| In this paper,we study the Riemannian analytic solution to quasilinear conservation laws with different initial discontinuous conditions.The Riemann problem of equation is the problem of differential equation with the initial discontinuous and with the different constant initial data.The study of the two-dimensional conservation laws is an important problem of nonlinear partial differential equations,the Riemann problem is one of the most important theoretical problems of the two-dimensional conservation laws,it is easier than the general Cauchy problem to find the basic nature of the solution and the solution of Cauchy problem can be approximated by the solution of Riemann problem.And because the Riemann solution is generally an explicit solution,it can be used as a standard solution to test the calculation format.This article consists of five chapters.The first chapter is the introduction.Firstly,the author introduces the initial problems and the Riemann problem and its development of the n-dimensional conservation laws,and introduces the current situation and research results of the Riemann problem of the two-dimensional.Secondly,we give the preliminary knowledge,and finally summarize the relevant results in this paper.The second chapter mainly study the solution of the quasilinear hyperbolic conservation laws with the initial discontinuity on the circumference of two disjoint units.Firstly,the initial position of the shock wave and its discontinuities,the starting position of the evacuation wave and the expression of the solution of rarefaction wave and its boundary are determined by using the H(H)condition.Secondly,we study the new shocks of the interaction of the basic waves,that is,two new shocks,the interaction between two pairs of shock waves and rarefaction waves,The characteristic line equations of the new shocks are given by R-H condition.From the intersection between the shock waves and the rarefaction waves boundary,the analytic expression of the new shock wave is obtained by using the constant variation method of the ordinary differential equation.The existence range of the waves is determined by the intersection between the shock waves,the rarefaction waves boundary and the new shock wave.Finally,the global distribution of the solution is given,and the large time behavior of the whole solution is given.Each time period has a structural distribution of the solution.In the third chapter,on the basis of two-dimension,we study the nonselfsimilar singularity structure and interaction of the three-dimensional quasilinear hyperbolic conservation laws with the initial discontinuities on two disordered unit spherical surfaces,and get the whole structure which Is the promotion of the previous chapter.At some time,the structure of the solution in three-dimensional space is very intuitive.In the fourth chapter,we use the geometric means to construct the solution of the discontinuity problem with the initial value in the concentric ring different from other parts.In the initial interruption line,the large circle and the small circle respectively produce a pair of shock wave and the rarefaction wave.The interactions between waves and waves in different time periods are also analyzed,and the intermittent or continuous rarefaction waves boundary are given,and give their corresponding existence range.This is useful for understanding the twodimensional Burgers equation,and can provide an example for the verification of numerical methods.In the fifth chapter,we construct the solution of the initial discontinuity on the non-convex curve.In this case,the small constant value disappears at a certain moment,and the second scattered wave disappears,and the large time behavior is described.In this chapter,we use some results in Chapters 2 and Chapters 4to fully verify the running mechanism of continuous movement and catch up the shock and the rarefaction wave,which will help to explore the application of shock and rarefaction wave. |
