Research On Global Solutions On Multi-dimensional Hyperbolic Conservation Laws

The research on multi-dimensional conservation laws is one of the most challenging direction in the field of nonlinear partial differential equation, it is very important and very difficult. For the non-homogeneous multi-dimensional scalar conservation laws, there is no existence theorem of global weak solution. In order to understand the weak solutions like piece-wise smooth solution, we must investigate the structures and properties of global smooth solution. This is the foundation of the research on global solution of conservation laws. The it is very hard if applied previous methods.In this paper, we will discuss the initial value problem of n dimensional non-homogeneous scalar conservation laws , in which initial function is unbounded. By applying another method, we get the sufficient condition of existence of global smooth solution, i.e. Theorem 1, the result is valuable and significant in theory.