Internal stabilization of the Oseen-Stokes equations by Stratonovich noise

(1) ∂X ∂t − ν∆X + (f · ∇)X + (X · ∇)g = ∇p in (0,∞)×O, ∇ ·X = 0 in (0,∞)×O, X(0, ξ) = x(t), ξ ∈ O, X = 0 on (0,∞)× ∂O. Here, O is an open and bounded subset of R, d = 2, 3, with smooth boundary ∂O and f, g ∈ C2(O;Rd) are given functions. In the special case g ≡ 0, system (1) describes the dynamic of a fluid Stokes flow with partial inclusion of convection acceleration (f ·∇)X (X is the velocity field). The same equation describes the disturbance flow induced by a moving body with velocity f through the fluid. Should we mention also that in the special case f ≡ g ≡ Xe, where Xe is the equilibrium (steady-state) solution of the Navier–Stokes equation