Large Friction Limit and the Inviscid Limit of 2D Navier–Stokes Equations Under Navier Friction Condition

Abstract

We show that as the friction coefficient tends to infinity fast enough, the Navier friction condition recovers the no-slip boundary condition for the 2D incompressible Navier–Stokes (NS) equations in a smooth bounded domain. We also show that as the viscosity and the (Navier) friction coefficient tend to zero and infinity, respectively, satisfying certain conditions, the incompressible NS equations tend to the incompressible Euler equations in $L^2$ for initial data in $H^2$. We give explicitly the rates of convergence in $L^2$ of these two limits in terms of the viscosity and the friction coefficient.