Global existence and asymptotic convergence of weak solutions for the one-dimensional Navier-Stokes equations with capillarity and nonmonotonic pressure

We construct global weak solution of the Navier-Stokes equations with capillarity and nonmonotonic pressure. The volume variable v(0) is initially assumed to be in H-1 and the velocity variable u(0) to be in L-2 on a finite interval [0, 1]. We show that both variables become smooth in positive time and that asymptotically in time u -> 0 strongly in L-2([0, 1]) and v approaches the set of stationary solutions in H-1 ([0, 1]). (c) 2008 Elsevier Inc. All rights reserved.