The Global Existence And Large Time Behavior Of The Boltzmann Equation In An Infinitely Expanding Ball

In this thesis, we focus on the global existence of the compressible gases in a slowly expanding ball. The flow is described by the Boltzmann equation. It is assumed that the expanding ball is described by Ωt={x∈R3:|x|<R(t), t≥0}, where t stands for the time and R(t)=(1+h2t2)1/2 for some positive constant h. From the physical point of view, due to the mass conservation of gas, the moving gas in the expansive ball will gradually become rarefactive and eventually tend to a vacuum state with the increasing of time. We shall confirm such a phenomena by rigorous mathematical proofs and simultaneously show that there are no appearances of vacuum domains in any part of the expansive ball. For the proof, changing the coordinates, we reformulate the problem by the Boltzmann equation with a potential term in a fixed ball. (?)τf+η·▽Vyf-h2y·▽ηf=cos2(hτ)Q(f,f).We use the L2-L∞ method to prove the global existence of the solution. For the L2-decay, the main difficulty is to get the estimate of the momentum term. To solve this, we split the momentum term into tangential part and normal part near the boundary. The estimates can then been derived by solving an elliptic system with mixed type boundary condition. The L∞-estimates then follows by integrating along the backward trajectory. The large time behavior is the result of the L∞-norm of the perturbation.