Two-dimensional Glimm Type Scheme And The Research On Blow-up And Singular Structures Of Solution For The Multi-dimensional Conservation Law

In this paper, we mainly study the Cauchy solution of two-dimensional nonlinear hyperbolic conservation law.In chapter 2, we first introduce some concepts and conclusions about two-dimensional scalar conservation law, and then we introduce some concepts and conclusions about two-dimensional T-C variation and bounded variation space.In chapter 3, we consider the Cauchy problem of two-dimensional scalar conservation law which its initial data has compact support, we construct our two-dimensional scheme by using Riemann solutions, and finally prove the limit of this scheme is the weak entropy solution of this problem, the process can be divide into five step. In section 2, we divide the time by time step At, and redefine the initial data in every beginning of time step, this redefinition makes the problem change into four-piece-constant Riemann problem, then we use the Riemann solution to indicate the solution in a time step At, from this we construct the two-dimensional scheme. In section 3, we estimate the T-C variation about space variables x, y of this two-dimensional scheme, using the weak entropy condition and the property of T-C variation, we prove the T-C variation of this two-dimensional scheme is bounded. In section 4, we consider the uniform continuity about time t of this two-dimensional scheme, we discuss it in two cases, one is in a time step and other is across many time steps, and we find out a uniformly appropriate inequality. In section 5, using the results of section 3 and section 4, we prove this scheme converge to a limit function u(x, y, t) in the sense of almost everywhere in R2 x R+. In section 6, we prove the limit function satisfies the weak entropy condition of this problem. Since we have made disturbance in every time step, we need to prove the control functional of all these disturbances approach to zero.In chapter 4, we apply our scheme to solve the Cauchy problem of a class of unbounded initial data uo{x, y) ∈ Lloc∞(R2) where uo(x, y) has bounded variation in every bounded area and satisfies where r is the radius of polar coordinate. The case of unbounded initial data is essentially different from the case of bounded initial data, and we proof the existence and uniqueness of the weak entropy solution of two-dimensional Cauchy problem in some conditions.In chapter 5, we study the Cauchy problem of n-dimensional nonhomogeneous conservation law and the structure of singular solution. In section 1, we introduce some relative concepts and the results which were already known. In section 2, we study the sufficient and necessary condition of blow-up about the smooth solution of Cauchy problem and the time when blow-up come into being, then we give a sufficient and necessary condition of the existence of global smooth solution. In section 3,we calculate two structures of two-dimensional nonhomogeneous Riemann solutions and its evolution.