The Study On Asymptotic Limit Of The Incompressible Boussinesq Equations And Related Models

Boussinesq system is a dynamical model describing atmospheric and ocean circulations.In mathematics,it is a coupling group of the vector velocity field of the fluid and the temperature(or the density).In this thesis,we study asymptot-ic limit problem for the following four cases:the boundary layer problem of the incompressible Boussinesq equations with the inhomogeneous Dirichlet boundary condition for the scalar temperature,the initial layer problem of the Boussinesq equations with ill prepared initial data condition for both the velocity field and the scalar temperature,the mixed layer problem of the Boussinesq equations with the no-slip boundary condition and ill prepared initial data condition for the ve-locity field and the inhomogeneous Dirichlet boundary condition for the scalar temperature and the initial layer problem of the Boussinesq equations with ill pre-pared initial data condition for the velocity field,the scalar temperature and the solute concentration.This thesis applies mathematical theories and methods such as,the matched asymptotic expansion method of singular perturbation theory,the truncation function method,the classical energy method and some impor-tant inequalities,such as Poincare inequality,Cauchy-Schwarz inequality,Holder inequality,Young inequality and Sobolev embedding theorem,etcIn Chapter 1,we mainly introduce the physical background of incompress-ible Boussinesq equations.Then we introduce the model and describe research progress.Finally the preparation of knowledge and the research content in this thesis are introducedIn Chapter 2,we study the boundary layer problem of the incompressible Boussinesq equations with the no-slip boundary condition for the velocity field and the inhomogeneous Dirichlet boundary condition for the scalar temperature,where the rectangular region is given by H=(0,L1)×(0,L2)×[0,1].Since there are two boundaries in the vertical direction,the system exists two boundary layers.Using the method of matched asymptotic expansions and the multi-scale analysis,we can obtain the inner equations at order 0,and the boundary layer equations at order 0,and then utilize the inner functions and boundary layer functions which have been obtained to construct the approximate solutions.At last,with the help of the classical energy method,we can get the convergence estimate of the approximate solutions as the thermal diffusion coefficient tends to zero.In Chapter 3,we study the initial layer problem of the nondimensional form of Boussinesq equations with ill prepared initial data condition for both the velocity and the temperature in three-dimensional cylindrical region Q=T ×[0,1],T=(R/2?)2 is the torus in R2.We use non-dimensionalization and the Boussinesq approximation to obtain the nondimensional form of Boussinesq equations for Rayleigh-Benard convection.It is found that the initial data condition of the nondimensional form is not matched with the initial data condition of the limit equations as the Prandtl number tends to infinity,then an initial layer occurs.By using the method of matched asymptotic expansions,we construct the inner equations at order 0 and order 1,and the initial layer equations at order 0 and order 1,and then utilize the inner functions and initial layer functions to construct the approximate solutions.We get the properties of the approximate functions and the error equations.Finally,we estimate the error equations and get the convergence estimate of the error equations by using Gronwall inequality.In Chapter 4,we study the mixed layer problem of the Boussinesq equations with the no-slip boundary condition and ill prepared initial data condition for the velocity and the inhomogeneous Dirichlet boundary condition for the temperature.We study the same simplified model as in chapter 3.It is found that the initial data condition for the velocity field and the boundary conditions for the scalar temperature of the limit equations are not matched with the corresponding initial boundary conditions of the simplified form as the Prandtl number tends to infinity,then an initial layer and the boundary layer occur.Firstly,we construct the initial layer for the field and the upper boundary layer,the lower boundary layer at order 0 for the temperature,then the approximate solution contains all the above terms.Using the method of matched asymptotic expansions,we get the properties of the approximate solution.Secondly,we use the approximate solutions to derive the error equations.Finally,using the energy estimates,we can get the convergence of the nondimensional form to the limit equations.In Chapter 5,we study the initial layer problem of the Boussinesq equations with ill prepared initial data condition for the velocity,the temperature and the so-lute concentration.We use non-dimensionalization and the Oberbeck-Boussinesq approximation to obtain the simplification of Boussinesq system for thermosolutal convection.Using the method of matched asymptotic expansions and the classical energy method,we get the asymptotic limit convergence of the solution of the nondimensional form as the Prandtl number tends to infinity.