PDEs with highly oscillatory coefficients: homogenization and beyond

这项研究旨在开发新的数学工具，用于研究决定这些现象和环境的参数在时间和/或空间上迅速变化的重要现象。例如，在材料科学、大气科学、燃烧、生物医学成像和金融市场动态中，这种现象无处不在。在这种情况下，速度是由尺度决定的：大气现象可能是几天或几小时，化学过程可能是几秒或毫秒。在数学术语中，这些现象可用具有振荡系数的偏微分方程组(PDE)来模拟。这项研究将通过数学分析和计算机模拟来促进对潜在偏微分方程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非均匀偏微分方程组；将评估方法在捕捉大尺度行为和关于解的涨落的某些微观尺度信息方面的性能。主要研究人员将开展以下研究：(I)发展哈密顿和/或扩散系数在空间和时间上高度振荡的动态随机环境中的Hamilton-Jacobi方程的齐化理论；特别是，将开发新的技术来克服由于胞元问题的时间相关性而导致的一致Lipschitz界的缺乏；(Ii)对于具有随机势的线性和半线性椭圆型方程，通过平均化给出其有效势，刻画齐化误差的极限概率分布，并将研究推广到包含非均匀系数的情况；(Iii)研究多尺度数值方法，评价它们在捕捉非均匀偏微分方程组的宏观尺度性质和关于解的某些微观尺度涨落信息方面的性能，并对这些方法进行改进。

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.