Zero-Viscosity Limit of the Navier–Stokes Equations in the Analytic Setting

2017 ◽  
Vol 224 (2) ◽  
pp. 555-595 ◽  
Author(s):  
Chao Wang ◽  
Yuxi Wang ◽  
Zhifei Zhang
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Zero Viscosity Limit ◽  
Viscosity Limit ◽  
Zero Viscosity

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

2018 ◽  
Vol 15 (01) ◽  
pp. 37-118 ◽  
Author(s):  
Tarek Elgindi ◽  
Donghyun Lee
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Stokes Equations ◽  
Local Existence ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Zero Viscosity Limit ◽  
Viscosity Limit ◽  
Zero Viscosity

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.

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