Perturbation Problems Of Continuous Subsonic-sonic Flows In Convergent Nozzles

In the thesis, we study perturbation problems of isentropic, irrotational, steadycompressible and continuous Euler subsonic-sonic flows in a two dimensional convergentnozzle.The main methods which we use to solve the problem of continuous subsonic-sonicflows in a convergent nozzle is the Schauder fixed point theorem and the method ofenergy estimates. First, we study the degenerate fixed boundary problem by the clas-sical elliptic theory, and then following from some properties, the degenerate boundaryproblem is solved. Precise estimates and optimal regularities are obtained by some tech-niques on degenerate elliptic equations. Then, we use the Schauder fixed point theoremto prove the existence of the continuous subsonic-sonic flow. As for the uniqueness ofthe continuous subsonic-sonic flow, by a suitable coordinates transformation, we fix thefree boundary into a fixed boundary and transform the nonlocal boundary conditioninto a common boundary condition. Then, the uniqueness theorem is established bythe method of energy estimate.The thesis is organized as follows.First, an introduction is given. It is devoted to introducing the physical back-ground and previous mathematical research works on subsonic-sonic compressible flows,especially important achievements by other scholars who treated Euler equations. The main problems we concerned, main results we obtained, and methods we utilized inthis thesis are also illustrated with comments.In the first chapter, it concerns continuous subsonic-sonic potential flows in atwo dimensional convergent nozzle with straight solid walls. It is shown that for thegiven inlet, which is a perturbation of an arc centered at the vertex of the nozzle,and the given incoming flow angle which is a perturbation of the angle of the innernormal of the inlet, and the given incoming mass flux belonging to an open intervaldepending only on the adiabatic exponent and the length of the arc, there is a uniquecontinuous subsonic-sonic flow from the given inlet with the given incoming flow angleand the given incoming mass flux. Furthermore, the sonic curve of this flow is a freeboundary, where the flow is singular in the sense that the speed is continuous and theacceleration blows up at the sonic state. The perturbation problem is solved in thepotential plane, where the flow is governed by a degenerate elliptic problem with twofree boundaries, on one of which the equation is degenerate and on the other of whichthe boundary condition is nonlocal. Then, the Schauder fixed point theorem, a suitablecoordinates transformation and the method of energy estimates, we get the existenceand uniqueness of the solution to the problem of continuous subsonic-sonic flows.In the second chapter, we investigate the well-posedness of a boundary value prob-lem for a continuous subsonic-sonic flows in a convergent nozzle with one straight solidwall and one wall which is a suitable perturbation of a straight solid wall. It is shownthat for the flow whose velocity vector is along the normal direction at the given inletbeing a perturbation of an arc and the outlet, which satisfies the slip condition on thewall and whose sonic curve has a fixed point, we should study a degenerate boundaryvalue problem with two free boundaries and two nonlocal boundary conditions in thepotential plane. Then, the existence and uniqueness of the solution to the problemare obtained by the same methods mentioned above with some appropriate conditions.Furthermore, this continuous subsonic-sonic flow is singular at the sonic state. In the last chapter, it is devoted to the study of well-posedness of a solution tothe continuous subsonic-sonic problem with three perturbations as mentioned before.According to the analysis and some discussion in the first two chapters, we formulate theperturbation problem in the potential plane which has three free boundaries and whoseboundary conditions at the inlet and the outlet are nonlocal. Following from somemethods and results as mentioned above, the existence and uniqueness of a solution tothe perturbation problem are proved under some appropriate conditions. Furthermore,it is shown that the speed of the continuous subsonic-sonic flow is continuous and theacceleration blows up at the sonic state.