Elliptic and Parabolic Equations for Partially Ordered Materials in Applied Fields

将分析超导和液晶材料在外加电场作用下的数学模型。首先，研究具有实际边界条件的稳态和时变非线性方程的金兹堡-朗道系统在外加磁场作用下的解的行为。其次，分析了应用领域中描述液晶系统的解的行为。这些问题的数学模型是偏微分方程组。本项目将采用的方法包括极大值原理、分岔理论、正则性、能量估计、偏微分方程的大时间分析以及数值方法。现代技术设备，如超导薄膜和液晶显示器，通常是在电场或磁场的存在下运行的。因此，有必要了解这些字段如何影响它们的性能。在超导体的情况下，存在外部磁场。这可能导致材料中的非超导位置（称为涡流）移动，从而降低设备性能。工程师们试图通过在材料中引入不均匀性来控制这些损失，这种不均匀性可以减少涡流的运动，但目前还不清楚所有这些影响是如何相互作用的，以及如何最好地利用不均匀性来实现可持续性。在液晶的情况下，磁场和电场被用来改变光学方向。这对于显示设备和可视屏幕的操作是必不可少的。这种情况下的材料是液晶和聚合物的混合物，导致比超导情况下更复杂的现象。然而，数学问题是密切相关的。该奖项将支持对数学模型的研究，这些模型将对这两类问题提供基本的见解，并有助于设计有效的数值模拟。

Mathematical models for superconducting and liquid crystal materialsin the presence of applied fields will be analyzed. First, thebehavior of solutions to the Ginzburg-Landau system of steady-stateand time-dependent nonlinear equations with realistic boundaryconditions in the presence of an applied magnetic field will beinvestigated. Second, the behavior of solutions to the systemdescribing liquid crystals in applied fields is to be analyzed.The mathematical models for these problems are systems of partialdifferential equations. Methods to be employed in this project include maximum principles, bifurcation theory, regularity, energy estimates, and large-time analysis of partial differential equations, as well as numerical methods. Modern technological devices such as superconducting films andliquid crystal displays typically are operated in the presence of electric ormagnetic fields. It is therefore necessary to understand how these fields affecttheir performance. In the case of superconductors, external magneticfields are present. This can result in nonsuperconducting sitesin the material (called vortices) which move and therefore degrade the device performance. Engineers have tried to control these lossesby introducing inhomogeneities in the material that curtail themotion of vortices, but it is not well understood how all of theseeffects interact and how best to use inhomogeneities to achievestability. In the case of liquid crystals, magnetic and electricfields are used to change optic orientations. This is essential to the operation of display devices and visual screens. The materials in this caseare mixtures of liquid crystals and polymers, resulting in more complicated phenomena than in the case of superconductivity. Still, the mathematicalissues are closely related. The award will support research on mathematical models that will give basic insights into both types of problems and help to devise effective numerical simulations.