Mathematical Sciences: Nonlinear Partial Differenial Equations

9622305菲利普斯 这个项目是从两个方面研究微分方程， 应用于超导的金兹伯格-朗道方程和与非线性弹性有关的变分法。特征 Ginzburg-朗道方程的解。 实验 他们所模拟的材料的证据和 方程本身表明，稳定的解决方案包含相干模式 形成（涡流阵列）。建议分析简单的解决方案 几何形状（例如圆柱棒和薄膜）。主要研究者 试图建立这些模式的存在，并了解 使其产生的机制。他还试图确定如何控制模式（钉涡分布）。这样做将导致更好地理解模型和估计的有效性 指一种材料在受到外加电流或磁场的作用时能产生超电流。建议的第二部分涉及的规则， 与非线性弹性有关的变分问题的解。P.鲍曼和主要研究者已经确定，某些问题的解决方案 二维弹性力学的边值问题是局部Lipschitz边值问题 连续同胚建议调查这些是否 解具有奇异性。从分析上讲，这是在问解决方案是否 处处可微。 %%% 科学家描述了一种给定的材料如何传导电流，或者弹性体如何扭曲和弯曲， 数学方程（称为偏微分方程） 基于每种情况下的物理基础。主要研究者 研究了这些解决方案的特点。例如，在一个 超导材料（一种可以导电的材料， 艾德认为，电流自然地绕着 对称的点阵列称为涡。还观察到，如果 电流足够强，阵列开始漂移， 然后失去超导性。模式的形成和 不稳定性是可以通过解来研究的特征。 理解并能够预测这些特征对于以下两个方面都很重要： 理论和工程应用。 ***

9622305 Phillips This project is a study of differential equations from two areas, the Ginzburg--Landau equations as they apply to superconductivity and the calculus of variations related to nonlinear elasticity. Features of solutions to the Ginzburg--Landau equations are to be studied. Experimental evidence for the materials they model and numerical simulations for the equations themselves indicate that stable solutions contain coherent pattern formations (vortex arrays). It is proposed to analyze solutions for simple geometries (e.g. cylindrical rods and thin films). The principal investigator seeks to establish the existence of these patterns and to understand the mechanisms that bring them about. He also seeks to determine how one can control the patterns (pin the vortex distributions). Doing this will lead to a better understanding of the models and to estimates for the effectiveness of a material to carry a supercurrent when subjected to an applied current or a magnetic field. The second part of the proposal deals with the regularity of solutions to variational problems related to nonlinear elasticity. P. Bauman and the principal investigator have established that solutions to certain boundary value problems from two dimensional elasticity are locally Lipschitz continuous homeomorphisms. It is proposed to investigate whether or not these solutions have singularities. Analytically this is to ask if the solutions are differentiable everywhere. %%% Scientists describe how a given material can conduct an electric current or how an elastic body can be twisted and bent by expressing such phenomena as solutions to mathematical equations (called partial differential equations) based on the underlying physics in each instance. The principal investigator studies the features of these solutions. For example in the case of a superconducting material (a material that can conduct electricity very efficiently) it is observ ed that the current naturally circulates around a symmetric array of points called vorticies. It is also observed that if the current is sufficiently strong the array begins to drift causing the current to die off and superconductivity is lost. Pattern formation and the onset of instability are features that can be investigated through the solutions. Understanding and being able to predict these features is important both for theory and engineering applications. ***