On very weak solutions of the stationary Navier-Stokes equations

We study the stationary Navier-Stokes equations in a bounded domain Ω of R3 with smooth connected boundary. The notion of very weak solutions has been introduced by Marušić-Paloka (Appl. Math. Optim. 41:365-375, 2000), Galdi et al. (Math. Ann. 331:41-74, 2005) and Kim (Arch. Ration. Mech. Anal. 193:117-152, 2009) to obtain solvability results for the Navier-Stokes equations with very irregular data. In this article, we prove a complete solvability result which unifies those in Marušić-Paloka (Appl. Math. Optim. 41:365-375, 2000), Galdi et al. (Math. Ann. 331:41-74, 2005) and Kim (Arch. Ration. Mech. Anal. 193:117-152, 2009) by adapting the arguments in Choe and Kim (Preprint) and Kim and Kozono (Preprint). © Università degli Studi di Ferrara 2009.