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.
|Effective start/end date||7/1/08 → 6/30/12|
- National Science Foundation (National Science Foundation (NSF))