**The Navier-Stokes Equations**A Class for Students who Love Flow and Mathematics

*A Comprehensive Course by Prof. S. A. E. Miller*

$$\frac{\partial \rho}{\partial t} + \frac{\partial \rho u_j}{\partial x_j} = 0$$

$$\frac{\partial \rho u_i}{\partial t} + \frac{\partial \rho u_i u_j}{\partial x_j} = ~- \frac{\partial p}{\partial x_j} \delta_{ij} + \frac{\partial \tau_{ij}}{\partial x_j}$$

$$\frac{\partial \rho e_o}{\partial t} + \frac{\partial \rho u_j e_o}{\partial x_j} = ~- \frac{\partial u_j p}{\partial x_j} – \frac{\partial q_j}{\partial x_j} + \frac{\partial u_i \tau_{ij}}{\partial x_j}$$

### Course Material

Here, I share a comprehensive set of notes that covers the state-of-the-art in understanding the Navier-Stokes equations. The course has approximately 1768 slides, over 2000 unique equations, 92 unique historical figures with pictures and fun facts, and many interspersed quotes of famous researchers. The course can be downloaded as a single PDF here …

Navier-Stokes Equations Course Notes Complete by Prof. S. A. E. Miller

### Course Objectives

The student will understand the history, physical meaning, and contemporary challenges within the field of theoretical fluid dynamics and turbulence.

### Course Description

Navier-Stokes Equations (NSE) History, derivation, physical meaning, classical solutions, stability, dynamical systems, existence, uniqueness, regularity, scales, ladder results, dissipation rates, Serrin’s blowup, capacitary approaches, mild solutions (Lebesgue, Sobolev, Besov, Morrey, BMO, Koch, Tataru), weak solutions, stochastic NSE, the Russian school, and invariant measures.

### Outline of the Course Material

- Introduction
- Introduction, Syllabus, Outline of Course, Clay, Interested in Solutions

- History
- Overview
- Detail

- Mathematical Review — Review, Notation
- Derivation of NS
- Physical Meaning – Physical Meaning of NS
- Classical Solutions – Classical solutions of the NS
- Stability – Introduction to linear and nonlinear stability
- Dynamical Dynamical Systems
- Intro Exist Unique Regularity
- Existence, Uniqueness, and Regularity
- Regularity and intro to length scales for the 2D and 3D NSE

- Ladder – Results
- Dissipation Rates – Energy dissipation rates of Fourier spectra – bounded flow
- Blowup – Criteria, Serrin’s criterion
- Capacitary approach of the NSE integral equations
- Differential and integral NSE
- Mild solutions
- Lebesgue / Sobolev spaces
- Besov or Morrey spaces
- BMO\(^{-1}\) and Koch and Tataru theorem

- Leray’s weak solutions
- Stochastics
- Statistics and Derivation of Stochastic NSE
- Probability and Statistical Theory of Turbulence
- Invariant measures and PDF
- Existence theory of swirling flow — existence of stochastic NSE

### What is this Class, or, What is it Not?

- What this class is
- Examination of classical and contemporary approaches for solutions and analysis of NSE
- How are these methods related to societal problems – and how can they be moved to predictive methods
- A lot of effort on the part of students to understand the material
- A lot of material

- What this class is not
- Derivation of trivial solutions of NSE
- Example – laminar flows with simplified boundary conditions (BC)s

- Easy to understand, but we will try our best to explain everything
- Review of earlier mathematics or fluids classes in CFD, turbulence, or mathematics
- If you don’t understand something, take the initiative and look it up

- Derivation of trivial solutions of NSE

### Recommended Textbooks and/or Software

- A complete set of course notes will be provided by the professor.
- There are no required textbooks for the class.