The Initial Value Problem For A Class Of Nonlinear Hyperbolic Systems Of Conservation Laws

In this article, we study the initial value problem for several classes of hyperbolic systems. One is the Riemann problem and the other is non-selfsimilar initial value problem. By use of the characteristic analysis method, the global solution for the hyperbolic system is constructively obtained.Section 2 introduces some useful concepts for the hyperbolic system firstly. Then some general theories about one-dimensional and two-dimensional hyperbolic systems are introduced in the next part of this section, respectively.The one-dimensional non-selfsimilar initial value problem, which contains three constant states, for a reduced model of Euler equations is studied in section three. The attention is paid to the interaction of the delta-shock and the other elementary waves. Due to the introduction of a suitable generalized Rankine-Hugoniot relation, delta shock gets a well depiction from velocity, location and weight. Then the unique solution to different initial conditions are constructively obtained. Under the entropy condition, a new phenomenon appears, which is the delta-shock will disappear and be replaced by a shock and a contact discontinuity suddenly when the delta shock penetrates the borderline of front rarefaction waves and back rarefaction waves.In section 4, we solve the Riemann problem with the initial date containing Dirac delta functions for a class of coupled hyperbolic systems of conservation laws. The elementary solutions include constant state, delta-shock and vacuum. Under the suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of solutions involving delta shock waves and vacuum are proved. We also get the global quality of the delta shock. Further, four kinds of different structures of solutions are established uniquely. Especially, when m₀= 0 in this paper is the results in [85], then it is the generalization of there. So the stability of the delta-shock to initial weight value perturbation is proved. Furthermore, when f(u) =u, the system which we studied is changed into the one-dimensional transportation (zero pressure flow) equations, when f(u) = u/((1+u²)^1/2), the system which we studied is nonstrictly hyperbolic system in astrophysics.In section 5, we solve the non-selfsimilar initial value problem for the two-dimensional transportation (zero pressure flow) equations. By use of the characteristic analysis method, we find a basic lemma to solving the two-dimensional transportation equations. Then the Space problem can be turned into a plane problem to solve. It makes the problem easier and the exactly solution can can be given. We study the cases that the initial value are divided into two constant states by a circular curve and an arbitrary smooth convex curve respectively and the global solutions are obtained constructively. The problem, what we studied, has no requiment of scale invariant, then the method, what we got, can be used to solve the initial value problem which is divided by an arbitrary curve.In section 6, the Riemann problem for a one-dimensional nonlinear degenerate wave equation system in elastic materials is solved constructively. When v≠0, the characteristic field is genuine nonlinearity, then the Riemann problem can be solved by simple waves and shock waves. For the stress function is not convex or concave and the shock condition is degenerate, we can define the degenerate shock S_d like paper [63]. Depending on the relative location between U_l and U_r, the global solution is obtained constructively by use of rarefaction wave R, shock wave S and degenerate shock S_d.