructor
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Spring 2024
Natasa Sesum
Subtitle:
Introduction to Ricci flow
Course Description:
Ricci flow has been studied extensively recently, as an analytic tool that can be used to prove important topological results about a manifold, whose metric we involve in time by Ricci flow. It is a parabolic equation, so it is expected that the properties of the initial metric improve in time, and in the best case that the flow converges to a metric of constant curvature.
We will learn how various geometric quantities evolve by the flow, discuss the short time existence of the flow (since it is only a weakly parabolic equation), discuss the regularity properties of the equation, we will characterize singular time of the equation. In geometric equations, like Ricci flow, it is very important to understand singularity formation, since that is the first step in possibly finding ways how to continue the flow past singularities. We will introduce Ricci solitons, that often model singularities of the flow, and talk about their properties. We will also introduce Perelman's monotonicity formula, and its important consequences, such as noncollapsing property of the flow. We will also talk about ancient solutions and canonical neighborhood theorem.
Text:
Introduction to Ricci flow by Peter Topping and various papers on Ricci flow
Prerequisites:
Differential Geometry and PDE knowledge is recommended
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Spring 2022
Yanyan Li
Subtitle:
The Yamabe problem and the σk σ k -Yamabe problem
Course Description:
In this course, We will discuss some partial differential equations arising from the study of conformally invariant quantities in Riemannian geometry. In particular we will give in-depth study on the $\sigma_k$-Yamabe problem and related topics. Emphasis will be given on methods and techniques in the study. A classical problem in this field is the Yamabe problem, that is to find on a compact Riemannian manifold a conformal metric which has constant scalar curvature. An extension of the Yamabe problem is the so-called $\sigma_k$-Yamabe problem, which is equivalent to solving certain fully nonlinear second order elliptic equations with certain conformal invariance structure. I will give a survey on results in the field, and will discuss main open problems in the field.
The above will be in the format of a survey, with outlines of proofs. Embedded in the middle of such a survey, I will give more detailed discussions of a number of important methods and techniques involved in the study or closed related to the study of these problems. Such methods and techniques have played very significant roles in the development of the study of nonlinear partial differential equations. Such methods and techniques will be selected among the following: Bernstein type arguments in making $C^1$ and $C^2$ priori estimates, Evans-Krylov estimates for second order uniformly fully nonlinear elliptic equations, theory of viscosity solutions, solving some $\sigma_k$-Yamabe problem using parabolic flows.
If time permits, I will also present studies on the Nirenberg problem, the $\sigma_k$-Nirenberg problem, and their corresponding parabolic flows. Open problems will be discussed.
Text:
There will be suggested papers
Prerequisites:
Math 517 or permission by instructor
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Spring 2020 - Natasa Sesum
Subtitle:
Geometric flows
Course Description:
I will talk about singularity formation in geometric flows, with the emphasis on the mean curvature flow and the Ricci flow. Ancient solutions, which are the solutions that exist for all times from negative infinity appear to be singularity models in above mentioned flows. Therefore, understanding and classifying those is very important in understanding the singularity formation. I will start with known results about ancient solutions in the semilinear heat equation and will talk about known classification results of ancient solutions in the Ricci flow and the mean curvature flow.
Text:
No particular textbook.
Prerequisites:
PDE. Knowing differential geometry is also recommended but not required. No knowledge on particular flows is needed, I will introduce all what I need during the course.
Schedule of Sections:
16:640:519 Schedule of Classes