Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spaces

Abstract

We investigate very weak solutions to the stationary Stokes and Stokes resolvent problem in function spaces with Muckenhoupt weights. The notion used here is similar but even more general than the one used in Amann (Nonhomogeneous Navier–Stokes equations with integrable low-regularity data. Int. Math. Ser., pp. 1–26. Kluwer Academic/Plenum Publishing, New York, 2002) or Galdi et al. (Math. Ann. 331, 41–74, 2005). Consequently the class of solutions is enlarged. To describe boundary conditions we restrict ourselves to more regular data. We introduce a Banach space that admits a restriction operator and that contains the solutions according to such data.