Infinite Boundary Friction Limit for Weak Solutions of the Stochastic Navier-Stokes Equations

We address convergence of the unique weak solutions of the 2D stochastic Navier-Stokes equations with Navier boundary conditions, as the boundary friction is taken uniformly to infinity, to the unique weak solution under the no-slip condition. Our result is that for initial velocity in $L_x^2$, the convergence holds in probability in $C_t{W}_x^{-\epsilon ,2}\cap L_t^2{L}_x^2$ for any $0<\epsilon$. The noise is of transport-stretching type, although the theorem holds with other transport, multiplicative and additive noise structures. This seems to be the first work concerning the large boundary friction limit with noise, and convergence for weak solutions, due to only $L_x^2$ initial data, appears new even deterministically.