Course outline: pdf file

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


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.


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


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

  2. Korobkov-Pileckas-Russo approach for 2D Boundary value problem
    1. Sard's theorem for Sobolev functions
    2. Bernoulli Law for stationary Euler equations in Sobolev spaces
    3. geometry of level sets of stream functions of 2D Euler equations in Sobolev spaces
    4. 2D existence for general boundary

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


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

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

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