Stability of deterministic and stochastic nonlinear PDEs of parabolic type with degeneracies

我们研究二阶和四阶（随机）偏微分方程，其中一个重点是研究由退化抛物型四阶偏微分方程（薄膜方程）描述的粘性薄膜的自由边界问题。这是申请人现有研究的延续，申请人在博士期间已编写了相应的出版物。在MPI莱比锡研究和博士后留在加拿大和美国。特别感兴趣的是在接触线，其中相液体，气体和固体合并的解决方案的规律性的理解。充分的正则性使得证明解的唯一性和定性性质成为可能。拟议的项目旨在改进现有的研究，从二维扩展到三维，并推广到更大的一类方程，对应于液体和固体之间界面的不同假设。解的时间和空间渐近性也是令人感兴趣的。这些项目的一部分将与Christian Seis博士（波恩大学）和Mircea Petrache博士（MPI Leipzig/波恩大学）共同进行。除了继续现有项目外，我们还希望将分析扩展到薄膜方程的随机版本。这个方程大约在十年前在物理学文献中提出，到目前为止还没有在数学期刊上发表。重点将放在温和的解决方案的存在性证明，我们希望在确定性的情况下，可以应用的分析。此外，我们想研究这个方程的时间渐近性。其目的是严格证明在随机情况下的时间渐近性的变化。该问题与自相似解的稳定性有关，申请人在确定性情形下对自相似解的稳定性进行了详细研究，并进一步研究了随机反应扩散方程行波解的稳定性.不同的稳定性概念将被研究的数值和分析，我们还想考虑在时间上的渐近。这将部分与Christian Kuehn博士合作。申请人希望与上述研究项目产生协同效应，在第3年，申请人希望开始长期研究项目，例如使用随机常微分方程系统描述液滴的粗化动力学，以及使用Evans函数研究随机偏微分方程的行波。

We are investigating (stochastic) partial differential equations of second and fourth order.One focus is placed on the investigation of free boundary problems for viscous thin films, described by a degenerate-parabolic partial differential equation of fourth order (thin-film equation). This is a continuation of existing research of the applicant, for which the respective publications have been prepared during the applicant's Ph.D. studies at the MPI Leipzig and postdoctoral stays in Canada and the United States. Specific interest lies in the understanding of the regularity of solutions at the contact line, where the phases liquid, gas, and solid coalesce. Sufficient regularity makes the proof of uniqueness and qualitative properties of solutions possible. The proposed projects aim at improving existing research regarding the extension from two to three dimensions and the generalization to a larger class of equations, corresponding to different assumptions at the interface between liquid and solid. The temporal and spatial asymptotics of solutions are of interest as well. Parts of these projects will be carried out jointly with Dr. Christian Seis (University of Bonn) and Dr. Mircea Petrache (MPI Leipzig/University of Bonn).In addition to a continuation of existing projects, we would further like to extend the analysis to a stochastic version of the thin-film equation. This equation has been proposed roughly ten years ago in the physical literature and has up to now not been addressed in mathematical journals. The focus will be placed on the proof of existence of mild solutions, where we expect that the analysis for the deterministic case can be applied. Furthermore, we would like to investigate the time asymptotics of this equation. The aim is to rigorously prove the change of the time asymptotics in the stochastic case. This question is connected to the stability of self-similar solutions, which the applicant has investigated in detail in the deterministic case.A further focus is placed on the analysis of stability of traveling waves for stochastic reaction-diffusion equations. Different notions of stability shall be investigated numerically and analytically, and we would further like to consider the asymptotics in time. This will be partially in collaboration with Prof. Christian Kuehn, Ph.D. The applicant expects synergies between this and the previously mentioned research projects.In the third year, the applicant would like to start working on long-term projects such as the description of the coarsening dynamics of droplets using systems of stochastic ordinary differential equations and the investigation of traveling waves for stochastic partial differential equations using the Evans function.