Solvability of the Navier-Stokes system with \(L^2\) boundary data (Q1568190)

The author proves the existence of a very weak solution of the Dirichlet problem for the stationary Navier-Stokes system in a bounded domain in \(\mathbb{R}^2\) or in \(\mathbb{R}^3\) with \(L^2\) boundary data. The uniqueness of those solutions is proved under smallness assumptions on the data. First, the penalization method is used to study the linearized problem; then, the Banach fixed point theorem is applied for the nonlinear problem with small boundary data. Results are extended to the case of large data by splitting them into a large regular and small irregular part.