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-复合材料该项目将侧重于推进来自物理学的偏微分方程（PDE）的“非经典”均匀化的一般理论，通常在尺度之间具有“强”相互作用。一种一般情况涉及强非均质介质，其可以特别是电磁、声学或弹性的，其中组成成分之间的对比度高。因此，一些组件可以响应于作为谐振器的所施加的动态频率，而其他组件将基本上表现为服从经典均化的准静态状态。采用适当的缩放导致相关PDE的高对比度均质化问题，其中对比度和异质性的微尺度的两个小参数被严格缩放。这种尺度之间的耦合性质导致了宏观尺度上许多有趣的物理效应。这些包括频率带隙，高色散，波局部化等该项目将旨在推进这些影响的数学理解，这将需要开发非经典工具的多尺度渐近分析偏微分方程。精确的主题预计将逐步发展，但一些具体的主题可能包括：分析具有预共振传播频率的高对比度周期性介质中的“耦合共振”的影响;调查由于统计分布的本征频率的共振器的捕获，高对比度共振器对波定位的随机性的影响;分析包括高各向异性在内的更一般的渐近简并模式的方向与频率滤波特性的影响，特别是对于诸如电磁学和弹性动力学中的矢量问题;高对比度晶格材料;通过电容型工具实现小型谐振器的均匀化。另一个有趣的挑战是试图发展一个均匀化理论的方法来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.