A local asymptotic expansion for a local solution of the Stokes system

We consider solutions of the Stokes system in a neighborhood of a point in which the velocity $u$ vanishes of order $d$. We prove that there exists a divergence-free polynomial $P$ in $x$ with $t$-dependent coefficients of degree $d$ which approximates the solution $u$ of order $d+\alpha$ for certain $\alpha >0$. The polynomial $P$ satisfies a Stokes equation with a forcing term which is a sum of two polynomials in $x$ of degrees $d-1$ and $d$. The results extend to Oseen systems and to the Navier-Stokes equation.