Homogenisation of Multiscale Partial Differential Equations from Physics

研究领域：连续介质力学、数学分析、数学物理和材料工程-复合材料该项目将专注于提出偏微分方程式(PDE)来自物理学的“非经典”齐次化的一般理论，通常具有尺度之间的“强”相互作用。一种一般情况涉及强非均质介质，特别是电磁、声波或弹性介质，其中组成成分之间的对比度很高。其结果是，某些组件可能响应于所施加的动态频率作为谐振器，而其他组件将基本上表现为服从经典均化的准静态状态。采用适当的尺度会导致相关偏微分方程组的高对比度均匀化问题，其中对比度和非均质性的微观尺度这两个小参数被临界标度。这种尺度之间的耦合的性质导致了宏观尺度上的一些有趣的物理效应。这些影响包括频率带隙、高度色散、波的局部化等。该项目的目的是促进对这些影响的数学理解，这将需要开发对偏微分方程进行多尺度渐近分析的非经典工具。预计会发展精确的主题，但一些特定的主题可能包括：分析具有预共振传播频率的高对比度周期介质中的“耦合共振”的影响；调查高对比度谐振器的随机性对由于具有统计分布的特征频率的谐振器捕获而引起的波局部化的影响；针对更一般的渐近简并模式(包括高各向异性，特别是矢量问题，如在电磁和弹性动力学中)，分析方向与频率滤波特性的影响；高对比度晶格材料；通过电容型工具对小尺寸谐振器的均匀化。另一个耐人寻味的挑战是试图开发一种齐次化理论方法来解释爱因斯坦的广义相对论方程。

Research areas: Continuum mechanics, Mathematical analysis, Mathematical physics and Material Engineering - compositesThe project will focus on advancing a general theory of ''non-classical'' homogenisation for partial differential equations (PDEs) coming from Physics, typically with a ''strong'' interaction between the scales. One general scenario relates to strongly heterogeneous media, which may in particular be electromagnetic, acoustic or elastic, where the contrast between the constituent components is high. As a result, some component may respond to an applied dynamic frequency as resonators while others will essentially behave as in a quasi-static regime amenable to a classical homogenisation. Adopting appropriate scaling results in a high-contrast homogenisation problem for relevant PDE, where the two small parameters of the contrast and of the microscale of the heterogeneity are critically scaled. The nature of such a coupling between the scales leads to a number of interesting physical effects at the macroscale. These include frequency bandgaps, high dispersion, wave localisation, etc. The project will be aiming at advancing mathematical understanding of these effects, which would require development of non-classical tools of multi-scale asymptotic analysis of PDEs. Precise topics are envisaged to evolve, but some specific topics may include: analysing the effect of ''coupled resonances'' in high-contrast periodic media with a pre-resonant propagating frequency; investigating the effect of randomness of high-contrast resonators on wave localisation due to trapping by the resonators with statistically distributed eigenfrequencies; analysing the effects of directional vs frequency filtering properties for more general patterns of asymptotic degeneracies including high anisotropies and particularly for vector problems such as in electromagnetism and elastodynamics; high-contrast lattice materials; homogenisation of small-size resonators via capacity-type tools. One additional intriguing challenge is to attempt to develop a homogenisation theory approach to Einstein's equations of General Relativity.