In this article, we prove the existence and obtain the expression of its solution formula of global smooth solution for non-homogeneous multi-dimensional（m-D） conservation law with unbounded initial value; our methods are new and essentially different with the situation of bounded initial value.
We investigate the Chapman-Jouguet model in multi-dimensional space, and construct explicitly its non-selfsimilar Riemann solutions. By the method we apply in this paper, general initial discontinuities can be dealt with, even for complex interaction of combustion waves. Furthermore, we analyze the way in which the area of unburnt gas shrinks.
We investigate Chapman-Jouguet models in three-dimensional space by means of generalized char- acteristic analysis. The interaction of detonation, shock waves and contact discontinuity is discussed intensively in this paper. If contact discontinuity appears, the structure of global solutions becomes complex. We deal with this problem when strength of detonation is small.
We use Hopf-Lax formula to study local regularity of solution to Hamilton- Jacobi （HJ） equations of multi-dimensional space variables with convex Hamiltonian. Then we give the large time generic form of the solution to HJ equation, i.e. for most initial data there exists a constant T 〉 0, which depends only on the Hamiltonian and initial datum, for t 〉 T the solution of the IVP （1.1） is smooth except for ~ smooth n-dimensional hypersurface, across which Du（x, t） is discontinuous. And we show that the hypersurface 1 tends asymptotically to a given hypersurface with rate t-1/4.
Two-dimensional Riemann problem for scalar conservation law are investigated and classification of global structure for its nomselfsimilar solution is given by analysis of structure and classification of envelope for non-selfsimilar 2D rarefaction wave. Initial data has two different constant states which are separated by initial discontinuity. We propose the concepts of plus envelope, minus envelope and mixed envelope, and some new structures and evolution phenomena are discovered by use of these concepts.
