Self-Similar Decay of Spatially Localized Perturbations of the Nusselt Solution for the Inclined Film Problem

The Nusselt solution for the flow of a viscous incompressible fluid with a free surface down an inclined plane is at best marginally stable, i.e., the linearization has essential spectrum at least up to the imaginary axis. Nevertheless, using a renormalization group approach here we establish the stability of the Nusselt solution in the full nonlinear system in case of linear stability by proving the self-similar decay of spatially localized perturbations. The asymptotic decay for is similar to the dynamics of localized perturbations of the trivial solution in the Burgers equation on the real line which is the amplitude equation of the problem.