The Steady Navier-Stokes System

Table of Contents

1. Frontmatter

2. Chapter 1. Preliminaries

   Mikhail Korobkov, Konstantin Pileckas, Remigio Russo

   Abstract

   In this chapter, we provide the main notation and facts from real and functional analysis. We introduce the spaces of smooth functions, the Sobolev spaces, and the spaces of divergence-free vector functions, which play a fundamental role in studying the Navier–Stokes equations. Additionally, we describe some results from general topology, geometric measure theory, and harmonic analysis that are used in this book. Finally, we formulate some basic properties of solutions to elliptic equations. Our aim is not to present an exhaustive treatment of the auxiliary results. Most of these facts are given without proof, and for more detailed study, we provide references to specialized literature. However, we do present proofs of selected assertions from geometric measure theory and harmonic analysis, which, in our opinion, are not as commonly found in the classical theory of differential equations. We have endeavored to present proofs that are not overly complicated, but the choice of statements to be proved depends on our preferences.

3. Chapter 2. Stokes Problem

   Mikhail Korobkov, Konstantin Pileckas, Remigio Russo

   Abstract

   This chapter focuses on studying the boundary value problem for the linear stationary Stokes system in a bounded domain. Definitions of weak solutions are provided, and their existence and uniqueness are proven. The regularity of solutions is examined, and the results related to the Stokes operator are presented. These results are classical and can be found in numerous books on the theory of the Navier–Stokes equations.

4. Chapter 3. The Stationary Navier–Stokes Problem in Bounded Domains

   Mikhail Korobkov, Konstantin Pileckas, Remigio Russo

   Abstract

   This chapter focuses on the analysis of the stationary nonlinear Navier–Stokes system in bounded domains. The well-known results regarding the existence and uniqueness of solutions are presented. Firstly, the existence and uniqueness of solutions are proven for the case of zero boundary values, followed by an extension to the general case of nonhomogeneous boundary data. Two methods for proving the existence of a solution under stringent outflow conditions are described: Hopf’s construction for boundary value extension and Leray’s method of obtaining a priori estimates through a contradiction. A counterexample is also provided to demonstrate that Hopf’s extension cannot be constructed if the stringent outflow condition is not satisfied. Furthermore, a counterexample is presented to show that the Navier–Stokes problem may have multiple solutions for large data. Lastly, the regularity of weak solutions is studied.

5. Chapter 4. The Case of Symmetric Two-Dimensional Domains: General Outflow Condition

   Mikhail Korobkov, Konstantin Pileckas, Remigio Russo

   Abstract

   In this chapter, we demonstrate that in a certain class of symmetric two-dimensional domains, Hopf’s lemma remains valid even when the strict outflow condition (3.2.12) is not satisfied. Instead, it is sufficient to satisfy the necessary general outflow condition (3.2.11).

6. Chapter 5. The Case of General Two-Dimensional Domains and General Outflow Condition

   Mikhail Korobkov, Konstantin Pileckas, Remigio Russo

   Abstract

   In this chapter, we study Leray’s problem in a two-dimensional bounded domain with multiply connected boundary. We prove that this problem admits at least one solution in the case of the general outflow condition with arbitrary fluxes. The only restriction on the flow domain is \(C^2\)-regularity of the boundary.

7. Chapter 6. The Case of Axially Symmetric Three-Dimensional Domains

   Mikhail Korobkov, Konstantin Pileckas, Remigio Russo

   Abstract

   In the last chapter, the Leray problem is studied in the axially symmetric three-dimensional domains with multiply connected boundaries.

8. Backmatter