Geometric and analytic issues of nonlinear equations modelling non-local phenomena

非局部现象建模非线性方程的几何和解析问题

• 批准号: 1523088
• 负责人: Nestor Guillen
• 金额: $ 3.85万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1523088&HistoricalAwards=false
• 关键词: Geometric; analytic; issues; nonlinear; equations

The purpose of this project is to understand from the point of view of geometry and mathematical analysis certain physical models that incorporate either free boundaries or nonlocal (long-range) effects. In terms of the mathematics, most of these problems consist of classical field equations from physics (e.g., fluid mechanics, electromagnetism, elasticity, kinetic theory) and exhibit at least one of two important characteristics. First, they might involve a "free boundary," namely, an unknown submanifold (interface) along which the field in question has a pointwise constraint (for instance, the temperature across along the interface of a metal might depend on the curvature). These submanifolds have as much physical interest as the other quantities, and their dynamics are strongly coupled to that of the fields. The second characteristic these equations might display is nonlocality, which arises when particles or "agents" interact at large (noninfinitesimal) scales, for example, in the Boltzmann equation or the quasigeostrophic equation. This always leads to equations involving integro-differential operators, such as fractional powers of the Laplacian. The specific models studied in this project present challenging analytical problems that are especially attractive in that they highlight the limits of our understanding of nonlinear partial differential equations. In particular, they pinpoint difficulties such as the following: obtaining useful pointwise bounds for solutions (without using comparison principles, or when equations are supercritical); deriving a priori regularity estimates for equations that are both nonlinear and nonlocal; understanding the physical validity of solutions (well-posedness and breakdown); handling nonlinear effects that dominate diffusion or dispersion (again supercriticality); analyzing multiscales and disordered media (homogenization).Nonlinear partial differential equations are ubiquitous in the natural sciences, as is well known. For this specific project, the richness of nonlocal equations and free boundary problems cover very diverse natural phenomena, for instance nucleation of phases, surface tension effects in fluids, crystal formation in metallurgy, droplet spreading, ocean-atmosphere interaction, and nonlocal electrostatics. All of these phenomena are relevant to science and engineering, for instance in materials science (composite design, dislocations), nanotechnology (microfluids, droplets), bioengineering (martensite or materials with memory), and biochemistry (nonlocal electrostatics, with great potential in medicine). A sound mathematical understanding of the respective equations would be highly beneficial to the development of these technologies.

该项目的目的是从几何学和数学分析的角度理解某些包含自由边界或非局部（远距离）效果的物理模型。在数学方面，这些问题中的大多数由物理学（例如流体力学，电磁，弹性，动力学理论）的经典场方程组成，并且至少具有两个重要特征之一。首先，它们可能涉及一个“自由边界”，即，沿该字段的未知子序（界面）具有侧重的约束（例如，沿金属界面的温度可能取决于曲率）。这些子手机具有与其他数量一样多的物理兴趣，并且它们的动态与田地的动力相结合。这些方程可能显示的第二个特征是非局部性，当粒子或“代理”在大型（非芬太尼）尺度上相互作用时，例如在玻尔兹曼方程或准真菌方程中。这始终导致涉及全差异操作员的方程式，例如拉普拉斯（Laplacian）的分数力量。该项目中研究的特定模型提出了挑战性的分析问题，这些问题特别有吸引力，因为它们强调了我们对非线性偏微分方程的理解的局限性。特别是，他们指出了以下困难，例如：获得解决方案的有用界限（不使用比较原理或方程式是超临界时）；得出非线性和非本地的方程式的先验规律性估计；了解解决方案的物理有效性（适合和分解）；处理主导扩散或分散的非线性效应（再次超临界）；众所周知，分析多尺度和无序培养基（均质化）（均质化）在自然科学中无处不在。对于这个特定的项目，非本地方程和自由边界问题的丰富性涵盖了非常多样化的自然现象，例如阶段的成核，液体中的表面张力效应，冶金中的晶体形成，液滴扩散，海洋 - 大层相互作用以及非局部静电剂。所有这些现象都与科学和工程有关，例如材料科学（综合设计，错位），纳米技术（微流体，液滴），生物工程（Martensite或具有记忆的材料）和生物化学（非局部电位静电学，具有巨大的药物潜力）。对各个方程式的合理数学理解将对这些技术的发展非常有益。