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

Abstract

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.