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

We consider initial-boundary value problems for the Navier—Stokes equations prescribing velocities, stresses, or normal component of the velocity and tangential stresses on the boundary. We show that they can be reduced to initial boundary value problems for systems of the form $v_t+Av+Kv=f$ where $A$ is a linear elliptic operator containing a non-local term and $K$ is a nonlinear operator. For these problems we prove a local existence theorem in Sobolev–Slobodetski spaces $W_2^{l,l/2}$.