Well-Posedness of Evolutionary Navier-Stokes Equations with Forces of Low Regularity on Two-Dimensional Domains

Abstract

Existence and uniqueness of solutions to the Navier-Stokes equation in dimension two with forces in the space $L^q((0,T);{\mathbf{W}}^{-1,p}(\Omega ))$ for $p$ and $q$ in appropriate parameter ranges are proven. The case of spatially measured-valued inhomogeneities is included. For the associated Stokes equation the well-posedness results are verified in arbitrary dimensions with $1<p,q<\infty$ arbitrary.