On Compressible Subsonic Flows In Ducts

The present dissertation is devoted to the study of ill-posedness or well-posedness of some boundary value problems for the steady complete Euler system and the potential flow equation which govern subsonic perfect compressible flows in two-dimensional or three-dimensional ducts.In our daily life, fluid flows are in most cases subsonic. For example, gases transported in various pipes are almost of lower velocity. Study of subsonic flows has significant physical as well as practical importance. It also plays a unique role in our understanding of more complex flow patterns in nature.For subsonic compressible flows around obstacle in unbounded domains, there is now a rather complete theory. However, there are few results about subsonic flows in ducts, especially for the Euler system and three-dimensional ducts cases. We investigate subsonic compressible flows in two-dimensional finite ducts for the steady complete Euler system, and in an infinitely long two-dimensional duct with periodic walls for the potential flow equation (with the stream function being the unknown). Then by utilizing maximum principle and tools like Harnack inequality, we study subsonic potential flows in three-dimensional finite duct, semi-infinite duct and infinite duct which have quadrate sections, as well as flows in the half space and whole space.The steady complete Euler system is of elliptic-hyperbolic composite type for subsonic flow. While the steady irrotational flows for compressible fluids are governed by second order quasi-linear partial differential equations whose coefficients are certain functions of the gradient of the unknown. So from the mathematical point of view, studies of subsonic compressible flows require us to solve boundary value problems of quasi-linear elliptic equations, or nonlinear elliptic-hyperbolic composite system.The dissertation is organized as follows.Chapter One is an introduction. It is devoted to introducing physical background and previous mathematical research works on subsonic compressible flows, especially important achievements by other scholars who treated similar problems or difficulties related to those of ours that are presented in the following chapters. The main problems we concerned, main results we obtained, and methods we utilized in this Ph.D. dissertation are also illustrated with comments.In Chapter Two we consider subsonic flows passing a two-dimensional finite duct for the steady compressible Euler system. If the Bernoulli constant is uniform in the flow field, the density at the entry and both the pressures at the entrance and the exit are given, we show that the problem is generally ill-posed; but if the pressure at the exit contains a constant to be solved, then under the same other conditions as above we establish the existence and local uniqueness of perturbed subsonic flows.In Chapter Three we deal with two-dimensional compressible potential subsonic flows in an infinitely long duct with periodic walls. It is shown that there exists a critical value of mass flux, if the incoming mass flux is less than the critical value, under suitable assumption we show that the flow is also periodic. We also obtain existence, uniqueness and regularity of the periodic solution by techniques of elliptic equations.In Chapter Four we study subsonic potential flows in three-dimensional finite duct, semi—infinite duct and infinite duct which have quadrate sections, as well as flows in the half space and whole space. We obtain an extreme principle of quasi-linear elliptic equation for solutions with bounded gradient in the above tubular-like unbounded domain; that is, the maximum and minimum can not be attained simultaneously at the positive infinity or negative infinity.