Boundary Asymptotic Analysis for an Incompressible Viscous Flow: Navier Wall Laws

We consider a new way of establishing Navier wall laws. Considering a bounded domain $\mathrm{\Omega }$ of R N , N=2,3, surrounded by a thin layer $\mathrm{\Sigma }ϵ$, along a part $\mathrm{\Gamma }$2 of its boundary $\mathrm{\partial }\mathrm{\Omega }$, we consider a Navier-Stokes flow in $\mathrm{\Omega }\cup \mathrm{\partial }\mathrm{\Omega }\cup \mathrm{\Sigma }ϵ$ with Reynolds’ number of order 1/$ϵ$ in $\mathrm{\Sigma }ϵ$. Using $\mathrm{\Gamma }$-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier law involving a matrix of Borel measures having the same support contained in the interface $\mathrm{\Gamma }$2. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider an optimal control problem within this context.