Rooted trees for 3d Navier-Stokes equation

We establish a representation of a class of solutions of 3d Navier-Stokes equations in $\R^3$ using sums over rooted trees. We study the convergence properties of this series recovering in a simplified manner some results obtained recently by Sinai and other known results for solutions in spaces of pseudo-measures introduced initially by Le Jan and Sznitman. The series representation make sense also in the critical case where there exists global solutions for small initial data and it allows the study of their long-time or small-distance behavior.