The instationary Stokes equations in weighted Bessel-potential spaces

Abstract

We investigate the solvability of the instationary Navier–Stokes equations with fully inhomogeneous data in a bounded domain \(\Omega \subset {{\mathbb{R}}^{n}} \). The class of solutions is contained 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. In this context we derive solvability for small data, where this smallness can be realized by the restriction to a short time interval. Depending on the order of this Bessel-Potential space we are dealing with strong solutions, weak solutions, or with very weak solutions.