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

可压缩流体（例如气体和等离子体）的运动的特征是冲击波的形成和传播，即在流体内产生薄的调整前沿，并且流体在其上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、航天器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。尽管对于一维（即平面）流的冲击波的形成和随后的传播存在良好的理论理解，但多个空间维度中的相应状况却远不那么令人满意。该项目的目的是开发一种新的几何框架和一种新的波浪运动数学描述，以便详细描述冲击波的形成以及多个空间维度中冲击波的后续动力学。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悬而未决的问题之一：在多个空间维度上，从平滑的初始数据形成流体动力冲击并以独特的方式传播。第一步称为“冲击形成”。这里，平滑的初始数据演化为第一奇点的尖点式欧拉时空超曲面，其中速度、压力、密度和能量的梯度变得无穷大，但这些场保留了 Holder 1/3 规律性。确定第一奇点的尖点状时空超曲面的位置和几何形状的 PI 方法依赖于平滑时空几何结构的构造，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学变量。第二步称为“激波发展”，其中使用第一奇点的尖点时空超曲面上的解的解析描述作为柯西数据，由此瞬时发展不连续的激波面。结合激波面，我们将确定所谓的弱特征不连续性的出现；这些是预激波同时（与激波一起）出现的特征表面，沿着这些表面，速度、密度和熵的梯度呈现单侧的保持器尖点。该框架可以研究更复杂的物理模型，例如等离子体流的磁流体动力学方程（MHD）。在这里，与气体动力学的单独经典压缩激波不同，我们的方法可以分析六种不同类型的 MHD 激波：快速激波、慢速激波和四种不同的中间激波。后者是由航行者号航天器在地球日光层中观测到的，但迄今为止，它们的数学存在仍然存在疑问。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力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.