Study On The Existence And Regularity Of Semi-hyperbolic Structures Of Euler Equations Under A Class Of Generalized Pressures

The well-posedness of the initial value problem for the compressible Euler equations in high dimensional is a very difficult problem.Semi-hyperbolic structures are not only common in numerical simulation of two-dimensional Riemann problems with compressible Euler equations,but also frequently appear in practical problems.This problem appears in the transonic flow problem of airfoil and Guderley reflection of Von Neumann paradox,it has important theoretical significance and application value.This paper mainly discuss the existence and regularity of smooth solutions of semi-hyperbolic structured to two-dimensional Euler equations under a class of generalized pressure.The main contents are as follows:Chapter 2 introduce characteristic angle α,β and sound velocity c to establish their characteristic decomposition.Chapter 3 use direct method,obtain the invariant region by characteristic decomposition of characteristic angle,then obtain the maximum norm estimate of α,β.Using the characteristic decomposition of sound velocity c and continuity method to establish the gradient estimates of solution,thereby proving the existence of global solution of semi-hyperbolic structures.Chapter 4 introduce a partial hodograph transformation,use the continuous induction method to establish the precise estimation of the sound velocity gradient up to the sound velocity boundary,so as to obtain the regularity of the solution on the partial hodograph plane.Finally returning to the self-similar variables to show that the global smooth solution up to the sonic line has uniformly C^1,1/6-regularity and the sonic line is also C^1,1/6 regular.