Uniform regularity for the incompressible Navier-Stokes system with variable density and Navier boundary conditions

We investigate the uniform regularity for the nonhomogeneous incompressible Navier-Stokes system with Navier boundary conditions and the inviscid limit to the Euler system. It is shown that there exists a unique strong solution of the Navier-Stokes equations in an interval of time that is uniform with respect to the viscosity parameter. The uniform estimate in conormal Sobolev spaces is established. Based on the uniform estimate, we show the convergence of the viscous solutions to the inviscid ones in L ∞ ( [ 0 , T ] × Ω ) L^\infty ([0,T]\times \Omega ) . This improves the result obtained by Ferreira et al. [SIAM J. Math. Anal. Vol. 45, No. 4, (2013), pp. 2576-2595], where L ∞ ( [ 0 , T ] , L 2 ( Ω ) ) L^\infty ([0,T],L^2(\Omega )) convergence was proved.