Nonlinear Partial Differential Equations Methods in the Study of Interfaces

这个项目研究了在纯数学和应用数学的几个领域出现的偏微分方程（PDE）。其中一个主要目标是物理模型的形式化推导，其数学描述是基于非局部型PDE的，描述了晶体材料中缺陷线的动力学。在另一个领域，PI将研究模拟偏析现象的非线性椭圆偏微分方程解的定性行为。该项目将通过弥合数学分析和物理学之间的差距，从根本上为这些领域做出贡献，并将使对物理现象的进一步数学理解成为可能。本项目为研究生和博士后提供科研培训机会。在技术层面上，本项目的目标是研究通过PDE方法形成界面表面。重点关注的主题领域是：(i)非局部反应-扩散方程，其主要目标是通过研究Peierls-Nabarro模型的渐近极限，推导出描述晶体（位错）缺陷线动力学的中尺度和宏观尺度模型，并证明由分数算子驱动的方程系统的异斜轨道和多碰撞轨道的存在；（ii）分离模型和非线性特征值问题。本文的主要目的是分析分离平稳构型的节点集及其与全非线性算子特征值问题的联系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project investigates partial differential equations (PDE) that arise in several areas of pure and applied mathematics. One of the main objectives is the formal derivation of physical models, whose mathematical description is based on PDE of nonlocal type, that describe the dynamics of defect lines in crystalline materials. In a different area, the PI will investigate the qualitative behavior of solutions to nonlinear elliptic PDE that model segregation phenomena. The project will fundamentally contribute to these fields by bridging the gap between mathematical analysis and physics and will enable further mathematical understanding of physical phenomena. The project provides research training opportunities for graduate students and postdoctoral researchers.At a technical level, the objective of this project is the study of forming interface surfaces through PDE methods. Thematic areas of focus are: (i) Nonlocal reaction-diffusion equations, where the main goals are to derive mesoscale and macroscopic scale models describing the dynamics of defect lines in crystals (dislocations) through the study of the asymptotic limits of Peierls-Nabarro models and to prove existence of heteroclinic and multibump orbits for systems of equations driven by fractional operators; and (ii) Segregation models and nonlinear eigenvalue problems. Here, the main objective is the analysis of nodal sets of segregated stationary configurations and their connection with eigenvalue problems for fully nonlinear operators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.