On the Existence of Steady-State Solutions to the Navier-Stokes System for Large Fluxes

In this paper we deal with the stationary Navier--Stokes problem in a domain $\Omega$ with compact Lipschitz boundary $\partial\Omega$ and datum ${\bmit a}$ in Lebesgues spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of $\partial\Omega$, with possible countable exceptional set, provided ${\bmit a}$ is the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux for $\Omega$ bounded.