Some Existence And Stability Problems Of The Boltzmann Equations

This thesis is concerned with some existence and stability theories on the Boltz-mann equations under certain conditions. The research for the Boltzmann equations has been one of the most important and challenging field in Partial Differential Equa-tions because of its rich physical background and practical applications. Thus, it is very important to reveal the properties of the Boltzmann equations mathematically.Fluid passing through porous media (e.g. the underground water passing through the earth) can be modeled by the Euler equations with frictional force which have been extensively studied. Since the Boltzmann equations are closely related to the equations of gas dynamics, we investigate in the first part of this thesis, the Boltzmann equa-tion with frictional force when the external force is proportional to the macroscopic velocity. We discuss the Cauchy problem of the Boltzmann equations with frictional force mainly for the hard sphere model. We give not only the existence theory but also the optimal time convergence rates of the solutions to the Boltzmann equations with frictional force towards equilibrium.In the second part, we consider the specular reflective boundary problem for the one-dimensional Boltzmann equations with soft potentials. It is shown that the solution converges to a global Maxwellian under certain initial conditions. Note that the result for hard potentials case has already been established, thus our result here is a good supplement of this problem.