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

该项目的起点是一个外延薄膜模型，用于在存在Schwoebel势垒的情况下生长晶体表面，该势垒不允许在表面扩散的沉积原子跳过表面的台阶。表面图形的数学模型是由一个四阶半线性抛物型随机偏微分方程组给出的，该方程受到一个小的加性时空白噪声的扰动，该噪声是由于沉积在表面上的材料的热起伏引起的。由于Schwoebel势垒，对于非常陡峭的表面，非线性很小。我们的主要目标是验证随机模型的行为不像建模所预测的那样。造成这一现象的主要原因是噪声引起的稳定效应。我们认为，噪声的空间粗糙度实际上消除了模型的所有非线性影响。确定性模型得到了相当好的研究，人们观察到了最初平坦表面上丘陵的生长，因为对于小的解，非线性表现为线性不稳定。然而，随机模型的解不够规则，不能用标准方法求解方程。这里必须考虑规则结构或副受控方法中的重整化效应。我们认为，任意小的白时空噪声的存在出人意料地消除了所有的非线性效应，导致了动力学的稳定，并抑制了这些模型中的山丘增长。然而，模型的离散化或规则化的噪声仍然可以导致模型中图案的形成，正如物理实验所预期的那样。这一点在确定性方程的分析和数值模拟中也很明显。但是，即使添加任意少量的噪声也可以消除所有的非线性效果，没有可见的图案形成，表面保持平坦。我们项目的一个主要目标是了解由噪声的粗糙度引起的山丘生长和平坦表面之间的这种过渡。

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.