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

Abstract

We consider initial-boundary value problems for systems of Navier-Stokes equations, prescribing on the boundary the velocities, the stresses or the normal component of the velocity and the tangential stresses. We show that they can be reduced to initial-boundary value problems for systems of the form vt + Av + Kv=f, where A is a linear elliptic operator, containing a nonlocal term, while K is a nonlinear operator. For these problems we prove a local solvability theorem in the Sobolev-Slobodetskii spaces W2l, l/2.