Global Weak Solutions of the Navier-Stokes System with Nonzero Boundary Conditions

Abstract

In memory of our colleague Prof. Tetsuro Miyakawa Abstract. Consider the Navier-Stokes equations in a smooth bounded domain W H R 3 and a time interval ½0; TÞ, 0 < T a y. It is well-known that there exists at least one global weak solution u with vanishing boundary values uj qW ¼ 0 for any, and satisfying the strong energy inequality. Our aim is to extend this existence result to a much larger class of global in time ''Leray-Hopf type'' weak solutions u with nonzero boundary values uj qW ¼ g A W 1=2; 2 ðqWÞ. As for usual weak solutions we do not need any smallness condition on g; indeed, our generalized weak solutions u exist globally in time. The solutions will satisfy an energy estimate with exponentially increasing terms in time, but for simply connected domains the energy increases at most linearly in time.

Global Weak Solutions of the Navier–Stokes Equations with Nonhomogeneous Boundary Data and Divergence

“…For the case of weak solutions with constant in time nonzero boundary conditions g see [4]. Further there are several independent results for smooth boundary values u j @V g T 0 in the context of strong solutions u if g or (equivalently) the time interval [0; T) satisfy certain smallness conditions, see [1], [3], [6], [10].…”