The Solutions For A Class Of Nonlinear Hyperbolic-parabolic Equations

The existence and uniqueness of the solutions for a class of nonlinear degenerate parabolic-hyperbolic equation are proved, then the error estimate of the solutions is studied in this paper. This kind of equations models many natural phenomenons, applied widely in filtration, phase transition, biochemistry and dynamics of biological groups, exhibiting both viscous and non-viscous phases, and reflecting the restriction between different materials in diffusion-convection process in porous medium, sedimentation-consolution process, convection of the biome in nature. For its widely used in physics, the study of the property for the equations are important.This paper consists of five parts.In the first chapter, we introduce the physical background of the degenerate parabolic hyperbolic equation, and the progress of the second-order nonlinear degenerate parabolic hyperbolic equation.In the second chapter, we will give the appropriate functional space, then we can introduce the definition of three kinds of solutions of degenerate parabolic hyperbolic equations. And the relations of three kinds of the solutions are compared in this part.In the third part, we give the theory of the existence of weak solutions to the initial and boundary value problem of a special simple degenerate parabolic-hyperbolic equation as follows: and it is proved by Galerkin method.In the fourth part, we study the uniqueness of entropy solutions to the Cauchy problem of the following equation: and the uniqueness of entropy solutions to derived forms of the equations can be proved in the same way.At last, we study the error estimates between entropy solutions and the smooth viscosity solutions.