PDEs with highly oscillatory coefficients: homogenization and beyond

具有高振荡系数的偏微分方程：均质化及其他

• 批准号: 1515150
• 负责人: Wenjia Jing
• 金额: $ 13万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1515150&HistoricalAwards=false
• 关键词: PDEs; highly; oscillatory; coefficients; homogenization

This research is directed to development of new mathematical tools for investigation of important phenomena where the parameters determining these phenomena and the environment change rapidly in time and/or space. Such phenomena are ubiquitous, for example, in material science, atmospheric science, combustion, biomedical imaging, and the dynamics of financial markets. In this context the rapidity is determined by the scale: it might be days or hours for atmospheric phenomena, or seconds and milliseconds for chemical processes. In mathematical terms these phenomena are modeled by partial differential equations (PDE) with oscillatory coefficients. This research will foster both qualitative and quantitative understanding of solutions of underlying PDE through mathematical analysis and computer simulations. In particular, the project will result in the design of multi-scale numerical methods to simulate heterogeneous PDEs; performance of the methods in capturing the large-scale behavior and certain micro-scale information about the fluctuations of the solutions will be evaluated.The Principal Investigator will carry out the following studies: (i) develop homogenization theory for Hamilton-Jacobi equations in dynamic random environments where the Hamiltonian and/or the diffusion coefficients are highly oscillatory in space and time; in particular, new techniques will be developed to overcome the lack of uniform Lipschitz bounds due to the time dependence of the cell problem; (ii) for linear and semi-linear elliptic equations with random potential, for which effective potential is given by averaging, characterize the limiting probability distribution of the homogenization error, and generalize the studies to the case where the differential operator involves heterogeneous coefficients; (iii) study multi-scale numerical methods and evaluate their performance in capturing not only the macro-scale property of the heterogeneous PDEs but also certain micro-scale information about the fluctuations of the solutions, and to improve such methods.

本研究旨在开发新的数学工具，用于调查重要现象，其中确定这些现象和环境的参数在时间和/或空间上迅速变化。这种现象普遍存在于材料科学、大气科学、燃烧、生物医学成像和金融市场动态等领域。在这种情况下，速度是由尺度决定的：对于大气现象来说，它可能是几天或几小时，对于化学过程来说，它可能是几秒或几毫秒。在数学术语中，这些现象由具有振荡系数的偏微分方程（PDE）建模。 这项研究将通过数学分析和计算机模拟促进对潜在PDE解决方案的定性和定量理解。 特别是，该项目将导致设计多尺度数值方法来模拟异质偏微分方程;将评估方法在捕获大尺度行为和关于解波动的某些微观信息方面的性能。首席研究员将进行以下研究：（i）在动态随机环境中发展Hamilton-Jacobi方程的均匀化理论，其中Hamiltonian和/或扩散系数在空间和时间上是高度振荡的;特别是，将开发新的技术来克服由于单元问题的时间依赖性而缺乏统一的Lipschitz界限;（ii）对于具有随机位势的线性和半线性椭圆型方程，其有效位势是通过平均给出的，刻画了均匀化误差的极限概率分布，并将研究推广到微分算子包含非齐次系数的情形;（iii）研究多尺度数值方法，并评估它们在捕捉非均匀偏微分方程的宏观尺度特性和某些微观特性方面的性能，尺度信息的波动的解决方案，并改善这种方法。