Continuous Subsonic-sonic Flows In Finitely And Infinitely Long Convergent Nozzles

This thesis aims to study the continuous subsonic-sonic flows in two-dimensional finitely and infinitely long symmetric convergent nozzle.The flow satisfies the slip conditions on the nozzle walls and its velocity vector is along the normal direction at the inlet.We also require that the outlet is sonic curve which touched a fixed point at the upper wall.The problems we consider can be described as the free boundary problem of nonlinear degenerate elliptic equations with nonlocal boundary value conditions and degeneracy at free boundary,whose free boundary is sonic curve.We show that there exists uniquely a subsonic-sonic flow in each nozzle,and the flow is singular in the sense that the speed is C1/2-Holder continuous and the acceleration blows up at the sonic state.We also determine the relation between the accurate rate that the continuous subsonic-sonic flow converges to the sonic state and the geometry of the walls.The symmetry of the nozzle yields that we only need to consider the part above the axis of symmetry.The thesis is divided into two parts.The fist part is an introduction,which lists the history and research status of many kinds of problems arising from sonic flow,such as the development and important results of the Euler equations of one-dimensional nonsteady polytropic gases,the outstanding achievements made by some scholars in investigating the problems related to the subsonic-sonic flows in nozzles which are studied in our thesis,the research progress of supersonic and transonic flows with or without shock waves,etc.Also,the work in this thesis and the structures of this thesis are contained in the first part.The remaining part,the main body of the thesis,is organized as the following three chapters:In Chapter 1,we deal with the continuous subsonic-sonic flow in the finitely long convergent nozzle on the weak condition that the nozzle is convergent near the outlet only.To establish the existence theorem,we must consider the boundary value problem of a degenerate elliptic equation with fixed boundary and given the speeds of the flow at the inlet and the upper wall.The lack of the constraint on the second derivative of the corresponding function of the upper wall leads to that the super-and subsolution to the problem with fixed boundary are difficult to construct.We are forced to show the existence of the solution to the problem with speed at the outlet replaced by sufficient small subsonic speed first,and then demonstrate that the outlet speed can reach the sonic state through establishing the estimate of the average speed of the flow in the nozzle.After choosing suitable function space,the Schauder fixed point theorem yields the existence of subsonic-sonic flow.The regularity can be prove by Harnack inequality,and the uniqueness can be obtained by energy estimates after choosing suitable coordinates.In Chapter 2,we consider the continuous subsonic-sonic flow in the special in-finitely long nozzle which is convergent at the outlet and flat at sufficient far fields.The problem can be described on the potential plane as a definite solutions problem in an unbounded domain.Firstly,we prove the existence and uniqueness of the solution to the truncated problem in bounded domain and obtain some important regularity estimates.Then we take the limit of the solutions to the truncated problems,to get the solution to the problem of continuous subsonic-sonic flow in infinitely long nozzle in the sense of locally uniform convergence.We should note that we need to get some positive constant independent of the length of truncated nozzle by use of the geometry of the walls such that each truncated problem admits a unique solution when the gradient of the length of vertical section is less than this constant.The regularity can be proved by Harnack inequality.Also,we need investigate the asymptotic behavior of the speed of the flow.Through rescaling and periodically extending the problem in some section at sufficient far fields,we get the asymptotic behaviors of the partial derivatives of the speed with respect to potential and stream by use of Harnack inequality.Also,we can prove that the flow must be uniformly subsonic at sufficient far fields and the relation among the mass flux,the limit of the speed of the flow at infinity and the limit of the length of the vertical section of the nozzle.At last,one can prove the uniqueness of the continuous subsonic-sonic flow in the nozzle which satisfies this kind of asymptotic behavior.by more-refined energy estimates after choosing suitable coordinates.In the final chapter,we study the continuous subsonic-sonic flow in the general infinitely long nozzle which is convergent near the outlet and convergent to a flat one at infinity by using a series of infinitely long nozzles which are flat at sufficient far fields to approach general nozzle.After making more-refined estimates on the partial derivative of the speed with respect to stream in virtue of the geometry of the walls,we get some constant independent of approximation nozzle,such that each approximation problem admits a unique solution when the gradient of the length of vertical section is less than this constant.We take the limit of the solutions to the approximation problems to get the solution to the problem of continuous subsonic-sonic flow in general infinitely long nozzle in the sense of locally uniform convergence.The regularity can be proved by Harnack inequality,too.To obtain the asymptotic behavior of the speed of the flow,we still need to rescale and periodically extend the problem in some section at sufficient far fields.But the boundary value condition on the upper wall is not homogenous Neumann condition,so we introduce a proper transformation to translate the boundary value conditions on the both walls into homogenous Neumann conditions.Although the translated equation is nonhomogeneous with a source term,the geometry of the wall ensures that the source term processes good properties after rescaling and periodically extending.Therefore,one can prove the asymptotic behaviors of the partial derivatives of the speed with respect to potential and stream,and then show that the flow must be uniformly subsonic at sufficient far fields,the relation among the mass flux,the limit of the speed of the flow at infinity and the limit of the length of the vertical section of the nozzle,as well as the uniqueness of the continuous subsonic-sonic flow which satisfies this kind of asymptotic behavior in the nozzle.