On Some Weighted Stokes Problems. Applications on Smagorinsky Models

In this paper we study existence and uniqueness of weak solutions for some non-linear weighted Stokes problems using convex analysis. The characterization of these considered equations is that the viscosity depends on the strain rate of the velocity field with a weight being a positive power of the distance to the boundary of the domain. These non-linear relations can be seen as a first approach of mixing-length eddy viscosity from turbulent modeling. A well known model is von Karman’s on which the viscosity depends on the square of the distance to the boundary of the domain. Numerical experiments conclude the work and show properties from the theory.