On isomorphism for the space of solenoidal vector fields and its application to the Stokes problem

We consider the space of solenoidal vector fields in an unbounded domain Ω ⊂ Rn whose boundary is given as a Lipschitz graph. It is shown that, under suitable functional setting, the space of solenoidal vector fields is isomorphic to the n − 1 product space of the space of scalar functions. As an application, we introduce a natural and systematic reduction of the equations describing the motion of incompressible flows. This gives a new perspective of the derivation of Ukai’s solution formula for the Stokes equations in the half space, and provides a key step for the generalization of Ukai’s approach to the Stokes semigroup in the case of the curved boundary.