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

University Of California-Davis, Davis CA

Investigators

Abstract

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.