Global Wellposedness Of The 2D Euler-Boussinesq System With Stratification Effects

In this paper,we consider the following two-dimensional incompressible Euler-Boussinesq system with stratification effects:where u = u(t,x)?(u1(t,x),u2(t,x))and ? = ?{t,x)denote the velocity and the temperature of the fluid respectively,the scalar function p = p{t,x)denotes the usual pressure.the positive function ?(?)is the thermal diffusivity and we shall assume here ? is a smooth function satisfying with some fixed positive constant C0>0.the ?e2 represents the buoyancy force.-u2N2(x2)stands for stratification effects,here the real quantity N(x2)(?)?T'0(x2)is called the buoyancy or Brunt-Vaisara frequency(stratification param-eter),where T0(x2)denotes a linear mean temperature profile.This paper is concerned with the Cauchy problem of the two-dimensional Euler-Boussinesq system with stratification effects,that is the model(0.2).We obtain the global existence of a unique solution to this system without any smallness conditions imposed on the data in Sobolev spaces.The thesis is arranged as follows:At first,we briefly describe the background of the Euler-Boussinesq system and the progress of the system are also mentioned both at home and abroad.Secondly,In the first place,we illustrate related definitions and notations.Next we list the lemmas and corollaries during the course of the proof.At last,we establish the Cauchy problem of the global existence of a unique solution to this system(0.2)without any smallness conditions.