Analysis of Mathematical Models of Materials

该项目旨在分析液晶和超导材料的数学模型，重点关注其中发生的电磁、机电和光电相互作用。要分析的超导材料的数学模型是基于金兹堡-朗道能量的，该项目将包括对数学解的定性行为的研究，这对高温、各向异性超导材料在外加电场和磁场存在下的应用很重要。要分析的液晶的数学模型是基于De Gennes的模型，该模型结合了Oseen-Frank能量项以及考虑近晶层（存在于手性材料中）、弹性和外加电场和磁场影响的术语。将研究包含越来越复杂的非线性相互作用的层次问题。特别强调将给予分析缩放从中尺度模型，其中物理被描述为宏观尺度模型在应用中使用。将采用数学建模、偏微分方程和有限弹性等技术。该项目将基于高温超导材料和液晶材料的微观结构，更好地理解它们对外加磁场和电场的响应。应用将包括为电力传输设计合适的超导体和创造强大的磁场。液晶研究的应用包括设计更小、更快、更精确的光开关、纳米级机电设备和液晶显示器。

The proposed project is aimed at analyzing mathematical models for liquid crystalline and superconducting materials focusing on the electro-magnetic,electromechanical, and electro-optical interactions that take place withinthem. Mathematical models for superconducting materials to be analyzedare based on the Ginzburg-Landau energy, and the project will include aninvestigation of the qualitative behavior of mathematical solutions which are important for applications such as high-temperature, anisotropic superconducting materials, in the presence of applied electric and magnetic fields. Mathematical models for liquid crystals to be analyzed are based on De Gennes' models incorporating the Oseen-Frank energy terms as well asterms accounting for smectic layering (present in chiral materials),elasticity, and the influence of applied electric andmagnetic fields. A hierarchy of problems will be studied incorporatingmore and more complex nonlinear interactions. Special emphasis will be given to the analysis of scaling up from meso-scale models where thephysics is described to macro-scale models used in applications. Techniquesfrom mathematical modeling, partial differential equations, and finiteelasticity will be employed. The project will result in a better understanding of the responses ofhigh-temperature superconducting materials and liquid crystallinematerials to applied magnetic and electric fields based on their microscopic structure. Applications will include the design of appropriate superconductorsfor power transmission and the creation of powerful magnetic fields. Applications of the liquid crystal research include the design of smaller, faster, and more accurate optical switches, nanoscale electro-mechanicaldevices, and liquid crystal displays.