On the Occurrence of "Vacuum States" for 2 $\times$ 2 Quasilinear Hyperbolic Conservation Laws

Abstract

We show that the solution Lo the Cauchy problem of 2 × 2 nonlinear conservation laws, in general, may go out the strictly hyperbolic region of the system in a finite time, here the initial data are given in the strictly hyperbolic region. In other words, in general, we can't confine our attention to solve the Cauthy problem of 2 × 2 nonlinear consenvation laws in strictly hyperbolic type. However, we can expect that it may be solved under the additional conditions (A) and (b).