Existence and stability of steady-state solutions with finite energy for the Navier–Stokes equation in the whole space

We consider the steady-state Navier–Stokes equation in the whole space driven by a forcing function f. We show that there exists a constant M_o > 0 such that for any M ⩾ M_o, provided the source terms f are sufficiently small in a natural norm (the smallness depending only on M), and the low frequencies of f are sufficiently controlled then there exists a solution U with bounds . These solutions are unique among all solutions with finite energy and finite Dirichlet integral. Using Fourier splitting tools, the constructed solutions will be shown to be stable in the following sense: if U is such a solution then any viscous, incompressible flow in the whole space, driven by f and starting with finite energy, will return to U.