On the existence for the Cauchy-Neumann problem for the Stokes system in the L-p-framework

The existence for the Cauchy-Neumann problem for the Stokes system in a bou nded domain Omega subset of R-3 is proved in a class such that the velocity belongs toW(r)(2,1) (Omega x (0, T)), where r > 3. The proof is divided in to three steps. First, the existence of solutions is proved in a half-space for vanishing initial data by applying the Marcinkiewicz multiplier theore m. Next, we prove the existence of weak solutions in a bounded domain and t hen we regularize them. Finally, the problem with nonvanishing initial data is considered.