Study On The Related Problems Of The 2-D Pseudo-steady Euler Equations

We mainly study the structure of centered simple waves of 2-D isentropic ir-rotational pseudo-steady flow around a convex corner and the expansion problem of polytropic gas into the vacuum at a convex corner.The paper is organized as follows.In Chapter 1,we mainly illustrate the background and the developments of 2-D compressible Euler equations.In particular,the detailed research developments of Riemann problem of 2-D Euler equations are shown.In Chapter 2,some basic concepts on quasilinear hyperbolic conservation equa-tions are given.The concepts include simple wave,Riemann problem,self-similar solutions,hyperbolicity and so on.In Chapter 3,the 2-D isentropic irrotational unsteady flow around a convex corner is considered.Firstly,we introduce the general characteristic analysis theory of the 2-D isentropic irrotational pseudo-steady Euler equations.Secondly,we dis-cuss the properties of the centered simple waves.At last,we discuss the structure of simple waves deduced from the supersonic flow around a convex corner.We prove that the supersonic oncoming flow can turn the convex corner by the centered rar-efaction wave or the centered compression wave and continue to flow along the rigid wall.In Chapter 4,we study an expansion of polytropic gases into vacuum which arises as two-dimensional(2-D)supersonic flow turns a convex corner and expands into a vacuum.The problem catches interaction of a centered rarefaction wave and a backward planar rarefaction wave,which deduces to a Goursat problem for 2-D self-similar Euler equations for compressible flow.By the methods of characteristic decomposition and invariant regions,we get the strict hyperbolicity in the wave interaction domain and priori C1 estimates of solutions to the Goursat problem.By the classical theory of C1 solution for quasilinear hyperbolic equations and the Heine-Borel Theorem,we obtain the global C1 solution on the interaction region.