Report on a local in time solvability of free surface problems for the Navier–Stokes equations with surface tension

Abstract

We consider the free boundary problem of the Navier–Stokes equation with surface tension. Our initial domain Ω is one of a bounded domain, an exterior domain, a perturbed half-space or a perturbed layer in ℝ n (n ≥ 2). We report a local in time unique existence theorem in the space with some T > 0, 2 < p < ∞ and n < q < ∞ for any initial data which satisfy compatibility condition. Our theorem can be proved by the standard fixed point argument based on the L p –L q maximal regularity theorem for the corresponding linearized equations. Our results cover the cases of a drop problem and an ocean problem that were studied by Solonnikov (Solvability of the evolution problem for an isolated mass of a viscous incompressible capillary liquid, Zap. Nauchn. Sem. (LOMI) 140 (1984) pp. 179–186 (in Russian) (English transl.: J. Soviet Math. 32 (1986), pp. 223–238)), Solonnikov (Unsteady motion of a finite mass of fluid, bounded by a free surface, Zap. Nauchn. Sem. (LOMI) 152 (1986), pp. 137–157 (in Russian) (English transl.: J. Soviet Math. 40 (1988), pp. 672–686)), Solonnikov (On nonstationary motion of a finite isolated mass of self-gravitating fluid, Algebra Anal. 1 (1989), pp. 207–249 (in Russian) (English transl.: Leningrad Math. J. 1 (1990), pp. 227–276)), Solonnikov (Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval, Algebra Anal. 3 (1991), pp. 222–257 (in Russian) (English transl.: St. Petersburg Math. J. 3 (1992) 189–220)), Beale (Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal. 84 (1984), pp. 307–352) and Tani (Small-time existence for the three-dimensional incompressible Navier–Stokes equations with a free surface, Arch. Rat. Mech. Anal. 133 (1996), pp. 299–331).