The Stokes System

Settings for the Stokes Problem As a first step to considering the Navier-Stokes equations, we will consider various settings for the Stokes problem of finding u⃗,p which satisfy −ν∇2u⃗ + ∇p = f⃗ in U ⊂ Rn 1 div u⃗ = 0. We could consider this system of equations on: 1. a. U = Rn : in this case we would have to impose some ”behavior at ∞" conditions such as u⃗ → ψ⃗ as | x⃗| → ∞ b. U ⊂ Rn, open bounded with smooth boundary ∂U. In this case we would impose conditions on ∂U such as u⃗ = 0⃗ on ∂U. The more smoothness we require of ∂U the easier the proofs become but the less realistic the problem becomes. c. U = 0,L = n − dim coordinate cell. In this case we impose periodic boundary conditions