Existence, Uniqueness and Regularity of Stationary Solutions to Inhomogeneous Navier-stokes Equations in R

For a bounded domain Ω ⊂ Rn , n > 3, we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system −∆u + u · ∇u +∇p = f , div u = k, u|∂Ω = g with u ∈ L q , q > n, and very general data classes for f , k, g such that u may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of papers by Frehse & Růžička, see e.g. Existence of regular solutions to the stationary Navier-Stokes equations, Math.Ann. 302 (1995), 669–717, where the existence of a weak solution which is locally regular is proved.