Research On Darboux Problemof Second Order Degenerate Hyperbolic Equations

The research of mixed degradation partial differential equation have been concerned for a long time, it is necessary to study the problems of definite solutions for mixed degenerate hyperbolic and degenerate elliptic equation, due to the needs of practical problems for modern science and technology, such as jet theory, high-speed aerodynamics, transonic and supersonic. Whether the degenerate hyperbolic or degenerate elliptic equations that can be used to eliminate the degradation by using the proper transformation, turn into the equation with singular line is studied. Euler-Poisson-Darboux equation is the earliest and most studied one. The Darboux equation is a kind of hyperbolic partial differential equation with singularities, hyperbolic equations is not any mention boundary condition on the entire border region compared with the elliptic equations.. The features for first problem of Darboux is that the value of the function itself is only given on a part of the border region, and the features for second problem is appear the directional derivative of outside normal boundaries on a part of the boundary.In this paper, we study the existence of solutions for the Darboux problem of two order degenerate hyperbolic equation, and mainly carry out the following research work:1) We obtain priori estimates for the expressions of established solutions and discussed in the existence and uniqueness of solution for the first Darboux problem in complex form, by using the Schauder fixed point theorem and Leray-Schauder fixed point theorem.2) We turn the second Darboux problem into the corresponding Riemann-Hilbert problem, in term of means of Riemann existence theorem, combined with the results of complex equations with same type under analytic and univalent functions mapping, through the appropriate coordinate transformation and eliminate the degradation, a partial differential equations with singular line is obtained.3) We obtain the existence and uniqueness of the solution for the two order degenerate hyperbolic equation with the boundary conditions of the oblique derivative and the point type conditions, by using the corresponding Riemann-Hilbert problem, further promotion of the second Darboux problem is further studied.