Research On Convection-Diffusion Models Of Air Pollutant And Its Inverse Problems

In recent years,air pollution problem has become very serious in our society.The composition and behavior,the accumulation process and the migration and transport law of the air pollutants are key scientific problems in the control and prevent for the air pollution.It is of important scientific meanings for the air pollution problems to put forwar suitable mathematical models to describe the migration and diffusion of the pollutants,and to determine the unknown parameters in the models utilizing the inverse problem method.In this paper,we deal with the convective-diffusion models of air pollutants and the coupling model of the air flow and contaminant diffusion,and investigate some inverse problems arising in the diffusion models utilzing the difference solution to the forward problem and the homotoy regularization algorithm.In Chapter 2,some preliminaries are introduced,including the variational adjoint method,the approximate control theory of parabolic type of patial differential equation,and the transport and diffusion models in the atmosphere.In Chapter 3,we deal with a two-dimensional convection-diffusion equation in a rectangular domain,and an inverse souce problem using the final observations is considered.The variational adjoint method is utilized to prove a conditional uniqueness for the inverse source problem based on an variatinoal identity connecting the known data with the unknown.Furthermore,the homotopy regularization algorithm is employed to solve the inverse source problem numerically with the help of numerical solution to the forward problem,and numerical inversions are presented.In Chapter 4 we mainly discuss the diffusion model in a general region in 2D case,and two kinds of inverse source problems are investigated by the boundary flux and the final observation,respectively.Also with the variational adjoint method,not only the conditional uniqueness is proved but also a conditional Lipschitz stability is construced,where the approximate control theory and the maximum principle for parabolic equations are also used.In Chapter 5 an inverse problem of determining the boundary flux in the two-dimensional convection-diffusion equation in a rectangular domain is discussed.Similarly to the method used in the previous chapters,a conditional uniqueness and Lipschitz stability for the inverse boundary problem are proved based on a variational identity and controllability to an adjoint problem.Moreover,the ADI scheme is applied to solve the forward problem,and numerical inversions are performed also utilizing the homotopy regularization algorithm.In Chapter 6,a coupling model for the atmospheric pollutants including the continuity equation and the momentum transfer differential equation is introduced by the theory of atmospheric dynamics and pollutant diffusion theory.The flow model is solved utilizing an iterative process in the 2D case,and some properties of the numerical solutions are given.