Global Smooth Solution And Non-selfsimilar Riemann Solution Of A Class Of Multi-dimensional Non-homogeneous Scalar Conservation Laws

This paper mainly investigate the global smooth solution of Cauchy problem and the global solutions of non-selfsimilar shock wave and rarefaction wave of Riemann problem for n dimensional non-homogeneous scalar conservation law;and also investi-gate the interactions of the solutions of n dimensional non-selfsimilar shock waves and rarefaction waves for n dimensional non-homogeneous Burgers equation with the initial data has two different constant states;and also investigate the interactions of waves of the Riemann solutions for two dimensional non-homogeneous Burgers equation with the initial data has three different constant states.In chapter 3,we mainly study the global smooth solution of n dimensional non-homogeneous scalar conservation law whose non-homogeneous term is a function of u and t with unbounded or bounded initial data.We prove the existence of global smooth solution and obtain its formula.The main tools used in this chapter are Implicit function theorem,Contraction principle and Principle of mathematical induction.In chapter 4,we construct a theory about solving the global solution and structure of Riemann problem for the n dimensional non-homogeneous scalar conservation law(?),where the initial data has two constant states separated by a n-1 dimensional smooth manifold.We first define the characteristic regions covered by characteristics starting from two initial areas,and construct the expressions of non-selfsimilar shock wave when characteristic regions interact with each other,and rarefac-tion wave when characteristic regions do not interact.And then,we obtain the unique non-selfsimilar global solution.To get the expressions of shock wave and rarefaction wave,we apply the Rankine-Hugoniot condition and Implicit function theorem,respec-tively.To get the uniqueness,we apply the Kruzhkov's Entropy condition.Finally,we give some applications of our theory.In chapter 5,we investigate the global structures of the non-selfsimilar solutions for n dimensional(n-D)non-homogeneous Burgers equation,in which the initial data has two different constant states,which are separated by a n-1 dimensional sphere.The case of this kind of closed initial discontinuity surface has not been investigated in other references and has new difficulty,in which coordinate transformation are hard to be applied,especially when n is big enough.Firstly,we get the expressions of solutions of n-D shock waves and rarefaction waves starting from the initial discontinuity.Sec-ondly,we discuss new kind of interactions of the relating elementary waves and obtain the global structures of non-selfsimilar solutions.Ingenious techniques are proposed to construct the n-D shock waves.New asymptotic behavior and structures are also discovered.In chapter 6,we investigate the global structures and wave interactions of non-selfsimilar solutions for two dimensional non-homogeneous Burgers equation,where the initial data has three constant states,separated by two disjoint circles.We first get the expressions of solutions of shock waves and rarefaction waves starting from the initial discontinuity.Secondly,we discuss the interactions of these elementary waves and find some new phenomena that the time of the interaction of shock wave and rarefaction wave has no critical point at which the structures of solutions begin to change,which are different from the homogeneous case.Finally,we construct the global structures of the non-selfsimilar solutions and find the new asymptotic behavior that the diameter of the region of elementary waves is bounded.