Small-time existence for the three-dimensional navier-stokes equations for an incompressible fluid with a free surface

Abstract

In 1980 BEALE [2] studied the problem of describing the motion of a layer of heavy, viscous, incompressible fluid lying above an infinite rigid bottom and having a non-compact free surface with no surface tension. He proved its local solvability in an anisotropic Sobolev space for any initial data. Its global solvability was discussed by SYLV~SXER [20]. But a crucial point of her proof does not appear clear to me: the regularity of the free surface. With surface tension taken into account, BEALE [31 and BEALE & NISHIDA [4] respectively studied the large-time existence and regularity and the large-time behavior of the solution to this problem with initial data near equilibrium. TERAMOTO [24, 25] considered these problems for fluids lying above an inclined plane. For any initial data the same problem was solved locally in time by ALLAIN [11 in the two-dimensional case. The aim of this paper is to establish the analogous result in the three-dimensional case. A similar problem describing the motion of a finite isolated mass of incompressible viscous fluid was studied thoroughly by SOLONNIKOV. He proved local solvability in a H61der space in [12] and global solvability in the space W 2' 1, p > n (where n is the dimension of the domain) in [16] without surface tension and in [-13 15, 17 19,21] with surface tension. Let us formulate our problem. Given an initial domain f~ _ R 3 with x3 being the vertical component and an initial velocity vector field Vo in f~, we want to know the domain f~(t), t > 0, occupied by the fluid, which is bounded by the fixed bottom SB and the free surface Se(t), the velocity vector field v = v(x, t) = (vl, v2, v3) and the pressure p = p(x, t) so that