Uniform regularity for free-boundary Navier–Stokes equations with surface tension

Abstract

We study the zero-viscosity limit of free boundary Navier–Stokes equations with surface tension in unbounded domain thus extending the work of Masmoudi and Rousset [Uniform regularity and vanishing viscosity limit for the free surface Navier–Stokes equations, Arch. Ration. Mech. Anal. (2016), doi:10.1007/s00205-016-1036-5] to take surface tension into account. Due to the presence of boundary layers, we are unable to pass to the zero-viscosity limit in the usual Sobolev spaces. Indeed, as viscosity tends to zero, normal derivatives at the boundary should blow-up. To deal with this problem, we solve the free boundary problem in the so-called Sobolev co-normal spaces (after fixing the boundary via a coordinate transformation). We prove estimates which are uniform in the viscosity. And after inviscid limit process, we get the local existence of free-boundary Euler equation with surface tension. Our main idea is to use Dirichlet–Neumann operator and time-derivatives.