Various Nonlinear Problems Related to Physics

物理科学中的许多现象都是由非线性偏微分方程（NLPDE）控制的。这一提议中的几乎所有问题都涉及“浓度峰值”；它们对应于空间中的小区域（通常被同化为点或线），其中一些变量可以取非常高的值。这样的峰值区域可以是数据的一部分，也可以是未知的一部分。例如，在核物理学中，正核是高密度的固定区域，周围漂浮着弥漫的电子云。一个令人惊讶的数学发现是，当高密度数据集中在“太小”的区域时，一些自然NLPDE不承认解决方案。在数学语言中，这可以表示为某些度量（例如，狄拉克质量）是不可接受的数据。主要研究者建议对一大类NLPDE的所有可接受的措施进行分类。在其他问题中，高度集中的区域是未知的一部分。当一个“温和”的外场作用于系统时，奇点可能会出现。与当前项目相关的这种现象的例子出现在液晶和超导体的物理学中。在这个项目中，首席研究员将在几个方向上继续他的研究，解决以下问题：需要什么样的外部作用来产生奇点？描述奇点的性质、强度和位置？在现实世界中，人们经常遇到极端强烈的现象，这些现象出现在很小的空间区域或只持续很短的时间间隔。一个简短的例子列表包括：流体力学和超导体中的涡流（类似于龙卷风）、固体力学中的断裂、非线性光学中的自聚焦光束（例如激光）、液晶中的缺陷、天体物理学中的黑洞。其中一个项目？S的目标是描述一大类非线性模型的所有可能的奇异行为。为了避免或加强这种“爆发”现象，了解造成这种现象的原因是很重要的。数学上类似的问题不仅出现在物理学中，也出现在许多其他数学领域。

Many phenomena in the physical sciences are governed by nonlinear partial differential equations (NLPDE). Almost all the problems in this proposal involve "peaks of concentration"; they correspond to small regions in space (often assimilated to points or lines) where some of the variables can take extremely high values. Such peaking zones can be part of the data or part of the unknown. For example, in nuclear physics, positive nuclei are fixed zones of high density surrounded by a diffuse cloud of electrons floating around them. A surprising mathematical discovery is that some natural NLPDE admit no solution when high-density data are concentrated in regions that are "too small." In mathematical language, this can be expressed by saying that some measures (e.g., Dirac masses) are not admissible data. The principal investigator proposes to classify all admissible measures for a large class of NLPDE. In other problems, the zones of high concentration are part of the unknown. Singularities may appear when a "mild" external field is applied to the system. The examples of this phenomenon that are relevant to the current project arise in the physics of liquid crystals and superconductors. In this project, the principal investigator will continue his research in several directions, addressing such issues as the following: What kind of external action is required to produce singularities? Describe the nature, the strength, and the location of singularities?In the real world, one often encounters phenomena of extreme intensity, which appear in small regions of space or persist only during a small time interval. A short list of examples includes the following: vortices (similar to tornadoes) in fluid mechanics and superconductors, fractures in solid mechanics, self-focusing beams in nonlinear optics (e.g., in lasers), defects in liquid crystals, black holes in astrophysics. One of the project?s goals is to describe all possible singular behaviors for a large class of nonlinear models. It is important to understand what causes such "blow-up" phenomena, in order either to avoid them or to enhance them. Mathematically similar problems occur not only in physics but in many other areas of mathematics.