| This paper is concerned with one-dimensional zero-pressure flow equations. Using a potential function and considering its minimizing value points, All characteristic lines are divided into two kinds, one never touches other efficient characteristics, and the other intersects a certain efficient characteristic within a finite time at a point which forms a shock wave.Assuming the intial functions are Ck smooth, we will attain following conclusions:when the initial function family is removed a subset, we will prove the set of minimizing value points of its corresponding potential function consists of finitely many path connected components. The number of Ck+1smooth shock curves in some neighborhood of the point depends on the number of the path connected components, and the solutions to the equations are unique and piecewise smooth in the neighborhood.We can define a shock mapping that from the set of singular points to a set on which potential function has more than one minimizing point or a unique degenerate minimizing point. It has shown the properties of the shock mapping. Based on these resuit, we study the globlal structure of the set of discontinuous solution for one-dimensional zero-pressure flow equations. It is shown that there exist a one-to-one correspondence between the path connected of a set of singularity points and the path connected components of a set of the function Φ(x) does not attain its minimum. |
