The global existence and the life span of smooth solutions to a class of complex conservation laws

In this paper, we consider the Cauchy problem for a class of complex conservation laws, which is introduced by P.D. Lax and essentially a class of quasilinear hyperbolic systems in two space dimensions. For a kind of flux functions, we prove the global existence of smooth solutions to the Cauchy problem with small initial data. On the other hand, for different kinds of flux functions, we give some estimates on the lower bound of the life span of smooth solutions.