Discontinuous Compressible Flows Onto Solid Objects

Description

The aim of this project is to develop the mathematical theory of the steady compressibleEuler equations, particularly focusing on the rigorous mathematical analysis of the discontinuouscompressible flows onto solid objects, and related problems governed by the steadycompressible Euler equations for the isentropic flows or the potential flows.The system of compressible Euler equations is one of the most important and fundamentalequations in mathematical fluid mechanics, as nature confront the observer with a wealth ofnonlinear wave phenomena. As one of its typical nonlinear weak solutions, steady obliqueshocks appear in many very important physical applications. The study of the well-posednesstheory related to this specific discontinuous solutions governed by the steady potential flowequation or the steady compressible Euler equations is always one of the most importanttopics in the study of mathematical fluid mechanics and conservation laws. The researchof this project contributes a lot to the mathematical theories of nonlinear equations (ofmixed-type and coupled system) with free boundaries.Since 1980s, there are many important results for the discontinuous solutions of thecompressible Euler equations in several space dimension although the general theory is terraincognita. So far almost all the research is focused on specific problems such as the existenceand stability of plane shocks and rarefaction waves, and the steady or self-similar solutionswith special structures, for example, the existence and stability of attached oblique shocksin supersonic steady compressible flows onto wedges.We have been working on some related problems for several years. Recently, we haveestablished the global existence of subsonic full Euler flows in infinite long nozzles, theuniqueness of transonic shocks onto a non-straight wedge, and the validity of the geometricoptic expansion for the Cauchy problem of systems of hyperbolic conservation laws. In thisproject, we will be mainly focused on a further development of the technics of proving theobtained results, to study some of the mentioned problems.So far, all the known mathematical results of the discontinuous compressible flows ontoa solid object are on the global existence and stability of solutions of steady attached shocksonto sharp wedges, which are slightly perturbation of a straight wedge. It is well-known thatthere are two configurations when a supersonic flow past a straight sharp wedge, both ofwhich satisfy the Rankine-Hugoniot conditions and the entropy condition across the shock.A nature question is how to find the physical one. It is an open problem proposed inCourant-Friedrichs’ book [21, p317–318]. Hence the results and techniques developed fromthis research project will also provide a better understanding of this problem.