A Class Of Subsonic Potential Flows Problem In Unbounded Domains

In our daily lives, most of the motion of the gas are subsonic and supersonic, such as the motion of the air, water, aircraft, space shuttle and so on. Motion rule of subsonic and supersonic flow is one of the fundamental problems in compressible fluid dynamics. So the study on the subsonic flow and supersonic flow is of great importance in our daily lives and the national defence and so on. When supersonic flow hits a sharp body, for example, cone and wedge, there will appear a shock. These problems arise not only in many important physical situations but also are fundamental in the mathematical theory of multidimensional compressible fluid flows. There are lots of experiments and numerical simulations involved in this field. The phenomenon that a supersonic flow past a given sharp wedge happens frequently in our daily life. The airfoil, for example, is designed like a sharp wedge.When a supersonic airplane flies in the air, if we consider that the plane is still, then there will be a supersonic flow past a sharp wedge and consequently there will appear a weak shock or a strong shock attached at the head of the wedge in terms of the different pressure states in the downstream region. In this field, there have been extensive works by Chen Shuxing, Liu Taiping, Xin Zhouping, Yin Huicheng, Xu Gang and so on. They have been given many local or global results under various cases, one can see [19,54,61,87,110] and the references therein. On the global existence of subsonic flow passing a body, half plane and finite or infinity long nozzle, under some different assumptions of the flow and the domain, there also have been extensive works by Guangchang Dong, Yin Huicheng, L.Bers, B.Bojarski, D.Gilberg, R.Finn. One can see for exempale [4,7,31,36,37,86,88,115] ,and the reference therein. For two dimensional flows, one of the most significant advances was due to Bers [4] who have proved that global subsonic potential flows exist if the Mach number of the freestream is small enough; furthermore,as the Mach number increases, the maximum flow speed will trend to the sound speed. Later on, Finn and Gilbarg [36] proved the uniqueness of subsonic flow past a profile by maximum principle and the asymptotic behavior of subsonic flows at the far field. Xie and Xin [99] study the global subsonic and subsonic flows through a general infinitely long nozzle and proved that exsits a critical value for the incoming mass flux so that a global uniformly subsonic flow exists in the nozzle as long as the incoming mass flux is less than the critical value. For the three dimensional flows, studies firstly by Finn and Gilbarg [37] and then by Dong and Ou [31], the final results are quite similar to those in the two dimensional case,that the subsonic flow exists globally if the freestream Mach number is suitably small;moreover the maximum flow speed will trend to the sound speed if the Mach number increases to some critical value. Xie and Xin[100] establish existence of global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles by combining variational method,many elliptic priori estimates and compactness method. However, we consider the well-posed problem of subsonic flow around 2D and 3D ramp domain.Our paper is motivated by the following descriptions given in Section 111 of [26]:For the flow around a sharp corner or body, if the oncoming flow is subsonic, then the problem involves potential flow, governed by an elliptic differential equation whose solution at any point depends on the boundary conditions even at remote parts of the boundary, and is more difficult to be treated than that in the case of supersonic flow.Our main results of this thesis are listed as follows:Firstly, if the flow is uniformly subsonic flow (q/c(q)<1-?,??0) and the gas is steady, isentropic, irrotational polytropic gas, that is, the movement of the flow can be described by the potential equation. By the use of Riemann mapping Theorem [92] we know that 2D ramp is conformal to a unit circle. So our problem is converted to look for a harmonic function which satisfies suitable boundary condition on the unit circle. We note that complex distorted velocity is conformal with respect to flow metric and two maps which are conformal with respect to the same flow metric can be considered to be an analytic function when one map takes as independent variable. Based on the above observe and carefully computation, we obtain a uniqueness result:q= 0 which is the same with [60]. Moreover, we can generalized the standard angular domain to a more general domain.Secondly, In 3D ramp domain, if the flow is steady potential flow and it is subsonic (q< c(q)) with the mass-flux condition at infinity and the ramp has a small period perturbation, we can obtain global existence uniqueness and stability by the use of separative variable method, Strum-Liouville theorem and scaling technique.The whole thesis is organized as follows:In Chapter 1, we give some physical background on the subsonic flow in un-bounded domain. Moreover, some newest study proceedings related to this paper are presented. Meanwhile, the main results and their significance in this thesis are illus-trated and commented.In Chapter 2, we study the steady,isentropic, irrotational polytropic gas through infinite long 2D ramp, we obtain a uniqueness result by use of complex method and we successfully remove the smallness condition.In Chapter 3, we study 3D potential flow equation in a 3D ramp domain which has a small period perturbation. We will prove global existence uniqueness and stability under mass-flux condition at infinity.