scholarly journals The inviscid limit for the incompressible stationary magnetohydrodynamics equations in three dimensions

2021 ◽  
pp. 2150006
Author(s):  
Weiping Yan ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Unbounded Domain ◽  
Three Dimensional ◽  
Viscosity Coefficient ◽  
Three Dimensions ◽  
Mhd Equations ◽  
Inviscid Limit ◽  
Magnetohydrodynamics Equations ◽  
Zero Viscosity Limit ◽  
Viscosity Limit ◽  
Zero Viscosity

This paper is concerned with the zero-viscosity limit of the three-dimensional (3D) incompressible stationary magnetohydrodynamics (MHD) equations in the 3D unbounded domain [Formula: see text]. The main result of this paper establishes that the solution of 3D incompressible stationary MHD equations converges to the solution of the 3D incompressible stationary Euler equations as the viscosity coefficient goes to zero.

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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