The Stationary Navier–Stokes Problem in Bounded Domains

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.