Similar Degree Of Estimation, A Number Of Model Equations In Fluid Mechanics

The present dissertation is devoted to the comparison of weak solutions of the system of quasi-one-dimensional isentropic gas flow and its irrotational approximation, the comparison of weak solutions of the system of quasi-one-dimensional non-isentropic gas flow and its non-isentropic approximation and the comparison of weak solutions of quasi-one-dimensional isentropic Euler-Poisson system and its irrotational approximation.Based on wave-front tracking algorithm and theory of semigroup,and using the precise comparison of h-Riemann solutions for different system,we can estimate the difference between the solutions for different models.Hyperbolic systems of conservation laws play an important role in many branch of physics.The typical one is the compressible Euler system.In many studying physical problems,one often make some assumptions on the fluid to simplify the equations.For instance,if the states of the flow are assumed to be independent of the time variable,we can get the steady Euler system.Furthermore,if the entropy of the flow remains as a constant,we can have the isentropic Euler system,and if the flow is assumed to be irrotational,we can have the potential flow equation. Since both the isentropic Euler systems and the potential flow equations are good approximate models for the Euler systems.Then the natural question is how to estimate the difference of the weak solutions to the systems describing different models,especially to estimate the weak solutions with shock waves.There are some results on the comparison of weak solutions of homogeneous hyperbolic system and its approximate model.But the comparison of weak solutions of hyperbolic system with source term and its approximate model is a new problem in the field of compressible flow equations,and is important in dealing with the inhomogeneous hyperbolic system.The result of present dissertation extends previous results,on the hyperbolic system without source term is present.The dissertation is organized as follows.Chapter One is an introduction.It is devoted to introducing physical back- ground and previous mathematical research works on the quasi-one-dimensional gas flow and Euler-Poisson system.The main problems,main results,and methods in this Ph.D.dissertation are also illustrated.In Chapter Two,under the assumptions that both initial data and the crosssection have sufficiently small total variation and that the initial data are supersonic (or are subsonic respectively),we prove that in any bounded domain the L~1 norm of the difference between solutions of the hyperbolic system of balance laws and the potential flow system of balance laws with the same initial data can be bounded by the cube of the sum of the total variations of the initial data and the cross-section.In Chapter Three,we consider the system of quasi-one-dimensional nonisentropic gas flow and the system of quasi-one-dimensional isentropic gas flow,because the number of equation in the system of quasi-one-dimensional non-isentropic gas flow is one more than the number of equation in the system of quasi-onedimensional isentropic gas flow,further treatment is necessary.To overcome this difficulty point,we have to add a new component for the pressure term in solution of the system of quasi-one-dimensional isentropic gas flow.By using the similar way in Chapter Two,with more complex computation,we get the conclusion:under the assumptions that both the initial data and the area of varying cross-sections have sufficiently small total variation,the L~1 norm of the difference between solutions of isentropic and non-isentropic balance laws with the same initial data in any bounded domain can be bounded by the cube of this total variation.In Chapter Four,we consider the bipolar Euler-Poisson system,it is a mathematical modeling for plasmas.Since the bipolar Euler-Poisson system is a hyperbolic and elliptic coupled system,we have to use new method to derive the estimate of the difference of the weak solutions to Euler-Poisson system and its "potential" approximation.That is,under the assumptions that both the initial data has sufficiently small total variation,the L~1 norm of the difference between solutions of the Euler-Poisson system and the potential flow system with the same initial data can be bounded by the cube of the total variation of the initial data.