Free boundary propagation and noise: analysis and numerics of stochastic degenerate parabolic equations

自由边界传播和噪声：随机简并抛物线方程的分析和数值

基本信息

批准号：
397495103
负责人：
Professor Dr. Günther Grün
金额：
--
依托单位：
Lehrstuhl für Angewandte Mathematik I
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2018
资助国家：
德国
起止时间：
2017-12-31 至 2020-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/397495103?language=en
关键词：
Free boundary propagation noise analysis

项目摘要

In a series of papers, Barbu, Da Prato, Gess, Kim, Röckner, and others recently studied existence and nonnegativity aspects of stochastic versions of second order degenerate parabolic equations. For stochastic porous-medium equations, finite propagation of the solution's support could be established as well - thus implicitly, these equations constitute free boundary problems. It is the scope of this proposal to investigate analytically and numerically the impact of noise on the propagation of free boundaries in stochastic variants of degenerate parabolic equations.It is based on our recent qualitative results about finite propagation and waiting time phenomena for stochastic porous-medium equations, our existence results for stochastic thin-film equations, and our convergence results for numerical schemes for stochastic porous-medium equations.As model equations, we intend to study stochastic porous-medium equations, stochastic parabolic p-Laplace equations, and stochastic thin-film equations. To guarantee the existence of almost surely globally nonnegative solutions, only multiplicative noise will be considered. It may arise inside a source-term or inside a convective term. Physically, the stochastic thin-film equation has been derived from stochastic Navier-Stokes equations to model the effects of thermal fluctuations on droplet spreading and on the dewetting of unstable liquid films. In particular on nano-scales, stochastic thin-film equations turn out to capture phenomena which cannot be described by their deterministic counterparts. Analytically, the investigation of second order equations is an important first step. In fact, in the deterministic setting, unifying analytical methods are available to obtain optimal results on propagation rates and on the size of waiting times for large classes of second and higher order degenerate parabolic equations. Accordingly, studies on stochastic versions of second order degenerate parabolic equations are expected to provide important methodological insight. In this spirit, we strive for quantitative estimates on the expected values of propagation rates and on the size of waiting times for second order equations. In situations where finite propagation and occurrence of waiting time phenomena are still open problems, we first look for qualitative results.Conceptually, the analytical approach is to adapt energy methods based on functional inequalities (like versions of Stampacchia's lemma) or differential inequalities to the stochastic setting.For stochastic thin-film equations for which so far only existence results for strictly positive solutions are known, we study convergent numerical schemes and we use them for Monte-Carlo simulations to obtain empirical evidence on the noise impact on the spreading of bulk droplets.

INA系列的论文，Barbu，Da Prato，Gim，Röckner和其他纸张的存在和随机版本的非晶格抛物线方程的随机版本的非成分方面。因此，这些方程式与自由边界问题相结合。随机多孔中等方程的数值方案。从身体上讲，在身体上，方程式是从随机的Navier-Stokes方程中得出的，以模拟热波动散布的影响和对不稳定液体薄膜的侵蚀。捕获确定性的现象，可以在确定性的环境中进行统一的分析方法预计将提供重要的方法论。差异不平等。随机薄膜的存在结果严格阳性溶液是已知的，从而获得了有关噪声影响散布液滴的噪声影响的经验证据。