Global estimates in weighted spaces of weak solutions of the Navier-Stokes equations in exterior domains

摘要

1. Introduction. Throughout the paper let Q <= R3 denote an exterior domain with boundary of class C~(2,u), 0 < n g 1. Then consider the nonstationary Navier-Stokes system. which describes the unknown velocity u = {ul, u2> u3) and pressure p of some fluid for xeQ and all times t § 0, while the external force f and the initial velocity u0 are given. It is well known that (1.1) has global weak solutions, but it is an open problem whether a weak solution is unique, globally strong or even classical. Therefore there is a great interest to obtain additional global properties of weak solutions besides the fact that the energy norms || u ||z M and || ▽u || 2 2 are finite; here || . ||_(qS) denotes the norm in the space U(U) = Ls(0, ∞; U(Ω), 1 ≤q, s≤ k∞.