Course Description

A major open problem for the 3D incompressible Navier-Stokes equations is the global existence of regular solutions for all smooth initial data with sufficient decay. A seminal result of Escauriaza, Seregin, and Sverak [2] shows regularity up to time T if the solution is bounded in $L^3(R^3)$ uniformly up to time T. Their main tool, besides scaling limits of the equation, is backward uniqueness and unique continuation for the heat equation based on Carleman estimates from harmonic analysis.

In this course I plan to go through the recent papers by Tao [3] and Barker-Prange [1] which study the hypothetical singularity of Navier-Stokes equations under Type I assumptions, i.e., the singular solutions satisfy certain scaling-invariant bounds. Tao replaces the backward uniqueness and unique continuation results of [2] by quantitative estimates, and hence he can avoid scaling limits and derive explicit lower bound for the blow up. Barker and Prange localize these estimates in space, using short time smoothing estimates for Navier-Stokes equations. Our main tool, the Carleman estimates, will be developed during the course. Introduction to the regularity problem will be given in the beginning.

Lecture summaries and some references will be available in a public owncloud folder, whose link will be given to the audience. The link will expire on May 31, 2022.

Prerequisites

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

Topics

  1. Introduction to the regularity problem

  2. The work of Escauriaza, Seregin, and Sverak

  3. The work of Tao

  4. The work of Barker and Prange

References

  1. T. Barker and C. Prange, Quantitative regularity for the Navier-Stokes equations via spatial concentration, Comm. Math. Phys. 385 (2021), no. 2, 717-792. (arXiv:2003.06717)

  2. L. Escauriaza, G. Seregin, and V. Sverak,
    • Backward uniqueness for parabolic equations. Arch. Ration. Mech. Anal., 169(2):147-157, 2003.
    • $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk 58 (2003), no. 2(350), 3-44, Translation in Russian Math. Surveys 58 (2):211-250, 2003.

  3. T. Tao, Quantitative bounds for critically bounded solutions to the Navier-Stokes equations (arXiv:1908.04958)

  4. T.-P. Tsai, Lectures on Navier-Stokes equations, AMS GSM 192, 2018
More references will be added during the term.

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.

Instructor and lectures

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

Lectures: MATH building room 202, Mon Wed Fri 10-10:50am

Office hours: by appointment.