Stabilization by rough noise for an epitaxial thin-film growth model

• Applicant: Professor Dr. Dirk Blömker
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 514726621

The starting point of the project is an epitaxial thin-film model for the growth of a crystalline surface in the presence of a Schwoebel barrier, which does not allow the deposited atoms, that are diffusing on the surface, to jump down over steps in the surface. The mathematical model for the graph of the surface is given by a fourth-order semilinear parabolic stochastic partial differential equation perturbed by a small additive space-time white noise due to thermal fluctuations in the material deposited on the surface. Due to the Schwoebel barrier the nonlinearity is small for very steep surfaces. Our main goal is to verify that the stochastic model does not behave as predicted by the modelling. The main reason for this are stabilization effects caused by the noise. We believe that the spatial roughness of the noise actually eliminates all non-linear effects of the model. The deterministic model is reasonably well studied, and one observes the growth of hills on an initially flat surface, because the nonlinearity behaves like a linear instability for small solutions. However, the solution for the stochastic model is not regular enough to solve the equation with standard methods. Renormalisation effects as in regularity structures or the paracontrolled approach must be taken into account here. We believe that the presence of arbitrarily small white space-time noise surprisingly eliminates all nonlinear effects, leading to a stabilization of the dynamics and suppressing the growth of hills in these models. Nevertheless, a discretisation of the model or a regularized noise can still lead to pattern formation in the model as expected from physical experiments. This is also evident in the analysis and numerical simulations for the deterministic equation. But even adding an arbitrarily small amount of noise removes all non-linear effects, no pattern formation is visible, and the surface remains flat. A main goal of our project is to understand this transition between the growth of hills and a flat surface caused by the roughness of the noise.

• DFG Programme: Research Grants