Nonlinear Wave Interactions & Singular Behavior

非线性波相互作用

• 批准号: 9501073
• 负责人: Esteban Tabak
• 金额: $ 8万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-01 至 1998-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9501073&HistoricalAwards=false
• 关键词: Nonlinear; Wave; Interactions; Singular; Behavior

9501073 Tabak The proposed work concentrates on open problems in wave propagation arising in the natural sciences. The first problem addresses the diffraction of weak shock waves, where an apparent contradiction, known as the von Neumann paradox, has been puzzling physicists and mathematicians for almost five decades. The paradox concerns the ubiquitous occurrence of three-shock intersections in a regime where, in principle, they are not allowed. It is proposed to study this problem within the context of a simplified asymptotic model. The results so far suggest that the solution to the paradox may involve an additional singularity arising at the triple-point. The second problem regards the long-scale behavior of winds, concentrating on the formation of fronts between warm and cold air masses in the atmosphere. The mathematical structure of this problem is surprisingly similar to the one associated with the question of formation of singularities in three-dimensional incompressible fluids. Both problems involve a combination of physical modeling, asymptotic analysis and analytical and numerical solution of partial differential equations. Both proposed problems have practical as well as theoretical significance. Shock diffraction occurs in a variety of scenarios, such as supersonic flight, flood waves and even medical applications, when focusing sound waves are used to destroy kidney stones. A better understanding of the diffraction process would therefore impact not only our knowledge of waves in nature, but also our capacity for predicting wave intensity amplification due to focusing and reflection in engineering devices. The formation of sharp fronts between air masses in the atmosphere, on the other hand, is an important and ubiquitous phenomenon. Understanding its mechanics would enhance our ability to predict weather and climate changes. The surprising relationship between this problem and the question of singularity for mation in incompressible fluids, on the other hand, provides a promising alternative way to study a mechanism which may play a fundamental role in turbulent intermittency.

小行星9501073 所提出的工作集中在波的开放问题 自然科学中产生的传播。第一 问题解决了弱冲击波的衍射，其中 一个明显的矛盾，被称为冯诺依曼悖论， 一直困扰着物理学家和数学家 五十年了这个悖论涉及到 三个冲击波交汇的区域，原则上， 他们是不允许的。 建议在以下范围内研究这个问题： 一个简化的渐近模型。迄今为止的结果 我认为解决这个悖论可能需要一个额外的 奇点出现在三重点。第二个问题 关于风的长期行为，集中在 冷暖气团之间锋面的形成 气氛这个问题的数学结构是 令人惊讶的是， 三维不可压缩流中奇点的形成 流体. 这两个问题都涉及到物理建模， 渐近分析及解析和数值解 偏微分方程 提出的这两个问题都具有实际意义和理论意义。 意义冲击波衍射发生在各种情况下， 例如超音速飞行、洪水甚至医疗 应用程序，当聚焦声波被用来破坏 肾结石更好地理解衍射过程 不仅会影响我们对自然界中波浪的认识 也是我们预测波浪强度放大的能力 这是由于工程设备中的聚焦和反射。的 在大气层中的气团之间形成锋面， 另一方面，是一个重要的和普遍存在的现象。 了解它的机制将提高我们的能力， 预测天气和气候变化。令人惊讶的关系 在这个问题和奇异性问题之间， 另一方面，不可压缩流体提供了一种有前途的 另一种研究机制的方法， 在湍流中的作用。