6 research outputs found
Universal Self-Similar Attractor in the Bending-Driven Leveling of Thin Viscous Films
We study theoretically and numerically the bending-driven leveling of thin
viscous films within the lubrication approximation. We derive the Green's
function of the linearized thin-film equation and further show that it
represents a universal self-similar attractor at long times. As such, the
rescaled perturbation of the film profile converges in time towards the
rescaled Green's function, for any summable initial perturbation profile. In
addition, for stepped axisymmetric initial conditions, we demonstrate the
existence of another, short-term and one-dimensional-like self-similar regime.
Besides, we characterize the convergence time towards the long-term universal
attractor in terms of the relevant physical and geometrical parameters, and
provide the local hydrodynamic fields and global elastic energy in the
universal regime as functions of time. Finally, we extend our analysis to the
non-linear thin-film equation through numerical simulations
Intermediate asymptotics beyond homogeneity and self-similarity: long time behavior for [formula]
"Vegeu el resum a l'inici del document del fitxer adjunt"
Semidiscretization and long-time asymptotics of nonlinear diffusion equations
We review several results concerning the long time asymptotics of nonlinear diffusion models based on entropy and mass transport methods. Semidiscretization of these nonlinear diffusion models are proposed and their numerical properties analysed. We demonstrate the long time asymptotic results by numerical simulation and we discuss several open problems based on these numerical results. We show that for general nonlinear diffusion equations the long-time asymptotics can be characterized in terms of fixed points of certain maps which are contractions for the euclidean Wasserstein distance. In fact, we propose a new scaling for which we can prove that this family of fixed points converges to the Barenblatt solution for perturbations of homogeneous nonlinearities for values close to zero