Regularity Analysis Of Solutions Of Two Hyperbolic Equations

We are concerned with the regularity analysis of the solutions to system of one dimensional zero-pressure fluid and Hamilton-Jacobi equation.When analyzing the global structure of the solutions to systems of one dimensional zero-pressure fluid,the sufficient and necessary condition for the potential function to obtain a unique non-degenerate minimum value on a characteristic are discussed.Moreover,it is shown that there exists a one-to-one correspondence between the path connected components of the set of singularity points and that of the complement set deduced by the initial data.When analyzing the Hamilton-Jacobi equation,we prove that if the initial data do not belong to the first category of Ck,then the solution is piecewise smooth.We study the structure of the solution in the neighborhood of the point where a shock generates.We show that there is a neighborhood such that a unique Ck+1 smooth shock exists.Moreover,we study the points where finitely many shocks collide to form a new shock and all of these shocks are Ck+1 smooth except the collision point.