Solvability of the Navier-Stokes system with L2 boundary data

We prove the existence of the very weak solution of the Dirichlet problem for the Navier-Stokes system with L-2 boundary data. Under the small data assumption we also prove the uniqueness. We use the penalization method to study the linearized problem and then apply Banach's fixed point theorem for the nonlinear problem with small boundary data. We extend our result to the case with no small data assumption by splitting the data on a large regular and small irregular part.