On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach

Using elementary ODE tools we show that for a class of dissipative PDEs, including Navier–Stokes equations on the plane with periodic boundary conditions, Galerkin projections are C1-convergent, i.e. the partial derivatives with respect to initial conditions for Galerkin projections converge giving rise to smooth dependence on initial conditions for full PDE.