Course Descriptions

16:640:509 - Topics in Analysis

Yanyan Li

Subtitle:

An introduction to incompressible Navier-Stokes equations

Text:

Galdi, G. P. An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems. Second edition. Springer Monographs in Mathematics. Springer, New York, 2011. Lectures on Navier-Stokes equations,  Graduate Studies in Mathematics 192,American Mathematical Society, 2018. Author: Tai-Peng Tsai

Lectures on Navier-Stokes equations,  Graduate Studies in Mathematics 192,American Mathematical Society, 2018, Author: Tai-Peng Tsai

Prerequisites:

640:517 or permission from instructor

Course Description:

In this course we will introduce some results on incompressible Navier-Stokes equations.Emphasis will be on incompressible stationary Navier-Stokes equations. For some parts, detailed proofs will be given, while for some other parts outlines of proofs will be given.

We will first introduce basic properties and existence of weak, strong, and mild solutions of incompressible nonstationary Navier-Stokes equations, and then introduce partial regularity.  We will use Chapter 3-6 of one of the text books by Tsai.

Then we will study classical and recent works on the existence of solutions to the nonhomogeneous incompressible stationary Navier-Stokes equations, on 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 periodic solutions of the incompressible stationary Navier-Stokes equations in dimension less or equal to 15, existence of regular solutions of the incompressible stationary Navier-Stokes equations with Dirichlet boundary data in dimension less or equal to 6.

Schedule of Sections: