Solutions of the Navier-Stokes equations (NSE) satisfy the same scaling invariance as the solutions of the heat equation. However, as opposed to the exponential decay of the heat kernel, the kernel of the solution operator of the linear problem associated with NSE (the Stokes system), has only polynomial decay. We consider a parabolic system that shares many features with NSE (scaling, energy estimate) and show that it may be thought of as an approximation of the Navier-Stokes equations. In particular, we address the problem of convergence of solutions to solutions of NSE and partial regularity questions.