The Investigation On The Global Existence And Asymptotic Behavior Of Solutions To M₁ Model

About M1 model,it is mainly used to describe the evolution process,characteristics and laws of radiation energy in the radiation transfer field.As an important radiative transfer model,it has attracted considerable attention of many scholars since it was proposed.Due to the complexity of M1 model,there are few mathematical researches related to it.In this paper,we focus on the existence and large-time behavior of M1 model in one-dimensional case.In order to make the study more convenient,by analyzing the structure of this model and combining the damped compressible Euler equation,we abstractively write it into a more general damped conservation law model(for the convenience of expression,we may still call it M1 model from Chapter 2).In this paper,we mainly study the global existence and asymptotic behavior of solutions to the Cauchy problem and the initial boundary value problem of this system in the one-dimensional case.Specifically speaking,the main contents of this paper are as follows:·In Chapter 1,we briefly introduce the physical background of this model and the research progress of related problems,describe the problems of interest in this paper and give the main results.Finally,we analyze the difficulties and innovations in the process of proof;·In Chapter 2,we consider the global existence and asymptotic behavior of the solutions to the Cauchy problem for M1 model near the constant states.Under the condition that the initial perturbation satisfies some smallness conditions,we prove that the solution of the Cauchy problem exists globally and converges asymptotically to the corresponding constant equilibrium state in time.In the proof method,we first use the basic L2 energy estimation to obtain the global existence and uniqueness of the solution,and then cleverly use the Green’s function method combined with weighted energy estimation to obtain the optimal convergence rates.Compared with the previous results,we have weaker requirements on the initial value,and the calculation process is simpler and clearer.·In Chapter 3,we consider the global existence and asymptotic behavior of the solutions to the Cauchy problem for M1 model near the nonlinear diffusion waves.Inspired by the articles[24,51],we first obtain the self-similar solution of the corresponding nonlinear parabolic equation,namely,the nonlinear diffusion wave,through the idea of asymptotic analysis.Then,under the assumption that the initial value perturbation and the wave strength are sufficiently small,the energy estimation method is used to prove that the solution of the Cauchy problem of the system exists globally and converges asymptotically to the nonlinear diffusion waves at the optimal convergence rates;·In Chapter 4,We consider the existence and the large time state of global solution to the initial boundary value problem for M1 model in the quarter plane.Under the condition that the initial perturbation satisfies some smallness assumptions,we prove that there exists a unique global solution to the initial boundary value problem of the system,which asymptotically converges to the solution of the corresponding nonlinear parabolic equation.The main method used in the proof process of this chapter is similar to that in Chapter 2.