Asymptotic Limit Problem Of A Group Of Hydrodynamic Equations

In this paper, the relation of Rayleigh-Bénard convection and its limit infinite Prandtl number model is studied by the asymptotic expansion method of the singular perturbation theory and the classical energy method.Rayleigh-Bénard convection model is confined by two parallel planes and heat- ed at the bottom plane. It can be described by the Boussinesq system, which consists of the incompressible Navier-Stokes equations for the fluid velocity, with a buoyan- cy force proportional to the temperature, coupled to the heat advection-diffusion equation, along with the bound conditions and the initial conditions.After nondimensional, the Boussinesq system can be seemed as a nonlinear diff- erential equation set with a small parameterε. For infinite Prandtl number model,we need to give the initial value of the temperature, while for the Boussinesq system, both the initial value of the temperature and the velocity must be given. Generally speaking, the initial value of the latter's velocity does not run to that of the former's velocity whenε→0.So this is a perturbation problem with an initial layer. References [2, 3] have made detailed study about this problem through effective dynamics, and obtained the convergence rateΟ(ε). On the basis of it, we get further results by the asymptotic expansion method of perturbation theory and the classical energy method. The approximating solution is divided into the outer functions(t >0) and the initial layer functions (near t =0), and we prove the convergence of the approximating solution and get the convergence rateΟ(ε3/2) and the optimal rateΟ(ε²). Furthermore, when the Boussinesq system's velocity is given a special initial value so that it runs to the initial value of the infinite Prandtl number model's velocity, the initial layer disappears. A N-order approximating solution is also proved to be convergent in H snorm by asymptotic expansion method, the classical energy method and insertion theory, and the convergence rate isΟ(ε^N+1).