Reduction of basic initial-boundary value problems for the Stokes equations to initial-boundary value problems for parabolic systems of pseudodifferential equations

It is shown that the initial-boundary value problems for the Stokes equations with the prescription of velocities $\vec{v}$, stresses, or of the normal component of velocity and tangential stresses on the boundary can be reduced to initial-boundary value problems for systems $\vec{v}_t+A\vec{v}=\vec{f}$ where $A$ is a linear operator containing a npa-local term (the so called singular Green operator). For solutions; of these problems coercive estimates in the sobolev space» $W_2^{\ell,\ell/2}$ and the estimate, of the norm of the resolving operator in $W_2^r$ are given.