Nonlinear Equations Of Mixed Type With Discontinuous Coefficients

In this paper, we mainly consider the existence and uniqueness of the Tricomi prob-lems for the nonlinear equations of mixed type. In terms of type, partial differential equa-tions can be divided into three kinds of types:elliptic equations, parabolic equations and hyperbolic equations. The theory for equations of only one type has been very mature, but so far, the theory for the coupled equations of two or three types has appeared formidable. Specially, the study of the coupled equations of elliptic type and hyperbolic type is very difficult, we call these equations as elliptic-hyperbolic equations of mixed type. For the linear elliptic-hyperbolic equations of mixed type, there exist many results, however, for the nonlinear elliptic-hyperbolic equations of mixed type, the theory is very little. In the other side, with the deep study of the physical problems, many mathematical models can be transformed into the boundary value problems for the nonlinear elliptic-hyperbolic e-quations of mixed type. Therefore, from the point of mathematics and physics, the study of boundary boundary value problems for the nonlinear elliptic-hyperbolic equations of mixed type is very important.The problems of this paper are based on the E-H Mach reflection. For the E-H Mach configuration, the corresponding equation is elliptic in one part, is hyperbolic in the other part, and the coefficients are discontinuous on the curve of transition of the type. The simplest model of this equation is Lavrentiev-Bitsadze equation (sgn y)uxx+uyy=0. In fact, the Lavrentiev-Bitsadze equation is simple and can not describe the original prob-lem. Therefore, in this paper, we study the two kinds of more general nonlinear equations which have the same properties as Lavrentiev-Bitsadze equation, we call these equations as nonlinear Lavrentiev-Bitsadze type equations.In chapter3of this paper, we mainly consider the Tricomi problem for the following equation uxx+(sgn y)(1+ux2)uyy=0. In the hyperbolic region, the equation is a second order nonlinear hyperbolic equation, by introducing Riemann method, we can give the formal solution. Therefore the original problem is reduced to a mixed boundary value problem for elliptic equation, and then we can prove the existence of the solution to the the original problem.In chapter4, we study the Tricomi problem for the following equation (sgn uy)uxx+uyy=1. When uy>0, the equation is elliptic; when uy<0, the equation is hyperbolic, and the curve of transition of type is unknown. By using partial hodograph transformation, the original problem is reduced to a mixed boundary value problem for elliptic equation, and then we can prove the existence and uniqueness of the solution to the the original problem.