权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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.

在一系列的文件，巴布，大普拉托，盖斯，金，Röckner，和其他人最近研究的存在性和非负性方面的随机版本的二阶退化抛物方程。对于随机多孔介质方程，也可以建立解的支撑的有限传播-因此隐含地，这些方程构成自由边界问题。本文基于我们最近关于随机多孔介质方程的有限传播和等待时间现象的定性结果，我们关于随机薄膜方程的存在性结果，作为模型方程，我们研究了随机多孔介质方程、随机抛物型p-Laplace方程和随机薄膜方程。为了保证几乎必然全局非负解的存在性，只考虑乘性噪声。它可能出现在源项或对流项内。从物理上讲，随机薄膜方程已从随机Navier-Stokes方程中推导出来，以模拟热波动对液滴扩散和不稳定液膜去湿的影响。特别是在纳米尺度上，随机薄膜方程能够捕捉到确定性方程无法描述的现象。从分析上讲，研究二阶方程是重要的第一步。事实上，在确定性的设置，统一的分析方法可获得最佳结果的传播速率和等待时间的大小为大类的二阶和高阶退化抛物方程。因此，对二阶退化抛物型方程的随机形式的研究有望提供重要的方法论见解。本着这种精神，我们努力的传播速率的预期值和二阶方程的等待时间的大小的定量估计。在有限传播和等待时间现象的发生仍然是开放问题的情况下，我们首先寻找定性的结果。概念上，分析方法是适应基于函数不等式的能量方法（类似Stampacchia引理的版本）或微分不等式到随机设置。对于随机薄膜方程，迄今为止只知道严格正解的存在性结果，我们研究收敛的数值方案，并使用它们进行蒙特-卡罗模拟，以获得关于噪声对散粒液滴扩散影响的经验证据。