Nonlinear Wave Interactions & Singular Behavior

9501073 Tabak拟议的工作集中于自然科学中出现的波传播的公开问题。第一个问题涉及弱冲击波的绕射，其中一个明显的矛盾被称为冯·诺伊曼悖论，近50年来一直困扰着物理学家和数学家。这一悖论关乎的是，在一个原则上不允许发生三次冲击交叉路口的政权中，这种情况随处可见。建议在一个简化的渐近模型的背景下研究这个问题。到目前为止的结果表明，这一悖论的解决方案可能涉及在三点处出现的另一个奇点。第二个问题与风的长期行为有关，集中在大气中冷暖气团之间锋面的形成。这个问题的数学结构与三维不可压缩流体中奇点的形成问题惊人地相似。这两个问题都涉及物理建模、渐近分析以及偏微分方程解的解析和数值解的组合。这两个问题的提出都具有现实意义和理论意义。当聚焦声波被用来摧毁肾结石时，冲击绕射在各种情况下都会发生，例如超音速飞行、洪水波，甚至医疗应用。因此，更好地了解绕射过程不仅会影响我们对自然界中波的知识，而且会影响我们预测由于工程设备中的聚焦和反射而产生的波强度放大的能力。另一方面，大气中气团间锋面的形成是一种重要而普遍的现象。了解它的机制将增强我们预测天气和气候变化的能力。另一方面，这一问题与不可压缩流体中的奇异性问题之间令人惊讶的关系，为研究可能在湍流间歇中起基本作用的机制提供了一种很有前途的替代方法。

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.