Some problems on Navier-Stokes equations and Boltzmann equations

Abstract

In this thesis, we study some problems on the Navier-Stokes equations and Boltzmann- Enskog equation which are in the active research area of applied mathematics. The system of Navier-Stokes equations is a typical example of the conservation laws. In the first part of the thesis, we study the global existence and convergence rates of solutions to the three-dimensional compressible Navier-Stokes equations without heat conductivity. The velocity is dissipative because of the viscosity, whereas the entropy is non-dissipative due to the absence of heat conductivity. The global solution is obtained by combining the local existence and a priori estimates if H3-norm of the initial perturbation around a constant state is small enough and its L1-norm is bounded. A priori decay-in-time estimates on the pressure and velocity are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained. The result has accepted for publication in Indiana University Mathematics Journal. On the other hand, the Boltzmann equation coming from the statistic physics is a fundamental equation in the kinetic theory for the rarefied gas. In the second part of this thesis, we study the half-space problem of the nonlinear Boltzmann-Enskog equation, assigning the Dirichlet data for the incoming particles at the boundary and a Maxwellian at the far field. It is an ongoing research project. The main part of the proof has been completed except Lemma 2.5. We want to show that if the far field Mach numberM1 < −1, there exists a unique smooth solution connecting the Dirichlet data and the far Maxwellian for any Dirichlet data sufficiently close to the far field Maxwellian. As a byproduct, the same holds for the linearized problem. The proof is essentially based on the macro-micro or hydrodynamics-kinetic decomposition of solutions and the energy estimate.

Date of Award	15 Feb 2008
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Tong YANG (Supervisor)