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

可压缩流体（如气体和等离子体）的运动以激波的形成和传播为特征，即在流体内部产生的薄调整锋，流体经历其状态变量的巨大变化。冲击波的例子在自然界和技术中比比皆是：商用和军用飞机产生的音爆，太空飞行器在重返大气层时产生的弓形激波，以及太阳风撞击行星时产生的弓形激波，等等。虽然对一维（即平面）流动的激波的形成和随后的传播有很好的理论认识，但在多个空间维度上的相应状态却不太令人满意。该项目的目的是开发一种新的几何框架和波浪运动的新的数学描述，以便在多个空间维度上详细描述激波的形成和随后的激波动力学。该项目还将为加州大学戴维斯分校和纽约大学的研究生和博士后提供研究机会和合作经验。该项目将开发分析和几何框架，以解决双曲偏微分方程和数学流体动力学领域中最重要的未解问题之一：在多个空间维度上，光滑初始数据的流体动力冲击的形成和独特传播。第一步被称为“激波形成”。在这里，光滑的初始数据演化为一个尖点样的欧拉时空超曲面，其中速度、压力、密度和能量的梯度变得无限，但这些场保持霍尔德1/3正则性。pi方法用于确定第一奇点的尖点时空超曲面的位置和几何形状，依赖于光滑时空几何结构的构建，以及声波传播的任意欧拉-拉格朗日（ALE）描述中的一组新的流体动力变量。第二步称为“激波展开”，其中使用第一奇异点的类尖时空超表面上解的解析描述作为柯西数据，从中立即发展出不连续的激波表面。结合激波面，我们将建立所谓的弱特征不连续的出现；这些是与预冲击同时出现的特征表面，沿着这些表面，速度、密度和熵的梯度表现出片面的霍尔德尖峰。这个框架使研究更复杂的物理模型，如等离子体流动的磁流体动力学方程（MHD）成为可能。这里，与气体动力学中单一的经典压缩冲击不同，我们的方法可以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。后者是由旅行者号宇宙飞船在地球的日光层观测到的，但它们的数学存在，到目前为止，仍然是一个问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.