BV Solution And Smooth Solution For A Radiating Gas Model With Nonlinear Radiative Source

In this paper,we are concerned with BV estimates L1 stability and some related conclusions of the solution to Lax-Friedrichs' scheme for the following Cauchy proplem of a hyperbolic-elliptic coupled system arising in the radiating gas with nonlinear radiative source(?) Here,(x,t)is an unknown function defined on R×(0,?),f(u):R?R is a given flux function,g(u):R?R is a differentiable nondecreasing function.There are three parts in the thesis:In the first one,we have two main theorems in the thesis.First,ly,it's a natural extention to the radiating gas model with linear radiative inhomogeneity.We find that the solution defined by Lax-Friedrichs'scheme is uniformly bounded by modified C-F-L condition and a simple but effective monotonic technic.At the same time,for any t E[0,T]the total variation is uniformly bounded and the solution satisfies the stability.In fact,the hyperbolic-elliptic coupled system can be converted into the equivalent scalar equation ut+f(u)x=-g(u)+K*g(u).Secondly,it's a natural extention of f(u)=1/2u2.We are concerned with the general f(u)and get the life cycle of smooth solution.Next,we also provide the research background and some related literature.Finally,two propositions are given,which play important roles in the proof of main theorems.In the second part,we present the proof of the first theorem.Firstly,we make a priori estimates of the solution unk defined by Lax-Friedrichs' scheme:uniformly bounded estimate,space estimate,time estimate,and stability estimate.Then,we get the solution u(x,t)by Helly's theorem.In the third part,we can get the other theorem by applying the comparison principle:life cycle of smooth solution.