** Course outline:** pdf file

** Lecture summary and references:** pdf
file (this file is also updated in the dropbox folder)

## Goal

This course studies the existence and bifurcation of several boundary value
problems for the
Navier-Stokes equations.
Two recent breakthroughs in the mathematical fluid mechanics are the existence theorem of
the boundary value problem in all bounded
2D and axisymmetric 3D domains of Korobkov-Pileckas-Russo, and the existence theorem of
large forward self-similar solutions of Jia-Sverak. We will present these results, starting
with the necessary background, and show one instance of their connection.
A common task of these topics is to obtain a priori bounds of the solutions.
Topics of study include Euler equations, Sard's theorem, Leray-Schauder degree and local
Leray solutions.
Most problems treated are either time-independent or time-periodic.

## Prerequisite

MATH 516. Other relevant materials will be reviewed during the
course.

## Topics

- Boundary value problem of stationery Navier-Stokes equations
- existence with zero boundary condition
- small data uniqueness, nonuniqueness
- existence when boundary is connected
- bifurcation of Navier-Stokes coupled with heat convection
- bifurcation of Couette-Taylor flows

- Korobkov-Pileckas-Russo approach for 2D Boundary value problem
- Sard's theorem for Sobolev functions
- Bernoulli Law for stationary Euler equations in Sobolev spaces
- geometry of level sets of stream functions of 2D Euler equations in Sobolev spaces
- 2D existence for general boundary

- Existence of large forward self-similar solutions
- Local Leray solutions in whole space and
a priori bounds
- Existence of self-similar solutions by Leray-Schauder degree
- Non-uniqueness conjecture
- Existence in half space using KPR approach
- Discretely self-similar solutions, strong and weak

## Evaluation

The course evaluation will be based on presentation. I will make a list
of papers
for you to choose from, and provide you the electronic files.
The presentation is about 30 minutes. You are expected to explain the main
results and the structure of the proof, but not the details of the proof.

## Instructor and lectures

**Instructor:**
Dr. Tai-Peng Tsai, Math building
room 109, phone 604-822-2591, ttsai at math.ubc.ca.

**Lectures:**
Mon 4pm-5pm, Wed 3pm-5pm, MATX 1102

**
Office hours:**
By appointment
(Tsai's schedule).