Existence and Regularity of Very Weak Solutions of the Stationary Navier–Stokes Equations

Abstract

We consider the stationary Navier–Stokes equations in a bounded domain Ω in Rn with smooth connected boundary, where n = 2, 3 or 4. In case that n = 3 or 4, existence of very weak solutions in Ln (Ω) is proved for the data belonging to some Sobolev spaces of negative order. Moreover we obtain complete Lq-regularity results on very weak solutions in Ln (Ω). If n = 2, then similar results are also proved for very weak solutions in \({L^{q_0} (\Omega )}\) with any q0 > 2. We impose neither smallness conditions on the external force nor boundary data for our existence and regularity results.