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

该项目的出发点是一个外延薄膜模型，用于在Schwoebel屏障存在下生长晶体表面，该屏障不允许在表面上扩散的沉积原子向下跳过表面中的台阶。的表面的图形的数学模型是由一个四阶半线性抛物随机偏微分方程扰动的一个小的添加剂时空白色噪声，由于在表面上沉积的材料的热波动。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预测的那样。其主要原因是噪声引起的稳定效应。 我们相信，噪声的空间粗糙度实际上消除了模型的所有非线性效应。 确定性模型是相当好的研究，和一个观察到的增长的山丘上的一个最初的平面，因为非线性行为像一个线性不稳定的小的解决方案。 然而，随机模型的解不够规则，无法用标准方法求解方程。这里必须考虑正则性结构或次受控方法中的重整化效应。我们相信，任意小的白色时空噪声的存在令人惊讶地消除了所有的非线性效应，导致稳定的动力学和抑制增长的山丘在这些模型中。然而，模型的离散化或正则化噪声仍然可以导致模型中的图案形成，正如物理实验所预期的那样。这在确定性方程的分析和数值模拟中也是明显的。但是，即使添加任意少量的噪声，也会消除所有非线性效应，看不到任何图案形成，并且表面保持平坦。我们的项目的一个主要目标是了解这种过渡之间的增长丘陵和平坦的表面所造成的粗糙度的噪音。

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.