The structure of the solution to the generalized Stokes equations and a new method for solving them

We show that the velocity solution $u$ to the generalized Stokes equations with Dirichlet boundary conditions in dimension two and three is the superposition of two divergence-free velocity fields $ω$ and $\thetaup$, $u=ω+\thetaup$. Both $ω$ and $\thetaup$ solve screened Poisson vector equations wit