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

Abstract

It is shown that an initial-boundary-value problem for Stokes' system, in which on the boundary one prescribes the vector field of velocities $$\vec v$$ , or the stress field, or the normal component of the velocity and the tangential stresses, reduces to an initial-boundary-value problem for a system of the form $$\vec v_t + A\vec v = \vec f$$ , where the operator A contains a nonlocal term (the so-called singular Green operator). For the solutions of these problems, coercive estimates in the spaces W2 l, l/2 and also estimates of the norm of the resolving operator in W2 r are obtained.