The Euler equations are recognized as a suitable model for multiphase fluid flows with moving boundaries and interfaces at large Reynolds number, and they serve as the basic mathematical model even when other physical phenomena are coupled to the fluid motion. Despite more than two centuries of mathematical analysis of these complicated nonlinear equations, the existence theory for these systems of hyperbolic moving free-boundary PDE, which model ideal compressible and incompressible fluid flow, remains a significant challenge. This includes a single mass of fluid, gas, or liquid, moving inside of a vacuum that creates degeneracy in the flow, as well as multiphase immiscible fluids separated by surfaces of discontinuity, across which velocity components experience jumps. Recently, the Principal Investigator of this project has been developing a novel set of analytical tools designed to establish existence theories and well-posedness theorems for multidimensional moving free-boundary hyperbolic problems, wherein the geometry of the free-surface interacts with the motion of the fluid at leading order. These analytical tools apply to the 3-dimensional incompressible and compressible free-surface Euler equations with or without surface tension on the boundary, and coupled fluid-structure interaction problems. The fundamental ideas rely on new anisotropic smoothing operators that permit approximations of the Euler equations that retain the geometric structures of transport and boundary regularity, and for which existence of smooth solutions is provable, and a new class of degenerate parabolic approximations to characteristic and degenerate hyperbolic systems of conservation laws. The proposal addresses the well-posedness of the motion of a multidimensional compressible gas in the so-called physical vacuum singularity, modeled by the free-boundary compressible Euler equations with sound speed vanishing at the boundary at the rate of the square-root of the distance to vacuum; well-posedness of supersonic 2-D vortex sheets and surfaces of discontinuity; and well-posedness for the motion of a relativistic fluid in vacuum, modeled by the Euler-Einstein equations.
Multiphase fluid flows with moving interfaces, modeled by the Euler equations, 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. The analytical understanding gained in this work may have important ramifications in the understanding of basic physical phenomena, which has hitherto been poorly understood. In addition to basic wave motion and mixing that occurs in the motion of interfaces between water and air, other conventional examples include the interface between air and helium under shock wave interaction, the so-called Richtmyer-Meshkov instabilities between two gases, the behavior of a gas bubble in a liquid in a shock wave, and liquid fuels that are usually burned by first atomizing a fuel jet to increase the surface area and hence the evaporation rate. We can also add the prediction of spray behavior, for which the initial atomization is both the most critical and the least understood aspect of the spray. Understanding the short-time nonlinear balance that occurs in the Rayleigh-Taylor instability should be quite important for the understanding of jets, which become unstable when capillary effects are large due to waves longer than the diameter, thus breaking up into a stream of relatively large drops.
The existence of solutions to gas dynamics with vacuum states (wherein the density of the gas vanishes) has been a major open problem in the mathematical theory of fluid dynamics, and was made stated explicitly in by John von Neumann at the conference Problems of Cosmical Aerodynamics, held in Paris in 1949. Vacuum states can be found in the expansion of gaseous stars, galaxies, and constellations, as well as in many models of atmospheric science and ocean modeling. The moving vacuum boundary, as a propagating surface of discontinuity in the compressible Euler equations, is the fundamental model for vacuum states in fluids, because any other mathematical model builds on this idealized inviscid limiting case. This moving physical vacuum boundary is resonant because the speed of the genuinely nonlinear waves coincide on this evolving hypersurface, and further degenerate because the density must vanish at this boundary with a prescribed rate. This latter degeneracy caused by the vanishing of the density creates manifest derivative loss, because coefficients in wave equations are no longer bounded from below away from zero. The PI has solved this longstanding open problem, and established the existence, uniqueness, and regularity of the Euler equations with moving physical vacuum boundary. A vacuum state is called ``physical'' when it permits the gas-vacuum boundary to accelerate, and induces a singularity in the pressure gradient, requiring the sound speed to vanish at a rate of the square-root of the distance to the vacuum. The vanishing of the sound speed, and hence the density of the gas, ensures that the Euler equations are a degenerate and characteristic hyperbolic free-boundary system of conservation laws, to which standard methods of symmetrizable multi-D conservation laws cannot be applied. It was widely conjectured that solutions for this problem must become singular near the vacuum boundary. Using a new Hardy-type inequality in conjunction with a unique parabolic-type regularization which does not destroy the transport structure of vorticity waves, the PI was able to construct solutions that are smooth (in classical Sobolev spaces) all the way to and including the moving vacuum boundary. The new analytical methods developed by the PI to study nonlinear hyperbolic systems such as gas dynamics have also been used to analyze classical parabolic free-boundary problems such as the Stefan problem. The PI was able to establish the nonlinear stability of equilibria for smooth solutions of the classical Stefan problem, a fundamental model of phase transition, such as the melting of icebergs. In particular, the PI proved the global existence of unique solutions with the optimal decay rate to an equilibrium state. The methodologies developed can be used for a number of other model problems, such as porous medium flow and Hele-Shaw cells, as well as for the analysis of fluid-structure interaction problems Additionally, the ideas developed by the PI to analyze gas dynamics have motivated the PI to develop a new, higher-order accurate, numerical algorithm for the propagation of shock waves and contact discontinuities in gas dynamics. This new method, called the C-method, is able to capture the contact discontinuity of the internal energy function in the very difficult so-called Leblanc Shock Tube experiment, wherein the internal energy varies by ten orders of magnitude.
