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

Abstract

Dedicated to Professor Vsevolod Alekseevich Solonnikovon the occasion of his 75th birthday.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 halfspace or a perturbed layer in R n (n ≥ 2). We report a local in time unique existence theorem in the space W 2,1) 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 Lp-Lq 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 [15,16,18,19], Beale [3] and Tani [21].

On the Rayleigh-Taylor instability for the two-phase Navier-Stokes equations

“…In case that Ω 1 (t) is a bounded domain, γ a = 0, and Ω 2 (t) = ∅, one obtains the one-phase Navier-Stokes equations with surface tension, describing the motion of an isolated volume of fluid. For an overview of the existing literature in this case we refer to the recent publications [17,19,20,21]. The motion of a layer of viscous, incompressible fluid in an ocean of infinite extent, bounded below by a solid surface and above by a free surface which includes the effects of surface tension and gravity (in which case Ω 0 is a strip, bounded above by Γ 0 and below by a fixed surface Γ b ) has been considered by [1,2,3,20,23,24].…”

On decay properties of solutions to the Stokes equations with surface tension and gravity in the half space

“…The study of free boundary problems with surface tension and gravity in the L p -L q maximal regularity class were started by Shibata and Shimizu [16]. We especially note that Abels [1] proved the local wellposedness of the finite depth problem with p = q > N , c σ = 0, and c g > 0.…”

On the two-phase Navier–Stokes equations with surface tension

“…Moreover, it is shown in [42] that if Ω 0 is sufficiently close to a ball and the initial velocity u 0 is sufficiently small, then the solution exists globally, and converges to a uniform rigid rotation of the liquid about a certain axis which is moving uniformly with a constant speed (see also [29]). More recently, local existence and uniqueness of solutions for (1.6) (in the case that Ω is a bounded domain, a perturbed infinite layer, or a perturbed halfspace) in anisotropic Sobolev spaces W 2,1 p,q with 2 < p < ∞ and n < q < ∞ has been established by Shibata and Shimizu in [39,40]. For results concerning (1.6) with σ = 0 we refer to the recent contributions [37,38] and the references therein.…”