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We study the zero-viscosity limit of free boundary Navier-Stokes equationswith surface tension in $\mathbb{R}^3$ thus extending the work of Masmoudi andRousset [1] to take surface tension into account. Due to the presence ofboundary layers, we are unable to pass to the zero-viscosity limit in the usualSobolev spaces. Indeed, as viscosity tends to zero, normal derivatives at theboundary should blow-up. To deal with this problem, we solve the free boundaryproblem in the so-called Sobolev co-normal spaces (after fixing the boundaryvia a coordinate transformation). We prove estimates which are uniform in theviscosity. And after inviscid limit process, we get the local existence offree-boundary Euler equation with surface tension. One of the main differencesbetween this work and the work [1] is our use of time-derivative estimates andcertain properties of the Dirichlet-Neumann operator. In a forthcoming work, wediscuss how we can take the simultaneous limit of zero viscosity and surfacetension [].

Uniform Regularity for free-boundary navier-stokes equations with surface tension