The Perturbed Riemann Problem For The Leroux System And The Simplified Chromatography System With Three Piecewise Constant Initial States

In this paper,the stability of Riemann solutions for some hyperbolic systems of conservation laws is considered,which include the Leroux system and the simplified chromatography system.Firstly,we study the Riemann problem for some hyperbolic systems of conservation laws.Next,the perturbed Riemann problem for the corresponding systems are considered by studying the wave interaction with the method of characteristics when the initial data are taken as three piecewise constant states.With the help of the geometrical structure,we can see the large-time asymptotic states of the global solutions to the perturbed Riemann problems.At last,it is concluded that the Riemann solutions for some hyperbolic systems of conservation laws are stable.The contents studied in the paper have broader physical background.It is very important in theoretical analysis and practical application for reason that the models studied here have broader physical background.This paper can be divided into six chapters as follows:In chapter one,were discuss the background and status of development about the main contents of this thesis,such as the hyperbolic conservation laws system,Riemann problem,wave interaction and method of characteristics.Besides,the main job of this thesis was also given to readers to have a general understanding.In chapter two,the basic definitions and theorems about the hyperbolic conservation laws system,Riemann problem,wave interaction and so on were presented as the foundation of this thesis.In chapter three,the formation and stability analysis of Riemann solutions to a scalar conservation law with a linear flux function involving discontinuous coefficients were considered,where the Riemann solution has a vacuum state in some particular initial data.The generalization Riemann problem was investigated with the method of characteristics by adopting the local linearization technique to treat the discontinuous coefficient.In addition,the stability of the Riemann solutions and the formation of vacuum state can be obtained in the limit process.In chapter four,the global solutions of the perturbed Riemann problem for the Leroux system are constructed explicitly under the suitable assumptions when the initial data are taken to be three piecewise constant states.The wave interactionproblems are widely investigated during the process of constructing global solutions with the help of the geometrical structures of the shock and rarefaction curves in the phase plane.In addition,it is shown that the Riemann solutions are stable with respect to the specific small perturbations of the Riemann initial data.In chapter five,the solutions of the perturbed Riemann problem for the chromatography system of Langmuir isotherm with one inert component are constructed in a completely explicit form when the initial data are taken as three piecewise constant states.The wave interaction problem is investigated in detail by using the method of characteristics.In addition,the generalized Riemann problem with the delta-type initial data is considered and the delta contact discontinuity is discovered.Moreover,the strength of delta contact discontinuity decreases linearly at a constant rate and then the delta contact discontinuity degenerates to be the contact discontinuity when across the critical point.In chapter six,we make a summary and outlook to the contents of this dissertation.