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 v0 is initially assumed to be in H1 and the velocity variable u0 to be in L2 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 L2([0,1]) and v approaches the set of stationary solutions in H1([0,1]).