Self-similar Solutions For The Euler Equations And The Related Models In Gas Dynamics

In this paper,we mainly study the multidimensional piston problem of the Euler equation with related models in gas dynamics.In the second chapter,we study the self-similar flow problems of the multidimen-sional full Euler equation caused by the uniformly expanding piston.When the piston isotropically expands outward at a constant speed,as a result,a shock wave forms and moves outward with constant velocity.Under the self-similar assumption,the problem can be reduced to a nonlinear ordinary differential equations with appropriate boundary conditions imposed on the piston surface and the shock front.We establish the global existence of solutions by analyzing the properties of the nonlinear ordinary differential system.In the third chapter,we consider the multidimensional piston problem of the isen-tropic compressible Euler equation with a two-constant equation of state.This is a rel-atively simple but widely used non-polytropic gas equation of state.In order to solve the difficulty caused by the equation of state,we first establish the global existence of smooth solutions to the boundary value problem for a new ordinary differential equa-tion system.By restricting the initial data of the new system,we get the solution to the original problem.In the fourth chapter,we use a natural method to study the piston problem.We use the two-dimensional nonlinear wave system as the model and transform the prob-lem into a boundary value problem of an autonomous nonlinear ordinary differential system by introducing a new set of variables.We analyze carefully the properties of the integral curves and use these properties to show the global existence of smooth positive solutions to the free boundary value problem of the autonomous ODE system.In addition,we also study the piston problem to the two-dimensional nonlinear wave system with a two-constant equation of state.