In an exterior domain Ω⊂R3 and a time interval [0, T), 0<T≤∞, consider the instationary Navier-Stokes equations with initial value u0∈Lσ2(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=g with non-zero time-dependent boundary values g. Although uniqueness for these solutions cannot be proved, we show the existence of at least one weak solution satisfying the strong energy inequality and a related energy estimate.
ASJC Scopus subject areas
- Applied Mathematics