Global Leray-Hopf weak solutions of the Navier-Stokes equations with Nonzero time-dependent boundary values

Abstract

In a bounded smooth domain Ω ⸦ ℝ3 and a time interval [0, T), 0 < T ≤ ∞, consider the instationary Navier-Stokes equations with initial value U0 ∈ L2σ(Ω) and external force f = divF, F ∈ L2(0, T;L2(Ω)). As is well known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray-Hopf weak solutions to the case when U|∂Ω= g with non-zero time-dependent boundary values g. Although there is no uniqueness result for these solutions, they satisfy a strong energy inequality and an energy estimate. In particular, the long-time behavior of energies will be analyzed.