The Full Nonlinear Navier-Stokes Equations

Abstract

In this chapter, $\Omega \subseteq {\mathbb{R}^n}$ means a domain with n = 2 or n = 3, and [0, T) is a fixed time interval with 0 < T ≤ ∞. The full nonlinear Navier-Stokes system in [0, T) × Ω has the form

$$\begin{array}{*{20}{c}} {{u_t} - v\Delta u + u \cdot \nabla u + \nabla p}& = &f&,&{div u = 0,} \\ {u|\partial \Omega }& = &0&,&{u(0) = {u_0}} \end{array}$$

(1.1.1)

where the boundary condition u|∂Ω = 0 is omitted if Ω = ℝⁿ. We refer to [Lad69, Chap. 6], [Tem77, Chap. III3], [Ama00] concerning weak solutions of these equations.