The Green's Function Method For Initial-Boundary Value Problem Of Boltzmann Equation

In this thesis, we take kinetic equations as examples to consider how the Green's func-tion method is applied to the initial-boundary value problem and equations with non-constantcoefficients. The thesis is arranged as follows:In Chapter 1, we review the physical background of kinetic theory and the history ofstudy for Boltzmann equation. We also introduce the problems we will study and the mainresults.In Chapter 2, we consider a special discrete Boltzmann equation, the Broadwell model.We study two different initial-boundary value problems of this model. When the physicalboundary is a supersonic one, Green's function combined with the boundary energy estimateyields the pointwise description of the solution. If the boundary is subsonic, we have toapply an iteration scheme to get the full boundary information. We also construct the Green'sfunction for initial-boundary value problem. Estimates for such a Green's function togetherwith consideration of nonlinear wave coupling result in the pointwise convergence rate ofthe solution for the nonlinear problem.In Chapter 3, we study the existence and stability problems of Knudsen layer for theBoltzmann equation. By using the time asymptotic method, we regain the existence theorywhich has been obtained in [78] and also get the estimate for boundary layer. One of theadvantages of choosing this method is that we can gain the stability of the boundary layerwhen Mach number is less than -1 in the proof of the existence. When Mach number is largerthan -1, we use the Green's function for equation which is linearized around the Maxwellianto study the equations with non-constant coefficients. Since linearization around boundarylayer can be considered as the combination of linearization around the Maxwellian and asmall term which also decays exponentially on spatial variable, we treat the extra small term by using the method which we take to treat the nonlinear term. When Mach numberis larger than 1, the Green's function for Cauchy problem is applied while when boundaryis subsonic, we will use the Green's function for initial-boundary value problem. Finally,we prove that the boundary layer is stable when boundary is not characteristic and get thepointwise convergence rate.In fact, such methods can also be used to treat the initial-boundary value problem forother systems with dissipasive structure. The study for Knudsen layer for hard-potentialmodel and half-space problems of multidimensional euler equations with damping and theconstruction of the Green's function for the Cauchy problem of the conservation laws withrelaxation can be found in [19, 20, 16].