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，激波是在流体内产生的薄调整前锋，流体在该前锋上经历其状态变量的巨大变化。冲击波在自然界和技术中比比皆是：商用和军用飞机产生的音爆，航天器在重返大气层时产生的弓形冲击波，以及太阳风撞击行星时产生的弓形冲击波，仅举几例。虽然对于一维(即平面)流动，激波的形成和随后的传播在理论上已经有了很好的理解，但在多个空间维度上的相应情况却不太令人满意。这个项目的目的是开发一种新的几何框架和一种新的波运动的数学描述，以便详细描述冲击波的形成和随后的冲击波动力学。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悬而未决的问题之一：从平滑的初始数据在多个空间维度中形成和独特传播流体动力激波。第一步被称为“震撼队形”。在这里，光滑的初始数据被演化成第一奇点的尖点状欧拉时空超曲面，其中速度、压力、密度和能量的梯度变得无限，但这些场保持Holder 1/3的正则性。确定这种具有第一奇点的尖点状时空超曲面的位置和几何形状的PI方法依赖于光滑时空几何的构造，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”，用第一奇点的尖点状时空超曲面上的解的解析描述作为柯西数据，由此瞬时发展出不连续的激波表面。结合激波表面，我们将建立所谓的弱特征不连续的出现；这些特征表面是从激波前同时出现的(与激波同时出现的)特征表面，速度、密度和熵的梯度沿着这些表面显示出单侧的Holder尖点。这一框架使得研究更复杂的物理模型成为可能，例如等离子体流动的磁流体动力学方程(MHD)。这里，与气体动力学中唯一的经典压缩激波不同，我们的方法可以分析六种不同类型的MHD激波：一种快激波，一种慢激波和四种不同的中间激波。后者是旅行者号航天器在地球日光层观测到的，但到目前为止，它们在数学上的存在仍然值得怀疑。这一奖项反映了美国国家科学基金会的法定任务，并通过使用基金会的智力优势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. 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.