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

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 ℝⁿ (n ≥ 2). We report a local in time unique existence theorem in the space W^2,1_{q, p} = L_p((0, T), W²_q(Ω)) ∩ W¹_q((0, T), L_q(Ω)) 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).