Some Research On Direct And Inverse Scattering Problem For Mathematical Physics Equations

Scattering refers to the phenomenon that an incident wave or incident particle deviates from the direction of propagation due to the influence of inhomogeneous medium in the propagation process.Scattering theory is of great significance in the study of mathematical physics.During the whole dissertation,we mainly consider the inverse scattering problem of the elastic wave equation,plate wave equation and the multiple cavities time-domain electromagnetic scattering problem.As we all know,these equations are some important kinds of partial differential equations in mathematics and physics.They play an important role in many scientific fields such as geological exploration,earthquake,nondestructive testing of materials,composite materials,metamaterial design,medical imaging,radar,sonar and so on.The time-domain anisotropic elastic wave equation can describe the vibration phenome--non of earthquake source propagating in complex geological strata.We focus on the study of the inverse scattering problem of the time-domain elastic wave equation with variable coefficients in a bounded domain in the first part.We use the Dirichlet-to-Neumann operator to establish an explicit reconstruction formula for the density.This formula is mainly based on the modified Boundary Control method and the Complex Geometric Optics solutions of the elastic wave equation.Moreover,since the inverse problem of the system is nonlinear,we assume the admissible set of Lamé coefficients and derive a direct Carleman estimate of the elastic wave equation,which shows that the system is stable observability.Then,under the stable observability of the elastic system,the Lipschitz stability at low frequencies and the logarithmic stability at high frequencies are estimated respectively by using some connect operators.The propagation of elastic waves in thin plates is usually referred to as plate waves.The plate equation is a significant portion of engineering construction that arises from the study of mathematical models in fluid mechanics and elasticity,also known as biharmonic equation.We consider the inverse scattering problem for the plate wave equation and establish the uniqueness of unknown density and the internal sources simultaneously by using the passive measurement in the second part.The proof is mainly rely on temporal Fourier transform to convert the time-domain problem into frequency domain.Then we try to make the differential equation is equivalent to a Lippmann-Schwinger integral equation.The asymptotic expansion of the solution of the frequency domain problem is derived,and some integral identities of the internal source function coupled with the density function are obtained.Finally,combining those integral identities and using harmonic analysis techniques with reasonable assumptions of density function and source functions,the uniqueness results are given.In view of the influence of damped in practical problems,we establish the stability estimate of the density function dependent on the damped coefficient.Considering the boundary value problem of the damped plate wave equation in a bounded domain,the stability of density is reconstructed by the Cauchy data set.Due to we only focus on the study of inverse scattering problem,so we use the Cauchy data instead of Dirichlet-to-Neumann data,which has the advantage that it does not need to prove the well-posedness of the direct scattering problem.Obviously,if the problem is well-posedness,the Cauchy data can be replaced by the Dirichlet-to-Neumann data.The proof is mainly depend on the complex geometric optical solution of the equation and the regularity estimate.Then under assumptions of the priori information about the Sobolev norm of the density and using Fourier analysis,a sharp estimate of the density with damped coefficient is obtained.Additionally,by changing the priori information for the density and the method proved in the low frequency part,we can optimize the stability index.In addition to the above linear elastic problem,we also consider the multiple cavities time-domain electromagnetic scattering problem.The basic equation to describe the electromagnetic wave propagation is Maxwell equations.And its scattering theory has important applications in both industrial and military fields.Maxwell equations with unbounded domain are used as the model in the third part.Assume that the multiple cavities,embedded in an infinite ground plane,are filled with inhomogeneous media characterized by variable electric permittivities and magnetic permeabilities.By using the invariance of the structure,we simplify the equations into a two-dimensional wave equation and propose the coupled transparent boundary condition.The multiple cavities scattering problem in the unbounded domain is equivalent to the initial boundary value problem in the bounded domain.And then the problem in the time-domain is transformed into a problem in the frequency domain by Laplace transform.The well-posedness and stability of the Helmholtz equation in the frequency domain and the wave equation in the time-domain are proved,respectively.Finally,a priori energy estimate for the electric field is obtained with minimum regularity requirement for the data and an explicit dependence on the time by studying the wave equation directly.