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

这个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(长期)效应的物理模型。在数学方面，这些问题大多由物理学(如流体力学、电磁学、弹性力学、动力学理论)中的经典场方程式组成，并至少表现出两个重要特征中的一个。首先，它们可能涉及“自由边界”，即一个未知子流形(界面)，所讨论的场沿该子流形具有逐点约束(例如，沿金属界面的温度可能取决于曲率)。这些子流形和其他量一样具有物理意义，并且它们的动力学与场的动力学是强耦合的。这些方程可能表现出的第二个特征是非定域性，当粒子或“代理人”在大(非无穷小)尺度上相互作用时，例如，在玻尔兹曼方程或准地转方程中，就会出现这种情况。这总是导致涉及积分-微分算子的方程，例如拉普拉斯的分数次幂。这个项目中研究的特定模型提出了具有挑战性的分析问题，这些问题特别吸引人，因为它们突出了我们对非线性偏微分方程组的理解的局限性。特别是，它们指出了以下困难：获得解的有用的逐点界限(不使用比较原理，或当方程为超临界时)；推导出非线性和非局部方程的先验正则性估计；了解解的物理有效性(适定性和破裂)；处理主导扩散或弥散的非线性效应(同样是超临界)；分析多尺度和无序介质(齐次化)。众所周知，非线性偏微分方程组在自然科学中普遍存在。对于这个特定的项目，丰富的非局部方程和自由边界问题涵盖了非常不同的自然现象，例如相的成核、流体中的表面张力效应、冶金中的晶体形成、液滴扩散、海洋-大气相互作用和非局部静电学。所有这些现象都与科学和工程有关，例如在材料科学(复合材料设计、位错)、纳米技术(微流体、液滴)、生物工程(马氏体或具有记忆的材料)和生物化学(非局部静电学，在医学上具有巨大的潜力)。对各个方程式的数学理解将对这些技术的发展大有裨益。

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.