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

我们研究的是二阶和四阶(随机)偏微分方程组，其中一个重点是研究由四阶退化抛物型偏微分方程(薄膜方程)描述的粘性薄膜的自由边界问题。这是申请者现有研究的继续，申请人在莱比锡MPI攻读博士学位期间以及在加拿大和美国的博士后逗留期间，为其准备了各自的出版物。特别感兴趣的是了解接触线上溶液的规律性，在接触线上，液体、气体和固体相结合。充分的正则性使得解的唯一性和定性性质的证明成为可能。拟议的项目旨在改进关于从二维扩展到三维的现有研究，并将其推广到一类更大的方程，对应于液体和固体界面上的不同假设。解的时间和空间渐近性也令人感兴趣。这些项目的一部分将与Christian Seis博士(波恩大学)和Mircea Petrache博士(MPI莱比锡/波恩大学)共同实施。除了继续现有的项目，我们还想进一步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就在物理文献中提出了，但到目前为止还没有在数学期刊上讨论过。重点将放在温和解的存在性的证明上，我们期望对确定性情况的分析可以应用于此。此外，我们还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时间渐近性的变化。这个问题与自相似解的稳定性有关，申请人已经在确定性的情况下进行了详细的研究。进一步的重点放在随机反应扩散方程的行波稳定性分析上。稳定性的不同概念应该用数值和解析的方法来研究，我们还想进一步考虑时间上的渐近性。这将部分与克里斯蒂安·奎恩博士合作。申请人希望这项研究与前面提到的研究项目之间有协同作用。在第三年，申请人希望开始进行长期项目，例如使用随机常微分方程组描述液滴的粗化动力学，以及使用埃文斯函数调查随机偏微分方程组的行波。

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.