The stationary Navier-Stokes equations in weighted Bessel-potential spaces

Abstract

We investigate the solvability of the instationary Stokes equations with fully inhomogeneous data in \({L^r(0,T;H^{\beta,q}_w(\Omega))}\) , where \({H^{\beta,q}_w(\Omega)}\) is a Bessel-potential space with a Muckenhoupt weight w. Depending on the order of this Bessel-potential space we are dealing with strong solutions or with very weak solutions. Whereas in the context of lowest regularity one obtains solvability with respect to inhomogeneous data by dualization, this is more delicate in the case of higher regularity, where one has to introduce some additional time regularity. As a preparation, we introduce a generalization of the Stokes operator that is appropriate to the context of very weak solutions in weighted Bessel-potential spaces.