Large Time Behavior Of Two Nonlinear Evolution Systems

Two nonlinear evolution systems are considered in this thesis, namely the multidimensional Euler Equations with damping and the Vlasov-Poisson-Boltzmann system with given magnetic field.In fluid dynamics, the Euler equations, named after Euler, which govern the inviscid flow, are written in the conservation form to emphasize the conservation of mass, momentum and energy. The further applications require researchers to study the viscosity and heat conduction effects or damping by the media or external force etc., which correspond to Navier-Stokes system, Euler Equations with damping, Euler-Poisson system, etc., respectively.The first problem consider in this thesis is the stability and L~p convergence rates of planar diffusion waves for multi-dimensional Euler equations with damping. The analysis relies on a newly introduced frequency decomposition and approximate Green's function based energy method. It is a combination of the L~p estimates on the low frequency component by using an approximate Green's function and L~2 estimates on the high frequency component through the energy method. By noticing that the low frequency component in the approximate Green's function has the algebraic decay which governs the large time behavior, while the high frequency component has the exponential decay but with singularity, their combination leads to global algebraic decay estimates. To use the decay property only of the low frequency component in the approximate Green's function avoids the singularity in the high frequency component. The energy estimates on the high frequency part can be closed due to its self-satisfied Poincare inequality. This new approach of the combination of the approximate Green's function and energy method through the frequency decomposition can also be applied to the hyperbolic-parabolic systems satisfying the Kawashima condition, and also those systems whose derivatives of the coefficients have suitable time decay properties. The fluid equations described the motion on the macro aspect. On the other hand, it is natural to think that the macro components of the fluid are governed by the motion of each molecule contained in the macro fluid. In principle, one can derive all the macro aspects if one knows all the details of each molecule in microscopic. But, confined by the capability to handle the huge number of the molecules, that is prohibitive, and also unnecessary. Actually, it is more reasonable to consider only the statistical description. The Boltzmann equation comes with the tide of fashion.In the description of motions of the ions and electrons under the influence of the electromagnetic fields, the electromagnetic force should be included. Vlasov type Boltzmann equations are used to describe the motion of particles under external force. There are two typical physical models, which are Vlasov-Poisson-Boltzmann system and Vlasov-Maxwell-Boltzmann system.The second problem considered in this thesis is the global existence of classical solution to the Vlasov-Poisson-Boltzmann system with given magnetic field when the initial data is a small perturbation around a global Maxwellian. This system is different from Vlasov-Maxwell-Boltzmann system, which studies the motion of electrified particles under the self-induced electromagnetic field. When the motion is slow, the self-induced magnetic field is negligible compared with the given magnetic field.The global existence of classical solution is obtained when the initial data is a small perturbation about a global Maxwellian. The proof is based on the macro-micro decomposition of the solution to the Boltzmann equation with respect to the local Maxwallian introduced by [Liu-Yang-Yu: Energy method for Boltzmann equation. PhysicaD, 188(3-4)(2004), 178-192]. Then the system is decomposed into compressible Navier- Stokes-Poisson equations with force for the macro components coupled with an evolution equation for the micro component. With the help of the celebrated H-theorem, the a priori estimate is arrived and the global existence followed from the local existence and continuity argument. The result shows the global existence of classical solution is independent of the given magnetic field.