### Fall 2022

### Yanyan Li

**Subtitle:**

An introduction to incompressible Navier-Stokes and Euler equations

### **Course Description:**

In this course we will introduce some results on incompressible Navier-Stokes equations. The course will be divided in equal number of lectures into two parts.

For Part 1, we will select materials from the text book [MB] as described below.

Basic properties and concepts related to solutions of the Navier-Stokes (NS) equations and Euler equations. Several exact solutions with physical backgrounds (Chapter 1 of [MB]). More on the construction of exact solutions, explain how to construct exact solutions of 2.5D NS and Euler, and exact solutions of axisymmetric NS and Euler. Recover the velocity and the velocity gradients from the vorticity, and the vorticity-stream formulation (Chapter 2 of [MB]).

More on the construction of exact solutions, explain how to construct exact solutions of 2.5D NS and Euler, and exact solutions of axisymmetric NS and Euler. Recover the velocity and the velocity gradients from the vorticity, and the vorticity-stream formulation (Chapter 2 of [MB]).Energy methods to prove the local existence and uniqueness theory in Sobolev space setting for the NS and Euler, and the global existence in the 2D case (Sections 3.1-3.3 in Chapter 3 of [MB]).Local existence theory in the Holder space setting for the 3D Euler via the approach of particle trajectory. Beale-Kato-Majda regularity criterion for the 3D Euler. The global existence result for the 3D axisymmetric Euler without swirl (Sections 4.1-4.3 in Chapter 4 of [MB]).Some lower dimensional models that share the vortex stretching mechanism with the 3D Euler vorticity equation, in particular the 1D model of Constantin-Lax-Majda (Sections 5.1-5.2 in Chapter 5 of [MB]).The result of Yudovich on the global existence and uniqueness of weak solutions to the 2D Euler equation with bounded initial vorticity (L^1\cap L^\infty) (Section 8.2 in Chapter 8 of [MB]).

Part 2 will be on incompressible stationary Navier-Stokes equations.

We will survey classical and recent works on the following topics. In addition, we will outline proofs or give detailed proofs for some results. The topics are: Leray's problem on the existence of solutions of the nonhomogeneous boundary value problem for the incompressible stationary Navier-Stokes equations in bounded domains in two and dimensions; Leray's problem of steady Navier-Stokes flow past a body in the plane; Liouville theorems for three dimensional incompressible stationary Navier-Stokes equations; existence of regular solutions of the incompressible stationary Navier-Stokes equations in Euclidean space and flat torus in dimension less than or equal to 15; existence of regular solutions of the incompressible stationary Navier-Stokes equations with Dirichlet boundary data in dimension less than or equal to 6.

### **Text:**

[MB] A.J. Majda and A. Bertozzi, Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002. |

### Prerequisites:

Math 501 or Permission from instructor

### ***********************************************************************************

### From Fall 2021 Semester:

### Yanyan Li

**Subtitle:**

Title: The Yamabe problem and the \(\sigma_k\)-Yamabe problem

A survey, and an exposition on methods and techniques in the study

### **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 survey results on the Yamabe problem and the \(\sigma_k\)-Yamabe problem. We will also give an exposition 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. This is equivalent to solving a conformally invariant semi-linear partial different equation of critical exponent. We will introduce the problem and give a review of the solution to this problem as well as results on compactness of the solution set. We will outline the proofs of these results.

An extension of the Yamabe problem is the so-called \(\sigma_k\)-Yamabe problem. I will survey results on the existence and compactness of solutions for the problem, 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 ampng the following: best Sobolev constant and extremal functions, symmetrizations and concentration compactness method, variational methods (lower semi-continuity and minimization in the Calculus of variations, the min-max method and the mountain pass theorem), Moser iterations and DeGorgi estimates, Liouville theorems for conformally invariant elliptic and degenerate elliptic fully nonlinear equations in Euclidean equations, the method of moving planes, small energy implies regularity, Bernstein type arguments in making \(C^1\) and \(C^2\) estimates, Evans-Krylov estimates for second order uniformly fully nonlinear elliptic equations. theory of viscosity solutions.

**Text:**

No text book

### Prerequisites:

Math517 or Permission by instructor

### ***********************************************************************************

### From Fall 2020 Semester:

### Natasa Sesum

**Subtitle:**

Mean curvature flow and ancient solutions

**Course Description:**

We plan to mention recent results about the classification of ancient solutions in the mean curvature flow which was used to prove the Mean convex neighborhood conjecture, which says that if a singularity is cylindrical, then there is a space time neighborhood around a singularity in which the mean curvature is strictly positive.

**Text:**

Different papers that will be shared with students throughout the semester. For the basics it is ' Regularity theory for Mean Curvature Flow' by Klaus Ecker.

### Prerequisites:

Partial differential equations and very basics of differential geometry (though I plan to introduce differential geometry concepts that we will use in the class).

**Schedule of Sections:**