Idealized fluid flows are modeled by the Euler equations of fluid dynamics; these are a coupled system on nonlinear conservation laws in (3+1)-dimensional space-time, and are the fundamental to all models of fluid motion. Compressible flows, in which the theory of sound is included, exhibit discontinuous wave profiles; examples include shock waves, contact discontinuities, moving material interfaces, moving vacuum boundaries, entropy waves, and a variety of additional discontinuous wave patterns. The analysis of such multi-dimensional wave patterns, and in particular, solutions of the Euler equations which propagate at least one surface of discontinuity, is fundamental to the understanding of basic physical phenomena. The goal of this research effort is to treat such multi-dimensional discontinuous wave profiles as moving free-boundary problems, and to develop a theory for the well-posedness of hyperbolic and degenerate hyperbolic systems, construct solutions which exhibit finite-time singularities wherein the propagating hypersurfaces collide with one another, study singular asymptotic limits such as vanishing viscosity limits and limits of zero surface tension, and develop a novel nonlinear stability theory for so-called hyperbolic-parabolic problems which are ubiquitous in models of phase transition.
Multiphase fluid flows with moving interfaces play a central role in a multitude of physical and engineering applications, ranging from the creation of hurricanes due to wind blowing on top of the ocean surface to the atomization of liquid fuel jets in combustion chambers to the motion of astrophysical bodies such as gaseous stars, and to fundamental predictions in atmospheric science and meteorology. The analytical understanding gained in this work may have important ramifications in the understanding of basic physical phenomena, which is heretofore, poorly understood. In addition to basic wave motion and singularities that occurs in the motion of interfaces between water and air, other conventional examples include the interface between air and water, the motion of cloud fronts, the melting of ice-bergs, basic instabilities between two compressed gases, the behavior of a gas bubble in a liquid in a shock wave, and liquid fuels which are usually burned by first atomizing a fuel jet to increase the surface area and hence the evaporation rate. This proposed research aims to analyze the motion of evolving surfaces of discontinuity in gases, liquids, plasmas, as well as in the context of classical phase transition models.
