ON THE CONVERGENCE OF 3D LERAY-α EQUATIONS WITH NAVIER SLIP BOUNDARY CONDITIONS

Abstract

We investigate the limit behavior of the solutions of the Leray-$\alpha$ equations with Navier slip boundary conditions in a 3D smooth bounded domain to the solution of Euler equations as the parameters $\alpha ,\nu$ tend to zero. Firstly, we focus on the convergence in $L^{\infty }(0,T;L^2(\mathrm{\Omega }))$ by the energy method. Secondly, we prove the convergence in $L^{\infty }(0,T;H^1(\mathrm{\Omega }))$ by choosing the proper expansion ansatz and constructing the corresponding boundary layer profile.