The Instationary Navier–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.