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

诸如气体和等离子体的可压缩流体的运动的特征在于冲击波的形成和传播，即，在流体内产生薄的调整前沿，并且流体在该前沿上经历其状态变量的大的变化。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冲击，以及太阳风撞击行星时产生的弓形冲击，仅举几例。尽管对于一维（即，平面）流动，多个空间维度中的对应状态就不那么令人满意了。该项目的目的是开发一个新的几何框架和一个新的波动数学描述，以便详细描述冲击波的形成和随后的冲击波在多个空间维度的动态。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PDE和数学流体动力学领域中最重要的未回答问题之一：在多个空间维度中，从光滑的初始数据中形成和独特的流体动力学冲击传播。第一步被称为“冲击形成”。在这里，光滑的初始数据被演化为具有第一奇点的尖点状欧拉时空超曲面，在那里速度、压力、密度和能量的梯度变得无穷大，但这些场保持保持器1/3正则性。PI的方法来确定的位置和几何形状的尖状时空超曲面的第一奇点依赖于一个光滑的时空几何结构的建设，在一起和一组新的流体动力学变量的任意欧拉-拉格朗日（ALE）描述的声波传播。第二步被称为“激波发展”，其中一个使用的解析描述的尖点状时空超曲面的第一奇性的柯西数据，从该激波表面的不连续性瞬间发展的解决方案。结合激波面，我们将建立所谓的弱特征不连续面的出现;这些特征面（与激波）同时从预激波中出现，并且沿其沿着，速度、密度和熵的梯度表现出单侧的保持器尖点。这个框架使得研究更复杂的物理模型，如磁流体动力学方程（MHD）的等离子体流。在这里，不同于气体动力学的孤独的经典压缩冲击，六种不同类型的MHD冲击可以用我们的方法进行分析：一个快速冲击，一个缓慢的冲击，和四个不同的中间冲击。后者是由旅行者号航天器在地球的日光层观测到的，但它们的数学存在，迄今为止，仍然是一个问题。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.