The Instationary Navier-Stokes Equations in Weighted Bessel-Potential Spaces

We investigate the solvability of the instationary Navier-Stokes equations with fully inhomogeneous data in a bounded domain Omega subset of R(n). The class of solutions is contained in L(r) (O, T; H(w)(beta,q) (Omega)), where H(w)(beta,q) (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.