Collaborative Research: Shock formation, shock development, and the propagation of singularities in fluid dynamics

合作研究：激波形成、激波发展以及流体动力学中奇点的传播

• 批准号: 2307681
• 负责人: Vlad Vicol
• 金额: $ 75万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2028-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307681&HistoricalAwards=false
• 关键词: Collaborative; Research; Shock; formation; shock

The motion of compressible fluids, such as gases and plasmas, is characterized by the formation and propagation of shock waves, i.e., thin adjustment fronts created within the fluid and across which the fluid experiences large changes of its state variables. Examples of shock waves abound in nature and technology: sonic booms generated by commercial and military airplanes, bow shocks generated by space vehicles upon re-entry through the atmosphere, and bow shocks created when the solar wind hits the planets, to name a few. Although a good theoretical understanding of the formation and subsequent propagation of shock waves exists for one-dimensional (i.e., planar) flows, the corresponding state of affairs in multiple space dimensions is much less satisfactory. The purpose of this project is to develop a new geometric framework and a new mathematical description of the wave motion that allows for a detailed description of shock formation and the subsequent dynamics of shock waves in multiple space dimensions. This project will also offer research opportunities and collaborative experiences for graduate students and postdocs at the University of California, Davis, and New York University.This project will develop the analytical and geometric framework for resolving one of the foremost unanswered questions in the fields of hyperbolic PDE and mathematical fluid dynamics: the formation and unique propagation of hydrodynamical shocks from smooth initial data, in multiple space dimensions. The first step is called "shock formation". Here the smooth initial data is evolved up to a cusp-like Eulerian spacetime hypersurface of first singularities, where the gradient of the velocity, pressure, density, and energy becomes infinite, but these fields retain Holder 1/3 regularity. The PIs approach to determining the location and the geometry of this cusp-like spacetime hypersurface of first singularities relies upon the construction of a smooth spacetime geometry, together and a new set of hydrodynamic variables in the Arbitrary Eulerian-Lagrangian (ALE) description of acoustic wave propagation. The second step is called "shock development" wherein one uses the analytical description of the solution on the cusp-like spacetime hypersurface of first singularities as Cauchy data, from which the shock surface of discontinuity instantaneously develops. In conjunction with the shock surface, we shall establish the emergence of so-called weak characteristic discontinuities; these are characteristic surfaces that emerge simultaneously (with the shock) from the pre-shock, and along which, gradients of velocity, density, and entropy exhibit one-sided Holder cusps. This framework enables the study of even more complicated physical models such as the magnetohydrodynamic equations (MHD) of plasma flow. Here, unlike the lone classical compressive shock of gas dynamics, six different types of MHD shocks can be analyzed with our approach: a fast shock, a slow shock, and four different intermediate shocks. The latter were observed by the Voyager spacecraft in Earth’s heliosphere, but their mathematical existence, to date, remains in question.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

兼容流体的运动，例如气体和等离子体，其特征是冲击波的形成和传播，即在流体中产生的薄调节阵线，并且流体在其状态变量中经历了很大的变化。自然和技术中的冲击波示例：商业和军事飞机产生的声音繁荣，通过大气再进入时太空车辆产生的弓形冲击，尽管对形成的理论有很好的理论理解以及随后的冲击波传播的理论理解，而冲击波的传播则存在于一维（即平面）的情况下，是多种多样的事实。该项目的目的是开发一个新的几何框架和对波动运动的新数学描述，该描述允许对冲击形成的详细描述以及多个空间维度中冲击波的随后动态。 This project will also offer research opportunities and collaborative experiences for graduate students and postdocs at the University of California, Davis, and New York University.This project will develop the analytical and geometric framework for resolving one of the foreign unanswered questions in the fields of hyperbolic PDE and mathematical fluid dynamics: the formation and unique propagation of hydrodynamical shocks from smooth initial data, in Multiple space dimensions.第一步称为“冲击形成”。在这里，平滑的初始数据被演变为类似风口尖的欧拉时期超表面的第一奇异点，其中速度，压力，密度和能量的梯度变得无限，但这些磁场保留了1/3规律性。确定这种类似尖齿的时空超表面的位置和几何形状的PI方法依赖于构建光滑的时空几何形状，以及在任意的Eulerian-lagrangian（ALE）中的一组新的流体动力变量。第二步称为“冲击开发”，其中，人们使用了溶液对尖端的时空超表面的分析描述作为cauchy数据，从中，不连续性的冲击表面即时发展。与冲击表面一起，我们将确定所谓的弱特征不连续性的出现。这些是从震动中简单地（带有冲击）出现的特征表面，并且沿途，速度，密度和熵的梯度表现出单方面的尖尖。该框架可以研究更复杂的物理模型，例如等离子体流动的磁流体动力方程（MHD）。在这里，与气体动力学的孤独经典压缩冲击不同，可以通过我们的方法分析六种不同类型的MHD冲击：快速冲击，慢速冲击和四种不同的中间冲击。在地球的地球领域中，旅行者航天器观察到了后者，但迄今为止，它们的数学存在仍然存在。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛影响的评估标准通过评估来获得的支持。