On A Class Of Transonic Shocks In Nozzles For Compressible Euler Flows

The present Ph.D. dissertation is concerned with the ill-posedness or well-posedness of a class of transonic shocks in nozzles for compressible Euler flows.Various nozzles are widely used in practice to transport fluids and control their motion. Experiments of aerodynamics and numerical simulations show that, for given supersonic flow at the entrance of a nozzle, if the pressure at the exit is appropriately large, then in the divergent part of the nozzle a shock may appear, and the supersonic flow will jump to subsonic after passing it, and the pressure increases to reach the given data at the exit. We call this type of shock waves as transonic shock. This dissertation concerns a special class of transonic shock in a straight pipe: both the states ahead and behind the transonic shock are small perturbations of certain uniform ones. The result is: Such shock is stable under perturbations of the upstream supersonic flow and the profile of the pipe if and only if the pressure at the exit is given with freedom one, that is, it contains an unknown constant to be determined simultaneously with the flow fields. (See Theorem 3.1 and Theorem 4.1.) The proof reveals the subtle relations between the ill-posedness and the requirement of conservation of mass in the pipe. We investigate the transonic shocks in two dimensional slowly varying nozzles and three dimensional ducts with rectangular sections respectively (for the latter, certain symmetry properties are required for the coming supersonic flow).The steady complete Euler system is used as the governing equations. Since it is of elliptic-hyperbolic composite type for subsonic flow and the transonic shock is also an unknown surface, from mathematical point of view, we are dealing with a free boundary problem of a quasi-linear elliptic-hyperbolic composite system.The whole dissertation is organized as follows.Chapter One is Introduction. This chapter is devoted to the physical and mathematical background of transonic shock problem in nozzles. The main results and methods of this Ph.D. dissertation are also illustrated with comments.Chapter Two establishes two special classes of transonic shocks for Euler system. The first class is the subject of this dissertation. A thorough understanding of the implications of it is important for investigating general transonic shocks.Chapter Three studies transonic shock problem in two dimensional slowly varying nozzles. We show that for given arbitrarily pressure (density, Mach number, entropy etc.), such a problem is in most cases ill-posed. The main tools used are Lagrange transformation and characteristic decomposition, by which we can reduce the Euler equations to a 2×2 system (it is elliptic for subsonic flow) and two algebraic equations. We used the Intermediate Schauder estimates developed by D. Gilbarg, L. Hormander and G. Lieberman to overcome the possible singularity near the wall due to the intersection of the shock and the wall.Chapter Four further studies transonic shock in three dimensional straight duct with square section under certain symmetric assumptions on the coming supersonic flow. The key point now is to decompose the original problem into four coupled sub-problems by suitable differentiations and linear combinations of the Euler equations by utilizing their special structures. At this stage, the ill-posedness is closely related to the solvability of an equi-valued boundary value problem for elliptic equations.