一类无界区域上的亚音速位势流问题
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摘要
在我们的日常生活中,气体的运动大多处于亚音速和超音速流运动状态,例如空气的运动,水的流动,飞机和航天飞行器的运动等等,探索亚音速和超音速流的运动规律也是流体动力学中的基本问题之一。因此研究亚音速流和超音速流对生产、生活、国防等方面有着广泛而重要的意义。超音速气流碰到尖锐物体时将会产生激波,特别对于锥状物体、锲型物体。这类问题不仅在物理中非常重要而且在高维可压缩流体的数学理论研究中也是很基本的。在这个领域中,有很多的实验和数值仿真。超音速流通过一个尖锐的楔形物体的现象经常发生在我们日常生活中。例如,飞机的机翼就被设计为一个尖锐的楔形形状。当我们考虑超音速飞机在空中飞行时,如果把飞机看做是静止的,那么这就是一个超音速流通过一个尖锐的楔形的例子,根据终端压力大小的不同在楔形的头部会出现弱激波或者强激波。对于这方面的研究,陈恕行、刘太平、辛周平、陈贵强、尹会成、许刚等做了一系列重要工作。他们在不同条件下给出了局部或全局的结果,可以参考[19,54,61,87,110]和其里所包含的文献。关于亚音速流通过一个障碍物、上半平面以及有限或者无限长管道的全局存在性问题,董光昌、尹会成、辛周平、L.Bers、B.Bojarski、D.Gilbarg、R. Finn等已经做出了一些非常重要的工作,见参考文献[4,7,31,36,37,86,88,115]等。关于二维流体,其中最重要的进展是由Bers[4]给出的,他证明了只要马赫数足够小全局的亚音速位势流是存在的,并且还得出,当马赫数增加时,流体的最大速度会趋向于声速。后来,Finn和Gilbarg[36]根据最大值原理和亚音速流在无穷远的渐进行为证明了通过障碍物的亚音速流是唯一的。Xie和Xin[99]研究了通过无限长的管道内的亚音速音速流问题,并且证明了存在关于来流的质量通量的一个临界值,如果质量通量小于这个临界值管道内全局一致亚音速流是存在的。至于三维流体情形,首先由Finn和Gilbarg [37]给出以及后来的Dong和Ou[31]给出的最终结果都和二维结果是类似的,即当马赫数适当的小,亚音速扰流整体存在。Xie和Xin[100]结合变分法、大量的椭圆先验估计以及紧性方法证明了亚音速音速流通过无限长的轴对称的管道的全局存在性。我们现在主要考虑亚音速流通过二维和三维的锲型物体的适定性,我们的论文主要受到Courant和Friedrichs的经典名著[26]的第111节的如下描述的启发：“流体通过尖锐的角点或者障碍物时,如果来流是亚音速的,与位势流相关的问题由二阶拟线性椭圆微分方程控制,在任何点的解依赖边界条件和边界的无穷远部分,并且比超音速流研究更加困难”。
本篇论文的主要结果如下：
第一、如果流体是一致亚音速的(q/c(q)<1—ε,0<ε<1),并且可以用位势流即流体是等熵无旋定常流来描述。运用复分析中的基本定理：黎曼映射定理[92]可知二维斜面与单位圆共形,我们的问题转化为在单位圆内找一个调和函数使它的边界满足合适的条件。并注意到复速度关于流体度量是共形的以及关于同一个度量共形的两个变换其中一个可以看做另一个的解析函数。基于以上的观察,我们得到类似于[60]中的结论：q三0.并且可以做到比角状区域稍一般的区域。
第二,对于三维斜坡,如果无穷远处加质量通量条件,流体是定常位势流并且是亚音速的(q 整篇论文组织如下：
第一章给出无界区域内亚音速流的一些物理背景,并介绍了与本论文有关的一些研究进展,同时对我们所做的工作的意义进行说明。
第二章研究的是二维定常无旋等熵多方气体,运用复方法得到二维位势流在二维角状区域内的唯一性,即：只有速度为零的解并且我们成功得去掉了小性条件。
第三章研究三维位势流方程当斜面做了周期性小扰动并在无穷远处提了一个合适的质量通量条件后,得到全局解的适定性。
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-ε,ε＞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.
本篇论文的主要结果如下：
第一、如果流体是一致亚音速的(q/c(q)<1—ε,0<ε<1),并且可以用位势流即流体是等熵无旋定常流来描述。运用复分析中的基本定理：黎曼映射定理[92]可知二维斜面与单位圆共形,我们的问题转化为在单位圆内找一个调和函数使它的边界满足合适的条件。并注意到复速度关于流体度量是共形的以及关于同一个度量共形的两个变换其中一个可以看做另一个的解析函数。基于以上的观察,我们得到类似于[60]中的结论：q三0.并且可以做到比角状区域稍一般的区域。
第二,对于三维斜坡,如果无穷远处加质量通量条件,流体是定常位势流并且是亚音速的(q
第一章给出无界区域内亚音速流的一些物理背景,并介绍了与本论文有关的一些研究进展,同时对我们所做的工作的意义进行说明。
第二章研究的是二维定常无旋等熵多方气体,运用复方法得到二维位势流在二维角状区域内的唯一性,即：只有速度为零的解并且我们成功得去掉了小性条件。
第三章研究三维位势流方程当斜面做了周期性小扰动并在无穷远处提了一个合适的质量通量条件后,得到全局解的适定性。
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-ε,ε＞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.
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