Remarks on vanishing viscosity limits for the 3D Navier-Stokes equations with a slip boundary condition

Abstract

The authors study vanishing viscosity limits of solutions to the 3-dimensional incompressible Navier-Stokes system in general smooth domains with curved boundaries for a class of slip boundary conditions. In contrast to the case of flat boundaries, where the uniform convergence in super-norm can be obtained, the asymptotic behavior of viscous solutions for small viscosity depends on the curvature of the boundary in general. It is shown, in particular, that the viscous solution converges to that of the ideal Euler equations in C([0, T];H ¹(Ω)) provided that the initial vorticity vanishes on the boundary of the domain.

Keywords

Poincaré inequality / Sobolev spaces with variable exponent

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